arX

iv:1

803.

0755

3v4

[m

ath.

NT

] 1

8 Ju

l 202

1

An intersection formula for CM cycles on Lubin-Tate spaces

QIRUI LI

ABSTRACT. We give an explicit formula for the arithmetic intersection number of CM cycles on

Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on

the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions

K1,K2 over a non-Archimedean local field F . Our formula works for all cases: K1 and K2 can

be either the same or different, ramified or unramified over F . This formula translates the linear

Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. As an example, we

prove the linear AFL for GL2(F ) in this article.

CONTENTS

1. Introduction 1

2. CM cycles on Lubin–Tate towers 8

3. An approximation for CM cycles on the infinite level 17

4. Intersection comparison 27

5. Computation of intersection numbers on the infinite level 33

6. Proof of main theorems 44

7. Proof of the linear AFL for GL2(F ) 53

References 62

1. INTRODUCTION

1.1. Global Motivation. In this article, we study an intersection problem of special cycles on

Lubin–Tate towers motivated by the Gross–Zagier formula and its generalizations to certain higher-

dimensional Shimura varieties.

The Gross–Zagier formula [GZ86] [YZZ13] relates the Neron–Tate height of Heegner points on

Shimura curves to the first central derivative of certain L-functions. There have been conjectural

generalizations of the Gross–Zagier formula to higher dimensional Shimura varieties, notably the

Gan–Gross–Prasad conjectures [GGP12] and the Kudla–Rapoport conjecture [KR14]. Recently,

Zhang has proposed another generalization [Zha17a] from which the intersection problem in this

paper arises.

We briefly recall the geometric construction in Zhang’s proposal. Let F0 = Q (in general,

a totally real field), and let F (resp. E0) be an imaginary (resp. a real) quadratic field. Let

E = E0 ⊗F0 F , a bi-quadratic field extension of F0. Let V be an n-dimensional vector space

over E with an E/E0-Hermitian form 〈−,−〉V . Consider the unitary group H (an algebraic group

over E0) associated to V and 〈−,−〉V , and denote by ResE0/QH (an algebraic group over Q) the

restriction of scalars of H . Let V ′ = V , viewed as an F vector space (of dimension 2n), and

let 〈−,−〉V ′ = trE/F 〈−,−〉V be the induced F/F0-Hermitian form. Let G be the unitary group1

associated to V ′ and 〈−,−〉V ′ . We have a natural embedding of algebraic groups over Q

ResE0/QH −→ G.

For a cuspidal automorphic representation π of G and φ ∈ π, we can consider the H-period

integral of φ. A conjecture in [XZ20] states that this period integral is related to the central value

of a certain L-function associated to π, analogous to the Waldspurger formula and the global Gan–

Gross–Prasad conjecture [GGP12].

We assume further that the signatures of V at the the two archimedean places of E0 are ((n −1, 1), (n, 0)). Then the signature of V ′ at the archimedean place of F0 = Q is (2n − 1, 1). The

Shimura varieties ShG and ShH associated to G and ResE0/QH have dimension 2n− 1 and n− 1respectively (see section 27 in [GGP12]). The algebraic group inclusion ResE0/QH −→ G induces

a cycle Z on ShG of codimension n. Then the height pairing of the π-isotypic part of the cycle Z is

conjecturally related to the first central derivative of a certain L-function associated to π, analogous

to the Gross–Zagier formula and the arithmetic Gan–Gross–Prasad conjecture [GGP12].

In [Zha12], Zhang proposed a relative trace formula (RTF) approach to the arithmetic Gan—

Gross-–Prasad conjecture. The approach leads to local conjectures, notably the arithmetic funda-

mental lemma (AFL) conjecture formulated by Zhang in [Zha12] (see [Zha21] for recent progress),

and the arithmetic transfer (AT) conjecture formulated by Rapoport–Smithling–Zhang [RSZ17,

RSZ18]. The AFL conjecture relates the special value of the derivative of a relative orbital integral

to an arithmetic intersection number on a Rapoport–Zink (RZ) formal moduli space of p-divisible

groups attached to a unitary group. In this paper, we study an analog of Zhang’s AFL conjecture

in this new setting. More precisely, let v be a place of F0 = Q that is split in F and inert in

E0. Then the local unitary group GF0,v (resp. HE0,v) is isomorphic to the general linear group

GL2n,F0,v (resp. GLn,E0,w for the unique place w of E0 above v). Then the corresponding RZ

spaces are the Lubin–Tate formal moduli spaces, and we will call the resulting cycle the CM cycle

(relative to the quadratic extension E0,w/F0,v). We then consider the intersection number of the

CM cycle with their translation in the ambient Lubin–Tate space. Zhang conjectured in his un-

published notes [Zha17b] that this intersection number is related to the first derivative of a relative

orbital integral. He called this conjecture the linear AFL, to be distinguished from the AFL con-

jecture [Zha12] in the context of the arithmetic Gan–Gross–Prasad conjecture. We will recall the

precise statement in §1.4.

In this article, we focus on the geometric side of the linear AFL and we will establish an explicit

formula for the arithmetic intersection numbers. In fact, we will work in a much more general set-

ting. First of all, our Theorem 1.4 calculates the intersection number on the Lubin–Tate space with

arbitrary Drinfeld level structure, and we allow both ramified and unramified quadratic extensions

in characteristic 0 or an odd prime. It also works in characteristic 2 for separable quadratic exten-

sions. Note that in the function field case, Yun and Zhang have discovered a “higher Gross–Zagier

formula” [YZ17], relating higher derivatives of L-functions to intersection numbers of special cy-

cles on the moduli space of Drinfeld Shtukas of rank two. Our theorem in the positive characteristic

case is related to a generalization of their theorem to the moduli space of Shtukas of higher rank

See Remark 4.1 of [Zha18]. Secondly, we obtain results for the intersection of CM cycles relative

to two different quadratic extensions (see Theorem 1.5). In the function field case, this is related

to the recent result of Howard–Shnidman on the Gross–Kohnen–Zagier formula [HS19].

Our formula can be expressed as an integral over GL2h(F ), where F is some non-Archimedean

local field. By using this formula in the case of h = 1, the author gives a new proof of Gross and2

Keating’s result on lifting endomorphisms of formal modules in [Li19]. Further, our formula re-

duces the linear AFL to an analytic identity between two integrals. In [Li20], we prove the identity

by direct calculation when h = 2, and hence verify the linear AFL in these cases. Moreover, we

could use our main theorem to verify various analogs of the arithmetic transfer conjecture in this

new setting. We will pursue these directions in the future. In the ongoing work of the author and

Howard, we conjectured that the linear AFL holds for more general settings, where we allow CM

cycles associated with different quadratic extensions [HL20].

1.2. Main Results — a baby case. We abandon all previous notations and start new ones. Let hbe a non-negative integer. Fix a non-Archimedean local field F (of any characteristic) with uni-

formizer π, ring of integers OF , residue field Fq, and algebraic closure F alg. Let K be a separable

quadratic extension of F with ring of integers OK and residue field FqK . Let ζK(s) = (1− q−sK )−1,

ζF (s) = (1−q−s)−1 be their zeta functions, and let |• |F , |• |K be their corresponding absolute val-

ues, which are normalized so that |x|−1F = #(OF/x) and |y|−1

K = #(OK/y) for any 0 6= x ∈ OF

and 0 6= y ∈ OK . Let Fq be the algebraic closure of Fq. We fix a field embedding FqK → Fq.

Let K (resp. F ) be the completion of the maximal unramified extension of K (resp. F ), with

ring of integers OK(resp. OF ). Let σ ∈ Gal(K/F ) be the non-trivial element, and denote by xσ

the Galois conjugate of x ∈ K. Let |DiscK/F |F be the norm of the discriminant of K/F , which

equals |(x − xσ)2|F for any x ∈ K such that OK = OF [x]. Let GF be a formal OF -module of

height 2h over Fq. It is well-known that

DF := End(GF )⊗OFF

is a central division algebra of invariant 12h

, which implies DF ⊗F Falg ∼= Mat2h×2h(F

alg). We fix

an OF -algebra map

(1.1) OK → End(GF )

in such a way that the induced action of OK on the Fq-vector space Lie(GF ) is through the reduction

map OK → FqK → Fq, which yields a formal OK-module GK of height h.

Denote the Lubin–Tate deformation space of GF (resp. GK) by M0 (resp. N0), which admits a

natural action of Aut(GF ) (resp. Aut(GK)). It is well known that M0 is the formal specturm of a

power series ring in 2h− 1 variables over OF . Deforming GF with the extra OK-module structure

through OK → End(GF ) gives rise to a morphism of formal schemes η00 : N0 → M0, which

yields a cycle Z0 := (η00)∗ON0 with h-dimensional support on M0. Translating Z0 by an element

γ ∈ Aut(GF ) yields another cycle Z0(γ). Consider the intersection number

χ(Z0 ⊗LM0Z0(γ)) =

∞∑

i=0

(−1)i lengthOF

(ToriM0

(Z0, Z0(γ))),

which is in fact well-defined and only has one non-zero term lengthOF(Z0 ⊗M0 Z0(γ)) for almost

all elements γ ∈ D×F . The main result of this article is a formula identifying χ(Z0⊗

LM0Z0(γ)) with

an integral involving invariant polynomials of γ, which can be described as follows.

Let K ⊂ DF be the embedding corresponding to (1.1). Denote by DK ⊂ DF the central-

izer of K. Note that DK ⊗F Falg ∼= Math×h(F

alg) × Math×h(Falg), and that DF ⊗F F

alg ∼=Mat2h×2h(F

alg). Up to conjugation, the embedding DK ⊗F Falg → DF ⊗F F

alg is identified with

the blockwise diagonal embedding

(1.2) Math×h(Falg)×Math×h(F

alg)→ Mat2h×2h(Falg),

3

which associates a degree h polynomial Pg to any element g ∈ GL2h(Falg) as described by the

following definition.

Definition 1.1 (Invariant Polynomial). For any A,B,C,D ∈ Math×h(Falg) be such that

g :=

(A BC D

)

is an element in GL2h(Falg), let X ′, X ′′ ∈ Math×h(F

alg) be such that

(1.3)

(X ′

X ′′

)=

(A

D

)(A BC D

)−1(A

D

)(A −B−C D

)−1

.

Then X ′ and X ′′ have the same characteristic polynomial, which we denote by Pg. Call Pg the

invariant polynomial of g with respect to the embedding GLh(Falg)×GLh(F

alg) ⊂ GL2h(Falg).

It is straightforward to see that Pg′ = Pg for any

g′ ∈(GLh(F

alg)×GLh(Falg))· g ·

(GLh(F

alg)×GLh(Falg)).

Therefore, Pg is defined up to the double quotient

GLh(Falg)×GLh(F

alg)\GL2h(Falg)/GLh(F

alg)×GLh(Falg).

For any γ ∈ D×F , we define the invariant polynomial of γ with respect toD×

K ⊂ D×F by regarding

it as an element in DF ⊗ F alg. The invariant polynomial of g ∈ GL2h(F ) with respect to the

embedding GLh(K) ⊂ GL2h(F ) is defined similarly. In fact, the coefficients of these polynomials

are in F (see Proposition 5.18). The main result of this article is the following formula and its

generalizations.

Theorem 1.2. Let Pγ be the invariant polynomial of γ with respect to D×K ⊂ D×

F . If Pγ is irre-

ducible and Pγ(0)Pγ(1) 6= 0, then χ(Z0 ⊗LM0

Z0(γ)) is finite and we have the following formula

χ(Z0 ⊗LM0Z0(γ)) =

(∏hi=1 ζK(i)

)2

∏2hi=1 ζF (i)

· |DiscK/F |−h2

F ·

∫

GL2h(OF )

|Res(Pγ, Pg)|−1F dg,

where Pg is the invariant polynomial of g ∈ GL2h(F ) with respect to an arbitrary embedding

GLh(K) ⊂ GL2h(F ) which restricts to GLh(OK) ⊂ GL2h(OF ), and Res(Pγ, Pg) is the resultant

of Pγ and Pg. Here dg is the normalized Haar measure of GL2h(OF ).

1.3. Main results — generalizations. Following Drinfeld [Dri74], there is a projective system of

formal schemes

M• = · · · →Mn →Mn−1 → · · · →M0

such that each member Mn parametrizes deformations G of GF with additional data of πn-level

structures defined by an OF -linear homomorphism

αn : O2h∨F /πnO2h∨

F −→ G[πn]

with some non-degeneracy conditions (See Definition 2.2). Here by O2h∨F we mean the OF -module

of 1 × 2h-matrices over OF . The action of the group GL2h(OF ) on πn-level structures induces a

surjective homomorphism

GL2h(OF ) −→ AutM0(Mn) ∼= GL2h(OF/πn)

4

with the kernel denoted by Rn for any n ≥ 0 . Clearly,

R0 = GL2h(OF ); Rn = I2h + πnMat2h×2h(OF )

for n ≥ 1. A pair

τ : Kh −→ F 2h,

ϕ : GK −→ GF(1.4)

consisting of an OF -quasi-isogeny and an F -linear isomorphism induces F -algebra embeddings

ϕ ·K · ϕ−1 ⊂ ϕ ·DK · ϕ−1 ⊂ DF ;

K ⊂ Math×h(K) ⊂ Mat2h×2h(F ).

Let O := K∩Mat2h×2h(OF ). Loosely speaking, up to some multiplicities, we define the CM cycle

Zn(ϕ, τ) on Mn by deforming GF with an extra O-module structure through ϕ·O·ϕ−1 ⊂ End(GF ),and with a πn-level structure αn chosen in such a way that O acts on αn through O ⊂ K ⊂Mat2h×2h(F ). Precomposing (1.4) with two elements γ ∈ D×

F and g ∈ GL2h(F ) yields another

cycle Zn(γϕ, gτ). Based on the discussion in §6.4, the cycle Zn(ϕ, τ) (which is also denoted by

Z(0)n (ϕ, τ) in this article) is either empty or isomorphic to a cycle Zn(ϕ

′, τ ′) with the pair (ϕ′, τ ′)normalized in the sense that Height(ϕ′) = Height(τ ′). Therefore, without loss of generality, for

the rest of the section, we may fix ϕ and assume Height(ϕ) = Height(τ) (See Definition 2.15).

Denote by Pγ the invariant polynomial of γ with respect to ϕ · DK · ϕ−1 ⊂ DF for any γ ∈ D×

F .

Denote by nrd(γ) the reduced norm of γ. We have the following generalization of our theorem.

Note that the restriction |nrd(γ)|−1F = | det(g)|F is necessary for (γϕ, gτ) to be an equi-height

pair.

Theorem 1.3. Suppose |nrd(γ)|−1F = | det(g)|F . If Pγ is irreducible and Pγ(0)Pγ(1) 6= 0, or if

Res(Pγ, Pxg) 6= 0 for any x ∈ Rn, then the intersection number of Zn(ϕ, τ) and Zn(γ · ϕ, g · τ) is

finite, and we have the following formula for the intersection numbers for n ≥ 1,

χ(Zn(ϕ, τ)⊗LMn

Zn(γ · ϕ, g · τ)) = |DiscK/F |−h2

F ·

∫

Rn

|Res(Pγ , Pxg)|−1F dx,

where Pxg is the invariant polynomial of xg ∈ GL2h(F ) with respect to the embedding GLh(K) ⊂GL2h(F ) induced by τ . Here dx is the normalized Haar measure of Rn.

This theorem generalizes the Theorem 1.2 by allowing g /∈ GL2h(OF ) as well as admitting level

structures. By allowing g ∈ GL2h(F ) one can construct more CM cycles. For example, when

n = 0, for any open OF -subalgebra O ⊂ OK , by an appropriate choice of (γ, g) ∈ D×F ×GL2h(F ),

one can construct the cycle Z0(γ · ϕ, g · τ) corresponding to the deformation of GF as a formal O-

module such that O acts on its Tate module through a preselected embedding O→ Mat2h×2h(OF ).Such a deformation is also known as a quasi-canonical lifting of the formal OF -module GF in the

literature (see [Gro86]).

Now we generalize our formula for correspondences. For any non-negative integer n, let

H (Rn\GL2h(F )/Rn) := f ∈ C∞c (GL2h(F )) : f(xg) = f(gx) = f(g) for any x ∈ Rn

be the set of doubleRn-invariant compactly supported real-valued functions on GL2h(F ). Suppose

g ∈ GL2h(F ) ∩Mat2h×2h(OF ). Let m be an integer such that g(O2hF ) ⊃ πmO2h

F , and let γ ∈ D×F

be an element such that |nrd(γ)|−1F = | det(g)|F . The pair (γ, g) induces a natural morphism

βm+nn (γ, g) : Mm+n −→Mn

5

for any n ≥ 0 (see (2.16)). Denote by βm+nn the transition map from Mm+n to Mn. Let

(1.5) fRn·g·Rn,m(x) :=deg( Mm+n

βm+nn // Mn )

Vol(Rn · g · Rn)1Rn·g·Rn(x) ∈H (Rn\GL2h(F )/Rn),

where 1Rn·g·Rn(x) is the characteristic function of Rn · g ·Rn. Then the correspondence associated

to the pair (γ, f) is defined by

Mn Mm+n

βm+nn (γ,g)

//βm+nnoo Mn .

To introduce a general formula, we fix the following constants

C0(K1, K2) :=

(∏hi=1 ζK1(i)

)(∏hi=1 ζK2(i)

)

∏2hi=1 ζF (i)

; Cn(K1, K2) := 1

for any n > 0, and any two quadratic extensions K1, K2 over F .

Theorem 1.4. Let f = fRn·g0·Rn,m and Cn = Cn(K,K). Suppose |nrd(γ)|−1F = | det(g)|F and

|nrd(γ0)|−1F = | det(g0)|F . If Pγ−1

0 γ is irreducible and Pγ−10 γ(0)Pγ−1

0 γ(1) 6= 0, or if

Res(Pγ−1

0 γ , Px−1g

)6= 0

for any x such that f(x) 6= 0, then the following intersection number is finite and given by the

formula

χ(Zn(ϕ, τ)⊗LMn

βm+nn∗ βm+n

n (γ0, g0)∗Zn(γ · ϕ, g · τ))

= Cn|DiscK/F |−h2

F

∫

GL2h(F )

f(x)∣∣∣Res

(Pγ−1

0 γ, Px−1g

)∣∣∣−1

Fdx,

where Px−1g is the invariant polynomial of x−1g ∈ GL2h(F ) with respect to the embedding

GLh(K) ⊂ GL2h(F ) induced by τ . Here the Haar measure dx is normalized by GL2h(OF ).

Note that when γ0 = id, g0 = id and m = 0, this formula specializes to Theorem 1.2 and

Theorem 1.3. The formula can also be generalized to the case of K1 6∼= K2. For i = 1, 2, we put

extra subscripts and denote by Zn(ϕi, τi) the cycle constructed by the data τi : Khi → F 2h and

ϕi : GKi→ GF , and we associate a matrix ∆(ϕi, τi) ∈ Mat2h×2h(DF ) to (ϕi, τi) (See Definitions

5.9 and 5.4). Again they are all equi-height pairs. The following theorem gives a formula in the

most general case without using invariant polynomials.

Theorem 1.5. Let f = fRn·g0·Rn,m and Cn = Cn(K1, K2). Suppose |nrd(γ)|−1F = | det(g)|F and

|nrd(γ0)|−1F = | det(g0)|F . If the integrand is bounded, then the following intersection number is

finite and given by the formula

χ(Zn(ϕ1, τ1)⊗LMn

βm+nn∗ βm+n

n (γ0, g0)∗Zn(γ · ϕ2, g · τ2))

= Cn|DiscK1/F |−h2

F

∫

GL2h(F )

f(x) |R(x)|−1F dx,

where R(x) is the reduced norm of a matrix R(x) ∈ Math×h(DF ), and

R(x) :=(0h Ih

)∆(ϕ1, τ1)

−1 · γ−10 γ · x−1g ·∆(ϕ2, τ2)

(Ih0h

).

Here the Haar measure dx is normalized by GL2h(OF ).6

1.4. The linear AFL. When K/F is an unramified extension with odd residue characteristic,

Zhang has a conjecture called the linear AFL, which gives another formula for the intersection

numbers of CM cycles. Recall Z0 := (η00)∗ON0 as defined in Section 1.2. Let γ ∈ D×F and

g ∈ GL2h(F ) be elements such that |nrd(γ)|−1F = | det(g)|F . Let f = fR0gR0,m as defined in (1.5).

The linear AFL predicts that the intersection number

χ(Z0 ⊗LM0βm0∗β

m0 (γ, g)∗Z0)

is given by the central derivative of a certain orbital integral OrbF (g(γ), f, s), which we describe

now.

Let G′ = GL2h(F ) and H ′ = GLh(F ) × GLh(F ). Consider H ′ as a subgroup of G′ by the

blockwise diagonal embedding. Let g(γ) ∈ G′ be an element having the same invariant polynomial

Pγ with γ. Let η be the non-trivial quadratic character associated to K/F . We regard η and |•|Fas characters on H ′ by precomposing them with (g1, g2) 7→ det(g−1

1 g2) (note the inverse on g1).Consider the following orbital integral

(1.6) OrbF (g(γ), f, s) =

∫

H′×H′

I(g(γ))

f(h−11 g(γ)h2

)η(h2)|h1h2|

sFdh1dh2,

where for any g ∈ G′,

I (g) = (h1, h2)|h1g = gh2.

Conjecture 1 (linear AFL). Let γ, g, f and g(γ) be as defined above. If Pγ is irreducible and

Pγ(0)Pγ(1) 6= 0, we have

(1.7) ± (2 ln q)−1 d

ds

∣∣∣∣s=0

OrbF (g(γ), f, s) = χ(Z0 ⊗LM0

βm0∗βm0 (γ, g)∗Z0),

where the sign is chosen so that the left hand side is non-negative.

In Section 7, we verify this conjecture for h = 1. By calculating both sides of this identity, the

author also proves the linear AFL in the case of h = 2 for the characteristic function of GL4(OF )in [Li20].

1.5. Outline of contents. The main idea is to raise the problem to the infinite level. We review

some history. In Theorem 6.4.1 of the paper [SW13] of Scholze-Weinstein, and also in the paper

[Wei16] of Weinstein, they show that the projective limit of the generic fiber of the Lubin–Tate

tower for GF is a perfectoid space M∞, which can be embedded into the universal cover of G2h,

where G is a certain deformation of GF . The main idea is to describe the infinite level Lubin–Tate

spaces by G2h.

In this article, after establishing definitions of Lubin–Tate towers and CM cycles in §2, we realize

their idea on the integral model of the Lubin–Tate tower on finite levels by a new construction.

In §3, we prove that the preimage of the closed point under the transition map Mn → M1 is

canonically isomorphic to G2hF [πn−1]. In other words, the following diagram is Cartesian (See

Proposition 3.9)

G2hF [πn−1] //

SpecFq

Mn

// M1.7

These heuristic examples let us regard G2hF as an approximation of the special fiber of Mn over M1

when n → ∞. Therefore, it is natural to construct CM cycles Z∞(ϕ, τ) on G2hF and formulate a

similar intersection problem. In §4, by Proposition 3.9, we show that the corresponding intersec-

tion number on G2hF is the same as the intersection number on the Lubin–Tate tower if the level is

sufficiently high. We calculated in §5 Proposition 5.20 that the intersection number of Z∞(ϕ, τ)

and Z∞(γϕ, gτ) is related to∣∣Res(Pγ, Pg)

∣∣−1

F. In §6, we prove our main Theorem 1.4 by using the

projection formula. The essential property for the method in §6 to work is that the transition maps

of the Lubin–Tate tower are generically etale. In §7, we prove the linear AFL for the case of h = 1as an application.

1.6. Acknowledgement. The author would like to thank Professor Wei Zhang, for suggesting

this topic and for his comments on this article. The author also thanks Professor Johan de Jong for

answering his several questions. The author would like to express his sincere gratitude to the three

anonymous referees for their reports which help the author make significant improvements on both

the mathematical contents and the exposition. Last but not least, the author thanks Professor Liang

Xiao for his encouragement, Ziquan Yang and Stanley Yao Xiao for their suggestions on writing.

2. CM CYCLES ON LUBIN–TATE TOWERS

We fix a non-Archimedean local field F with ring of integers OF and residue field Fq. Let

K be a finite separable extension of F of degree k (of any characteristic), with standard discrete

valuation vK , ring of integers OK , and residue field FqK . We fix a uniformizer π of F , which is

not necessarily a uniformizer of K. Let K (resp. F ) be the completion of the maximal unramified

extension ofK (resp. F ) with ring of integers OK (resp. OF ). Denote by C (resp. CK) the category

of complete Noetherian local OF -(resp. OK-)-algebras with residue field Fq. For R = K,F,OKand OF , denote by Rn the R-module of n× 1 matrices over R, and by Rn∨ the R-module of 1× nmatrices over R. The pairing

(2.1) 〈−,−〉 : Rn∨ ⊗R Rn → R

is given by the matrix multiplication.

We first construct the Lubin–Tate towers, which admit an action by D×F × GLkh(F ) where DF

is the F -central division algebra of invariant 1/kh. We then construct CM cycles corresponding

to algebraic group embeddings D×K ⊂ D×

F and GLh(K) ⊂ GLkh(F ). Here DK is the K-central

division algebra of invariant 1h

. Denote by Fq the algebraic closure of Fq.

2.1. The Lubin–Tate tower. In this subsection, we give a precise definition of the Lubin–Tate

tower associated to a formal OK-module GK of height h over FqK .

2.1.1. Formal modules. Let A be a complete Noetherian local algebra over a local ring B with

structure map s : B −→ A and residue field Fq. A formal B-module over A is a one dimensional

formal group law G endowed with a B-action i : B −→ End(G), where we insist that the induced

action on the Lie algebra

B −→ End(LieG) ∼= A

is identical to the structure map s. A homomorphism of two formal B-modules β : G1 −→ G2

over A is a homomorphism of formal group laws that commutes with actions of B. When β is

surjective and pA = 0 , the height of β is defined by

(2.2) Height(β) := logqB(deg β)8

where qB is the cardinality of the residue field of B. Furthermore, if B is a discrete valuation ring

with uniformizer πB , then we define the height of G as the height of the endomorphism induced

by πB ∈ B ⊂ End(G). Furthermore, we denote by [b]G : G −→ G the action of b ∈ B, [+]G :G × G −→ G (resp.[−]G) the addition (resp. subtraction) of G, and G the base change of G to

the residue field of A. For any homomorphism of formal modules ψ : G1 −→ G2, we denote its

reduction by ψ : G1 −→ G2.

Definition 2.1. A deformation of GK over A ∈ CK is a pair (G, ι) such that

• G is a formal OK-module over A,

• ι : GK −→ G is an OK-quasi-isogeny.

We define level structures following Drinfeld [Dri74].

Definition 2.2. For any x ∈ OK , an x-level structure of a formal OK-module G over A ∈ C is a

homomorphism of left OK-modules α : Oh∨K −→ G(A) such that

• α factors through Oh∨K −→ Oh∨K /xOh∨K −→ G(A),• α satisfies the non-degeneracy condition that the power series [x]G(X) is divisible by

∏

v∈Oh∨K /xOh∨

K

(X [−]Gα(v)) .

Definition 2.3. For any x ∈ OK , an x-level deformation is a triple (G, ι, α) obtained by attaching

an x-level structure α to a deformation (G, ι) of GK over A ∈ CK .

Definition 2.4. Two x-level deformations (G1, ι1, α1) and (G2, ι2, α2) are equivalent if there is an

isomorphism ζ : G1 −→ G2 of formal OK-modules such that the following two diagrams commute

GKid //

ι1

GK

ι2

G1ζ // G2

Oh∨Kid //

α1

Oh∨K

α2

G1

ζ // G2.

We denote by [G, ι, α]n the equivalence class of (G, ι, α), where the subscript n is indicating that αis an x-level structure with vK(x) = n · vK(π).

Note that for any x, x′ ∈ OK with vK(x) = vK(x′), an x-level structure is an x′-level structure.

Therefore the subscript n is sufficient to describe the level of α in the triple [G, ι, α]n. For simplicity,

we will call the x-level structure as the πn-level structure. The use of πn is symbolic and it may

not be a well-defined element in OK when n is not an integer.

Theorem 2.5 (Drinfeld, Lubin–Tate [Dri74] [Lub66]). The functor sending A ∈ CK to the set of

equivalence classes of πn-level deformations of GK over A is representable by a formal scheme

Nn∼ of dimension h over Spf OK . This formal scheme decomposes as

Nn∼ =

∐

j∈Z

Nn(j)

where Nn(j) is the open and closed formal subscheme on which ι has height j, and there is a

regular local OK-algebra An such that Nn(j) = Spf (An) for every j.

9

Definition 2.6. The Lubin–Tate tower N•∼ associated to GK is a projective system consisting of

Nn∼ for n ∈ 1

vK(π)· Z≥0 with transition maps, functorial in A ∈ CK , given by

Nn+m∼(A) −→ Nn

∼(A)

[G, ι, α]n+m 7−→ [G, ι, [πm]G α]n

for any integer m ≥ 0. We denote this transition map by θm+nn : Nm+n

∼ −→ Nn∼.

These transition maps are finite flat, and they keep the superscript of components — they induce

maps between Nn+1(j) and Nn

(j) for every n and j. We denote by N•(j) the subtower of N•

∼ with

superscript (j).

Remark 2.7. The subscript n of Nn(j) refers to the πn-level structure. Since π is not necessarily a

uniformizer of OK , a fractional subscript like Nnk

(j) can make sense if K/F is ramified. But we do

not need these spaces with fractional-subscripts in our article. From now on, all subscripts n will

be a non-negative integer.

2.2. Maps between Lubin–Tate towers. Let GF be the formal OF -module of height kh obtained

by only remembering the OF -action of GK , and denote its associated Lubin–Tate tower by M•∼,

which is a regular formal scheme of dimension kh over Spf OF . Denote the degree of the corre-

sponding residue field extension of K/F by

r := [FqK : Fq] = logq qK .

From now on, we work over the base Spf OF and we regard each formal scheme as a functor from

C to sets. We can extend the domain of the functor Nn∼ from CK to C as follows. For anyA ∈ C, we

define Nn∼(A) to be the set of πn-level deformations [G, ι, α]n over A with extra endomorphisms

by OK . The action of OK on Lie(G) ∼= A gives A an OK-algebra structure, therefore Nn∼(A) 6= ∅

implies that A ∈ CK .

A map from N•∼ to M•

∼ over Spf OF can be constructed by choosing a pair of homomorphisms

(ϕ, τ)

ϕ : GK −→ GF ;

τ : Kh −→ F kh,(2.3)

where τ is an F -linear isomorphism and ϕ is a quasi-isogeny of formal OF -modules. In fact, this

τ is purely symbolic.

Our later definitions and proofs will work instead with the dual map

τ∨ : F kh∨ −→ Kh∨,

which is defined by l 7→ τ∨(l) where τ∨(l) ∈ Kh∨ is the unique element such that 〈l, τ(v)〉 =trK/F 〈τ

∨(l), v〉 for any v ∈ Kh. Here the paring 〈−,−〉 is defined in (2.1). The uniqueness of

τ∨(l) requires the non-degeneracy of the form trK/F 〈−,−〉, which is valid because the extension

K/F is separable.

Roughly speaking, the map N•∼ → M•

∼ is obtained by identifying special fibers of deforma-

tions of GK with the ones of GF through ϕ, and identifying their level structures through τ∨. We

prefer working with τ∨ since the action of GLh(OK) on the πn-level structures is a right action,

while the action of GLh(OK) on τ is a left action.10

Note that the data τ and ϕ induce algebraic group embeddings GLh(K) ⊂ GLhk(F ) and D×K ⊂

D×F , which will be the essential data parametrizing our CM cycles. Therefore, for our convenience,

among those τ which induce the same group embedding, we can select one such that

(2.4) τ∨(Okh∨F ) ⊂ Oh∨K .

Denote the set of morphisms f : X −→ Y of two formal schemes over Spf OF by MorOF(X, Y ).

Remark 2.8. In our article, a map for two towers

η•∼•∼(ϕ, τ) : N•∼ −→M•

∼

means an element in

lim←−j

lim−→i

MorOF(Ni

∼,Mj∼).

This kind of element is uniquely determined by a system of compatible morphisms Nm+n∼ →Mn

∼

over Spf OF for all n ≥ 0 with some fixed m ≥ 0. In such a system we denote each member by

(2.5) ηn+mn (ϕ, τ) : Nm+n(j) −→Mn

(k)

where the superscript is intentionally chosen to specify the domain, and the subscript is chosen for

the codomain.

Next we define maps of Lubin–Tate towers induced by the pair (ϕ, τ). We start with the simplest

case that τ∨(Oh∨K ) = Okh∨F , in which the corresponding algebraic group embedding GLh(K) ⊂GLhk(F ) restricts to GLh(OK) ⊂ GLhk(OF ). Moreover, if α is a πm-level structure for G, then so

is α τ∨.

Definition 2.9. If τ∨(Oh∨K ) = Okh∨F , we define the map η•∼•∼(ϕ, τ) : N•∼ −→M•

∼ by the following

compatible morphisms, functorial in A ∈ C,

(2.6)ηnn(ϕ, τ) : Nn

(j)(A) −→ Mn(rj+Height(ϕ−1))(A)

[G, ι, α]n 7−→ [G, ι ϕ−1, α τ∨]n

for each n ≥ 0. In other words, ηnn(ϕ, τ) maps [G, ι, α]n to [G, ι′, α′]n such that the following two

diagrams

(2.7) GKϕ //

ι

GF

ι′

Gid

G // G

Okh∨Fτ∨ //

α′

Oh∨K

α

G G

idGoo

commute.

This definition is obviously well-defined as this map carries equivalent triples to equivalent

triples. In general, if τ∨(Oh∨K ) 6= Okh∨F , the map α τ∨ may not be a well-defined level structure.

We fix this issue and generalize our definition in the remaining part of this subsection.

Definition 2.10. For any F -linear map τ : Kh −→ F kh, the conductor cond(τ) of τ is the minimal

integer m such that

τ∨(Okh∨F

)⊃ πmOh∨K .

11

The idea to generalize Definition 2.9 is allowing the identity maps in the diagram (2.7) to be

replaced by certain isogenies, which we describe now. For any pairs (ϕ, τ), any integer m ≥cond(τ), and any equivalence class of πm+n-level deformation [G, ι, α]m+n over A, we construct a

formal OK-module G′ and an isogeny ψ : G −→ G′ over A as follows. Note that

(2.8) τ∨(Okh∨F ) ⊃ πmOh∨K .

Put

(2.9) V = τ∨(πnOkh∨F )/πn+mOh∨K .

Since α : Okh∨F −→ G(A) is a Drinfeld πn+m-level structure, by definition, it induces a morphism

α : Okh∨F /πn+mOkh∨F −→ G(A).

Consider a power series ψ defined by

(2.10) ψ(X) =∏

v∈V

(X [−]Gα(v)).

By Serre’s construction, there exists a formal OF -module G′ over A such that ψ : G −→ G′ is an

isogeny. Note that this construction is functorial in A ∈ C.

Definition 2.11. We call the isogeny ψ : G −→ G′ in (2.10) Serre’s isogeny associated to m,(ϕ, τ)and G.

Lemma 2.12. Let ψ : G −→ G′ be Serre’s isogeny associated to m, (ϕ, τ) and G for some m ≥cond(τ). Suppose α is a πm+n-level structure for G, then ψ α τ∨ is a πn-level structure for G′.

Proof. It suffices to show that [πn]G′(X) is divisible by∏

v∈Okh∨F /πnOkh∨

F

(X [−]G′ψ α τ∨(v)

).

Since the construction of Serre’s isogeny is functorial in A ∈ C, without loss of generality, we

assume that G is the universal formal OK-module over A where A is chosen so that Nm+n(0) =

Spf A. As A is a regular local ring and, in particular, a unique factorization domain, it suffices to

prove that ψ α τ∨(v) are distinct solutions of [πn]G′(X) = 0 for v ∈ Okh∨F /πnOkh∨F . First, for

any v,w ∈ Okh∨F /πnOkh∨F such that v 6= w, we have ψ α τ∨(v) 6= ψ α τ∨(w) because

τ∨(v)− τ∨(w) /∈ ker(ψ α

)= τ∨(πnOkh∨F ).

Second, X = ψ α τ∨(v) is a solution for [πn]G′(X) = 0 for any v ∈ Okh∨F /πnOkh∨F . Indeed,

[πn]G′(ψ α τ∨(v)) = ψ([πn]G α τ∨(v))

= ψ α τ∨(πnv)

= 0

as desired. This completes the proof.

Definition 2.13. For any pair (ϕ, τ) as described in (2.3), and any interger m ≥ cond(τ), we

define the map η•∼•∼(ϕ, τ) : N•∼ −→ M•

∼ by the following compatible morphisms over Spf OF ,

functorial in A ∈ C,

(2.11)ηn+mn (ϕ, τ) : Nn+m

(j)(A) −→ Mn(rj+Height(ψπ−m)+Height(ϕ−1))(A)

[G, ι, α]n+m 7−→ [G′, ψ ι π−mϕ−1, ψ α τ∨]n12

for any n ≥ 0, where ψ : G −→ G′ is Serre’s isogeny associated to (ϕ, τ), m, and G. In other

words, ηn+mn (ϕ, τ) maps [G, ι, α]n+m to [G′, ι′, α′]n such that the following diagrams commute

(2.12) GKπmϕ //

ι

GF

ι′

Gψ // G

′,

Okh∨Fτ∨ //

α′

Oh∨K

α

G′ G.

ψoo

This definition is well-defined by previous lemmas and propositions. It is clear that the in-

duced map η•∼•∼(ϕ, τ) does not depend on the choice of m since one sees that ηn+m+1n (ϕ, τ) =

ηn+mn (ϕ, τ) θn+m+1n+m for any m ≥ cond(τ), where we recall that θn+m+1

n+m : Nn+m+1 −→ Nn+m is

the transition map. Moreover, one can show that those morphisms are finite by Proposition 3.9.

From now on, by saying Serre’s isogeny attached to ηn+mn (ϕ, τ) and G, we mean Serre’s isogeny

associated to m, (ϕ, τ) and G. When no confusion can arise, we will simply say Serre’s isogeny

attached to ηn+mn (ϕ, τ).

Corollary 2.14. If ηn+mn (ϕ, τ)[G, ι, α]n+m = [G′, ι′, α′]n, then there exists an isogeny ψ : G −→ G′

such that the diagram (2.12) commutes.

Proof. Let G′′ be the formal module obtained from Serre’s isogeny ψ′ : G −→ G′′ attached to

ηn+mn (ϕ, τ) and G, and write

ηn+mn (ϕ, τ)[G, ι, α]n+m = [G′′, ι′′, α′′]n.

Note that G′′ may not equal G′. Instead, there is an isomorphism ζ : G′′ −→ G′ translating

[G′′, ι′′, α′′]n to [G′, ι′, α′]n. Letting ψ = ζ ψ′, we have the following diagrams

GKπmϕ //

ι

GFidGF //

ι′′

GF

ι′

G

ψ

66❲ ❬ ❴

ψ′

// G′′ ζ // G

′

Oh∨K

α

Okh∨Fτ∨oo

α′′

Okh∨Fidoo

α′

G

ψ′

//

ψ

55❳ ❬ ❴ G′′

ζ // G′

where for each diagram, the left square commutes because of Definition 2.13, and the right square

commutes because of Definition 2.4. The corollary follows from the commutativity of outer

squares.

2.3. Heights. Note that the superscript of the codomain of the map in (2.11) is shifted by

Height(ϕ−1) + Height(ψ π−m),

where the shifting contributed by τ is Height(ψπ−m). For ease of exposition, we call this number

the height of τ . Note that

Height(π−mψ) = Height(π−m) + logq#V = logq Vol(τ∨(Okh∨F )).

Definition 2.15. Recall that q is the cardinality of the residue field of OF . For an F -linear map

τ : Kh −→ F kh, define the height of τ by

(2.13) Height(τ) = logq Vol(τ∨(Okh∨F )),

where the volume function Vol(•) is normalized so that Vol(Oh∨K ) = 1.13

2.4. Group actions on Lubin–Tate towers. From now on, we fix an isomorphism End(GF )⊗OF

F ∼= DF . The Lubin–Tate tower M•∼ admits a natural action of the group D×

F × GLkh(F ),which can be described as follows. An element (γ, g) ∈ D×

F × GLkh(F ) induces the following

homomorphisms

γ : GF −→ GF ;

g : F kh −→ F kh,(2.14)

where g is an F -linear isomorphism and γ is a quasi-isogeny of formal OF -modules. We may

assume further that

(2.15) g∨(Okh∨F ) ⊂ Okh∨F .

Therefore, the pair (γ, g) induces a map functorial in A ∈ C

(2.16)

βm+nn (γ, g) : Mn+m

(j)(A) −→ Mn(l)(A)

[G, ι, α]n+m 7−→ [G′, ψ ι π−mγ−1, ψ α g∨]n

in a similar way as in Definition 2.13, where l = j − Height(γ) + Height(g), and ψ : G −→ G′ is

Serre’s isogeny attached to βm+nn (γ, g). Here Height(g) is defined similarly as in (2.13). Note that

the construction of ψ and βm+nn (γ, g) are essentially the same as in Definition 2.13 if we replace

K/F by the trivial extension F/F . Therefore the same arguments prove that the map βm+nn (γ, g)

is well-defined, and hence we obtain a map β•∼•∼(γ, g) : M•

∼ −→M•∼ for the Lubin–Tate tower.

If both γ and g are identity, we abbreviate βn+mn (id, id) to

βn+mn : Mn+m∼ −→Mn

∼,

which coincides with the transition map between Mn+m∼ and Mn

∼ in the Lubin–Tate tower.

2.5. CM cycles on the Lubin–Tate tower. In this subsection, we introduce CM cycles on the

Lubin–Tate tower M•∼ induced by

(2.17) η•∼•∼(ϕ, τ) : N•∼ −→M•

∼.

In this article, we construct cycles as coherent sheaves. To define a system of compatible cycles,

we need to uniformize their multiplicities. Therefore we need to consider things like F⊕ 12 for

some coherent sheaf F on a formal scheme X . This motivates us to define the following concept.

Denote by K (X) the free abelian group generated by isomorphism classes of coherent sheaves of

OX -modules, modulo the relations by [F⊕n] = n[F] for any coherent sheaf F of OX -modules on

X , where we use [F] to denote the corresponding element in K (X) associated to F.

Definition 2.16. Let η•∼•∼(ϕ, τ) be the map between Lubin–Tate towers induced by (ϕ, τ) as in

Definition 2.13. The corresponding CM cycle Z∼• (ϕ, τ) is a collection of cycles

Z(j)n (ϕ, τ) ∈ K

(Mn

(j))⊗Z Q

for each n and j, where each cycle Z(j)n (ϕ, τ) is defined as follows: Suppose the map η•∼•∼(ϕ, τ)

from Nn+m∼ to Mn

∼ is given by

ηn+mn (ϕ, τ) : Nn+m(l) −→Mn

(j)

14

where l = jr− Height(τ) + Height(ϕ). If l is an integer, we define

(2.18) Z(j)n (ϕ, τ) :=

1

deg(Nn+m

(l) → Nn(l))[ηn+mn (ϕ, τ)∗ONn+m

(l)

]

where the map Nn+m(l) → Nn

(l) is the transition map. Otherwise, we define Z(j)n (ϕ, τ) := 0.

Remark 2.17. The Definition 2.16 does not depend on m because each transition map

θm+1m : Nm+1

(l) −→ Nm(l)

is a finite flat map of formal spectra of regular local rings, and therefore θm+1m∗ O

Nm+1(l)∼= Od

Nm(l)

for d := deg(Nm+1

(l) → Nm(l))

. Further, since each (γk, gk) ∈ D×K × GLh(K) induces an

isomorphism of the tower N•∼, one finds that Z∼

• (ϕ γk, τ gk) = Z∼• (ϕ, τ) by an analogue of

Lemma 2.22. Therefore, the cycle only depends on the algebraic group embeddings D×K → D×

F

and GLh(K)→ GL2h(F ) induced by ϕ and τ∨.

2.6. Classical Lubin–Tate spaces. Recall that the Lubin–Tate tower is a disjoint union of com-

ponents indexed by integers

M•∼ =

∐

j∈Z

M•(j).

Note that the action of an element ω ∈ D×F with valuation j maps Mn

(j) isomorphically onto Mn(0),

which reduces all problems to a single space Mn(0). From now on, without loss of generality, we

only consider Lubin–Tate spaces with superscript (0) for ease of exposition, whereas we come

back to the general situation later in §6.4.

Definition 2.18. We call a space in the Lubin–Tate tower with superscript (0) ( for example Nn(0)

or Mn(0)) a classical Lubin–Tate space. For simplicity, we omit their superscript and denote them

by Nn or Mn.

When considering CM cycles on Mn, we may also omit the superscript (0) of a cycle and simply

denote it by

Zn(ϕ, τ) := Z(0)n (ϕ, τ).

2.6.1. The height restrictions for CM cycles on classical Lubin–Tate spaces. To induce a mor-

phism from N• to M•, a pair (ϕ, τ) must be restricted so that the superscript (0) remains un-

changed after the map η•∼•∼(ϕ, τ) (see Definition 2.13). This is equivalent to requiring Height(τ) =Height(ϕ).

Definition 2.19. We call a pair of morphisms in (2.3) an equi-height pair if

Height(τ) = Height(ϕ).

Denote by Equih(K/F ) the set of equi-height pairs.

Remark 2.20. Similarly, applying Definition 2.19 to the trivial extension F/F , one obtains that

Equikh(F/F ) is the set of elements (γ, g) ∈ D×F ×GLkh(F ) such that Height(γ) = Height(g), or

equivalently,

| det(g)|F = |nrd(γ)|−1F ,

where nrd(•) is the reduced norm of DF .15

2.7. Composition of maps of Lubin–Tate spaces. In this subsection, we introduce several prop-

erties of compositions of maps between Lubin–Tate towers for our applications later.

Proposition 2.21. Letting (ϕ, τ) ∈ Equih(K/F ), (γ, g) ∈ Equikh(F/F ), m1 ≥ cond(τ) and

m2 ≥ cond(g), we have

βn+m2n (γ, g) ηn+m2+m1

n+m2(ϕ, τ) = ηn+m1+m2

n (γ · ϕ, g · τ).

In other words, the following diagram commutes

Nn+m1+m2

ηn+m1+m2n (γ·ϕ,g·τ)

((

ηn+m2+m1n+m2

(ϕ,τ)

uu

Mn+m2

βn+m2n (γ,g)

// Mn.

Proof. For any A ∈ C, and any [G, ι, α]n+m1+m2 ∈ Nn+m1+m2(A), let

[G′, ι′, α′]n+m2 := ηn+m2+m1n+m2

(ϕ, τ)[G, ι, α]n+m1+m2 ;

[G′′, ι′′, α′′]n := βn+m2n (γ, g)[G′, ι′, α′]n+m2 ,

(2.19)

and denote Serre’s isogenies attached to ηn+m2+m1n+m2

(ϕ, τ) and βn+m2n (γ, g) by

ψ1 : G −→ G′,

ψ2 : G′ −→ G′′.

By Definition 2.13 and using (2.12), we have the following commutative diagrams

(2.20) GKπm1ϕ //

ι

GFπm2γ //

ι′

GF

ι′′

Gψ1 // G

′ ψ2 // G′′

Oh∨K

α

Okh∨Fτ∨oo

α′

Okh∨F

α′′

g∨oo

Gψ1 // G′

ψ2 // G′′.

Letting ψ3 : G −→ G′′ be Serre’s isogeny attached to ηn+m1+m2n (γ · ϕ, g · τ), it suffices to prove

ψ3 = ψ2 ψ1.

By definition of ψ1 : G −→ G′ and ψ2 : G′ −→ G′′, we have

(2.21) ψ2(ψ1(X)) =∏

v∈U(g)

(ψ1(X)[−]G′α′(v))

where U(g) = g∨(πnOkh∨F )/πn+m2Okh∨F . Using the diagram (2.20), we have α′ = ψ1 α τ∨,

which allows us to write each factor of (2.21) as

ψ1(X)[−]G′α′(v) = ψ1(X)[−]G′ψ1(α(τ∨(v))).

Denote w := τ∨(v). Since ψ1 is an isogeny from G to G′, the factor further simplifies to

ψ1(X)[−]G′ψ1(α (w)) = ψ1 (X [−]Gα(w)) =∏

v∈U(τ)

(X [−]Gα(w + v)) ,

where U(τ) = πn+m2τ∨(Okh∨F )/πn+m1+m2Oh∨K , and the last equality is obtained from the definition

of ψ1. Putting these simplified factors back to the original product (2.21), we obtain

ψ2 ψ1(X) =∏

w∈U(gτ,τ)

∏

v∈U(τ)

(X [−]Gα(w + v)) ,

16

where U(gτ, τ) = τ∨ g∨(πnOkh∨F )/πn+m2τ∨(Okh∨F ). Therefore, by further simplifying this ex-

pression we obtain

ψ2 ψ1(X) =∏

v∈U(gτ)

(X [−]Gα(v)) = ψ3(X),

where U(gτ) = (gτ)∨(πnOkh∨F )/πn+m1+m2Oh∨K . This implies ψ3 = ψ2 ψ1, which completes the

proof of the lemma.

The next lemma shows that the action of (γ, g) translates the cycle Zn(ϕ, τ) to Zn(γ · ϕ, g · τ).

Lemma 2.22. For any n ≥ 0, any (ϕ, τ) ∈ Equih(K/F ), any (γ, g) ∈ Equikh(F/F ), and any

m ≥ cond(g), we have

(2.22)1

deg(Nn+m

// Nn

)βn+mn (γ, g)∗Zn+m(ϕ, τ) = Zn(γ · ϕ, g · τ).

Proof. Let M = cond(τ). By Definition 2.16, it suffices to prove that

1

deg(Nn+m+M

// Nn+m

)deg

(Nn+m

// Nn

)βn+mn (γ, g)∗ηn+m+Mn+m (ϕ, τ)∗

[ONn+M+m

]

=1

deg(Nn+m+M

// Nn

)ηn+m+Mn (γ · ϕ, g · τ)∗

[ONn+M+m

],

which follows directly from the observation that

deg(Nn+M+m

// Nn

)= deg

(Nn+m+M

// Nn+m

)deg

(Nn+m

// Nn

),

and that

βn+mn (γ, g) ηn+m+Mn+m (ϕ, τ) = ηn+m+M

n (γ · ϕ, g · τ)

by Proposition 2.21.

3. AN APPROXIMATION FOR CM CYCLES ON THE INFINITE LEVEL

In this section, we keep our general assumption that [K : F ] = k, and use the same notation

as in the previous section. As we described in Definition 2.13, an element (ϕ, τ) ∈ Equih(K/F )induces a system of compatible morphisms

ηn+mn (ϕ, τ) : Nm+n −→Mn

for all n, and all m ≥ cond(τ). Let GhK (resp.GkhF ) be the h-(resp.kh-)-fold self-direct product of

GK (resp.GF ) over SpecFq. In this section, we will define a similar map

η∞+m∞ (ϕ, τ) : GhK −→ GkhF

in Definition 3.1, and use it to construct CM cycles Z∞(ϕ, τ) on GkhF . We intentionally use∞ as

our subscript since Z∞(ϕ, τ) should be thought of as an approximation of Zn(ϕ, τ) when n→∞.

Our main result in this section is Proposition 3.11, which shows a comparison identity

imm+n∗Zn+m(ϕ, τ) = im∞

∗Z∞(ϕ, τ) ∈ K(G2hF [πm]

)⊗Z Q

for the following closed embeddings for any n > 0,

Mm+n G2hF [πm]

im∞ //

imm+noo GkhF ,17

which will be defined in (3.18) and (3.19).

We need to introduce some constructions before proving our main results. Our strategy is to

interpret GkhF and GhK as moduli spaces of level structures of GF and GK respectively by writing

GhK∼= HomOK

(Oh∨K ,GK)

and

GkhF∼= HomOF

(Okh∨F ,GF ),

where by HomOK(Oh∨K ,GK) we mean the functor defined by assigning each A ∈ C to the OK-

module

HomOK

(Oh∨K ,GK(A)

),

and the meaning for HomOF(Okh∨F ,GF ) is similar.

Since all formal OK-modules of height h are isomorphic over Fq, throughout this subsection,

without loss of generality, we may assume that

[π]GK(X) = Xqkh.

Moreover, by C⊗Fq we mean the full subcategory of C consisting of those A ∈ C such that π = 0in A. In this section, all pairs (ϕ, τ) and (γ, g) are equi-height pairs, and the integer m is chosen

so that m ≥ cond(τ). Therefore

Height(ϕ−1) = −Height(τ) = − logq Vol(τ∨(Okh∨F )) ≥ 0,

and

Height(πmϕ) = −Height(πm(τ)−1) = logqVol(τ∨(Okh∨F ))

Vol(πmOh∨K )≥ 0,

which implies that ϕ−1 : GK −→ GF and πmϕ : GK −→ GF are actual isogenies.

3.1. Maps from GhK to GkhF .

Definition 3.1. For any (ϕ, τ) ∈ Equih(K/F ) and m ≥ cond(τ), let η∞+m∞ (ϕ, τ) : GhK −→ GkhF

be the quasi-homomorphism of formal OF -modules defined functorially in A ∈ C by

(3.1)

η∞+m∞ (ϕ, τ) : HomOK

(Oh∨K ,GK(A)) −→ HomOF(Okh∨F ,GF (A))

f 7−→ πmϕ f τ∨.

Moreover, let GhK [πmϕτ ] be the group scheme sitting inside the following exact sequence

0 // GhK [πmϕτ ] // GhK

η∞+m∞ (ϕ,τ)

// GkhF .

Remark 3.2. For any (γ, g) ∈ Equikh(F/F ) and m ≥ cond(g), functorially in A ∈ C, we define

the map β∞+m∞ (γ, g) by

(3.2)

β∞+m∞ (γ, g) : HomOF

(Okh∨F ,GF (A)) −→ HomOF(Okh∨F ,GF (A))

f 7−→ πmγ f g∨.

Moreover, GkhF [πmγg] is the group scheme sitting inside the following exact sequence

0 // GkhF [πmγg] // GkhFβ∞+m∞ (γ,g)

// GkhF .18

Proposition 3.3 (Analogue to Proposition 2.21). For any (ϕ, τ) ∈ Equih(K/F ), (γ, g) ∈ Equikh(F/F ),m1 ≥ cond(g) and m2 ≥ cond(τ), we have

β∞+m1∞ (γ, g) η∞+m2

∞ (ϕ, τ) = η∞+m1+m2∞ (γ · ϕ, g · τ).

In other words, the following diagram commutes

GhKη∞+m1+m2∞ (γ·ϕ,g·τ)

&&

η∞+m2∞ (ϕ,τ)

xxqqqqqqqqqqqqq

GkhFβ∞+m1∞ (γ,g)

// GkhF .

Proof. For any A ∈ C, and any f ∈ GhK(A), by Definition 3.1, we have

β∞+m1∞ (γ, g) η∞+m2

∞ (ϕ, τ)(f) = β∞+m1∞ (γ, g)(πm2ϕ f τ∨) = πm1+m2γ ϕ f τ∨ g∨,

which equals

πm1+m2(γ · ϕ) f (g · τ)∨ = η∞+m1+m2∞ (γ · ϕ, g · τ)(f)

as desired.

3.2. Thickening comparison. This part is the technical core of the article. Loosely speaking,

thinking of GhK as the deformation space on the infinite level, we temporarily denote N∞ := GhKand M∞ := GkhF . Then the exact sequence in Definition 3.1 is

0 // GhK [πmϕτ ] // N∞

η∞+m∞ (ϕ,τ)

// M∞.

Since intuitively M∞ is an approximation of Mn when n is sufficiently large, one may carelessly

expect an exact sequence

0 // GhK [πmϕτ ] // Nn+m

ηn+mn (ϕ,τ)

// Mn

for sufficiently large n. Going back to the rigorous math, the exact sequence here does not make

any sense since Mn is not a group scheme. Nevertheless, we realize this idea in Proposition 3.9 by

identifying GhK [πmϕτ ] with the preimage of the closed point Spec(Fq) under the map ηn+mn (ϕ, τ) :

Nn+m −→Mn, which is the main result of this subsection.

Before the construction of the map GhK [πmϕτ ] −→ Nm+n, we need some lemmas for level

structures of GK .

Lemma 3.4. All elements in

Hom(Oh∨K ,GK [π

n](A))

are πn+1-level structures of GK over A for any A ∈ C⊗ Fq.

Proof. Suppose A ∈ C⊗ Fq. We have π = 0 ∈ A. Let f ∈ Hom(Oh∨K ,GK [π

n](A)). It suffices to

verify the non-degeneracy condition for f , that is, to show that

(3.3)∏

w∈Oh∨K /πn+1Oh∨

K

(X − f(w)) = [πn+1]GK(X).

19

We prove it by induction on n. If n = 0, the expression (3.3) is clearly true since f = 0. Assuming

the following induction hypothesis

(3.4)∏

w∈Oh∨K /πnOh∨

K

(X − f(w)) = [πn]GK(X).

Abbreviating [πn]GKto [πn], the right hand side of (3.3) simplifies to

[πn+1](X) = [πn]([π](X)),

which, by the induction hypothesis (3.4), equals

∏

w∈Oh∨K /πnOh∨

K

([π]X − [π]f (w)) .

Since [π](X) = Xqkh , each factor of the preceding product simplifies to

[π]X − [π]f (w) = Xqkh − f(w)qkh

= (X − f(w))qkh

.

Putting the simplified factors back into the product, we obtain

∏

w∈Oh∨K /πnOh∨

K

(X − f(w))qkh

=∏

w∈Oh∨K /πn+1Oh∨

K

(X − f(w)),

which is the left hand side of (3.3). Here the last equality is obtained from the observation that

f(w + v) = f(w) + f(v) = f(w) for any v ∈ πnOh∨K . This completes the proof.

Lemma 3.5. All elements f ∈ GhK [πmϕτ ](A) are πm+n-level structures of GK over A for any

integer n > cond(τ), any integer m ≥ cond(τ) any (ϕ, τ) ∈ Equih(K/F ), and any A ∈ C⊗ Fq.

Proof. Since Height(ϕ−1) ≥ 0, the map

(3.5) ϕ−1 : GK −→ GF

is an actual isogeny. Since n ≥ cond(τ) + 1, we have τ∨(Okh∨F ) ⊃ πn−1Oh∨K , which implies that

πn−1(τ∨)−1(Oh∨K

)⊂ Okh∨F .

Therefore πn−1(τ∨)−1 restricts to a map

(3.6) πn−1(τ∨)−1 : Oh∨K −→ Okh∨F .

By Definition 3.1, f ∈ GhK [πmϕτ ](A) implies that πmϕ f τ∨ = 0, which can be put into the

following commutative diagram

Okh∨Fτ∨ //

0

Oh∨K

f

GF GK .

πmϕoo

20

Extending this diagram using the maps in (3.5) and (3.6), we have the following commutative

diagram

Oh∨K

πn−1

++ ❵ ❴ ❫ ❪ ❭ ❩ ❨ ❳

πn−1(τ∨)−1//

0

Okh∨F τ∨//

0

Oh∨K

f

GK GF

ϕ−1

oo GK .πmϕoo

πm

kk ❵❴❫❪❭❩❨❳

Note that this diagram implies that [πm+n−1]GK f = 0, and so that f is a πm+n-level structure by

Lemma 3.4 , which completes the proof.

The formal scheme GhK can be naturally thought of as a functor from C to the category of sets by

extending its definition from C⊗ Fq to C, where we define GhK(A) = ∅ for any A /∈ C⊗ Fq.

Definition 3.6. For any A ∈ C ⊗ Fq, by GK/A (resp. GF/A) we mean the formal OK (resp.OF )

module obtained by the base change of GK (resp. GF ) to A.

Note that GK is defined by a lot of power series describing its addition and its scalar multiplica-

tion by OK . Those power series have coefficients in Fq. The formal module GK/A has exactly the

same power series with GK , but we view those coefficients as coefficients in A through Fq ⊂ A.

Therefore, any endomorphism γ : GK −→ GK can be directly viewed as an endomorphism of

GK/A, and by abuse of notation, we use the same symbol γ : GK/A −→ GK/A to represent this

endomorphism.

Definition 3.7. We define the map GhK [πmϕτ ] −→ Nm+n functorially in A ∈ C by

(3.7)

GhK [πmϕτ ](A) −→ Nm+n(A)

f 7−→ [GK/A, id, f ]m+n

for any n > cond(τ) and any (ϕ, τ) ∈ Equih(K/F ).

The following lemma will be used in the proof of Proposition 3.9.

Lemma 3.8. For any A ∈ C⊗ Fq and [GK/A, γ, f ]n ∈ Nn(A), we have

[GK/A, γ, f ]n = [GK/A, id, γ−1 f ]n.

Proof. Note thatNn is an abbreviation of Nn(0) (for notation see Theorem 2.5). Since [GK/A, γ, f ]n ∈

Nn(0)(A), the morphism γ : GK −→ GK/A is of height 0, and thus an isomorphism. Since

GK/A = GK , the isomorphism γ is an automorphism of GK . By base change to A, the isomorphism

γ : GK/A → GK/A identifies the triple (GK/A, id, γ−1 f) to the triple (GK/A, γ, f). Therefore these

two triples are equivalent.

Proposition 3.9. In the following diagrams, all unlabeled vertical maps are defined by the same

way as in Definition 3.7. We have:21

(1) For any (ϕ, τ) ∈ Equih(K/F ), any m ≥ cond(τ), and any n > cond(τ), the following

diagram

(3.8) GhK [πmϕτ ] //

SpecFq

Nm+n

ηn+mn (ϕ,τ)

// Mn

is Cartesian. In particular, the vertical maps are closed embeddings.

(2) For any (ϕ1, τ1) ∈ Equih(K/F ), any (γ, g) ∈ Equikh(F/F ), anym1 > cond(τ1), and any

m2 > cond(g), we denote (ϕ3, τ3) := (γϕ1, gτ1) ∈ Equih(K/F ), and m3 := m1 +m2.

Then the following diagram

(3.9) GhK [πm3ϕ3τ3]

η∞+m1∞ (ϕ1,τ1) //

GkhF [πm2γg]

Nn+m1

ηn+m1n (ϕ1,τ1) // Mn

is Cartesian for any n > m2 + cond(τ3).

Proof of statement (1). It suffices to prove that the diagram (3.8) is Cartesian. Then the morphism

GhK [πmϕτ ] −→ Nm+n is automatically a closed embedding since it is obtained by base change of

the closed embedding SpecFq → Mn. In other words, it suffices to show that for any A ∈ C and

any [G, ι, α]m+n ∈ Nm+n(A),

(3.10) ηn+mn (ϕ, τ)[G, ι, α]m+n = [GF/A, id, 0]n

if and only if

(3.11) [G, ι, α]m+n = [GK/A, id, f ]m+n for some f such that πmϕ f τ∨ = 0.

We prove that (3.11) implies (3.10) as follows. Let n′ = maxn,m. Let ψ : GK/A −→ G′ be

Serre’s isogeny attached to ηm+nn′ (ϕ, τ) and GK/A. By definition,

ψ(X) =∏

w∈V

(X [−]GK/A

f(w)),

where V = τ∨(πn′

Okh∨F )/πn+mOh∨K . Since πmϕ f τ∨ = 0 and Height(ϕ−1) ≥ 0, we have

f τ∨(πn′

v) = πn′−m · ϕ−1(πmϕ f τ∨(v)) = πn

′−m · ϕ−1(0) = 0

for any v ∈ Okh∨F , which implies that f(w) = 0 for any w ∈ V . Therefore

ψ(X) =∏

w∈V

(X [−]GK/A

f(w))= X#V .

Since

ψ [π]GK/A(X) = Xqkh+#V

= [π]GF/A ψ(X),

we have G′ = GF/A. Thus we can write ηm+nn′ (ϕ, τ)[GK/A, id, f ]n+m = [GF/A, ι, α

′]n′ for some

πn′

-level structure α′ : Okh∨F → GF/A(A) and some automorphism ι : GF → GF . Then

ηn+mn (ϕ, τ)[GK/A, id, f ]n+m = βn′

n ηn+mn′ (ϕ, τ)[GK/A, id, f ]n+m = βn

′

n [GF/A, ι, α′]n′ = [GF/A, ι, α]n,

22

where α = α′ πn′−n. By Corollary 2.14, we obtain an isogeny ψ′ : GK/A −→ GF/A such that the

following diagrams commute

(3.12) GKπmϕ //

id

GF

ι

GK

ψ′

// GF ,

OhK

f

OkhFτ∨oo

α

GK/A

ψ′

// GF/A.

Note that the lifting of endomorphisms of GF is unique, so that ψ′ = ψ′. From the left diagram of

(3.12), one sees that

ψ′ = ι πmϕ.

Since πmϕ f τ∨ = 0, from the right diagram of (3.12), we obtain that

α = ψ′ f τ∨ = ι πmϕ f τ∨ = ι 0 = 0,

which implies that

ηn+mn (ϕ, τ)[GK/A, id, f ]n+m = [GF/A, ι, 0]n.

Furthermore, note that Height(τ) = Height(ϕ). Therefore πmϕ : GF/A −→ GK/A and ψ′ :GK −→ GF are elements of the same height in End(GF ), which implies that ι = ψ′ (πmϕ)−1

is an isomorphism of GF . Therefore, by Lemma 3.8, [GF/A, ι, 0]n = [GF/A, id, 0]n, which proves

(3.10).

Conversely, let

ϕ0 : GK −→ GF , τ∨0 : Okh∨F −→ Oh∨K

be natural forgetful isomorphisms defined by only remembering OF -actions. Then there exists

γ ∈ DF and g ∈ GLkh(F ) such that the following two diagrams

GKϕ0

∼=//

ϕ''

GF

γxx♣♣♣

♣♣♣♣♣♣♣♣♣♣

GF ,

Kh∨ F kh∨τ∨0∼=

oo

F kh∨τ∨

ggPPPPPPPPPPPPP g∨

77♥♥♥♥♥♥♥♥♥♥♥♥♥

commute. Since Height(τ0) = Height(ϕ0) = 0 and Height(τ) = Height(ϕ), the pair (γ, g) is an

equi-height pair. Letting u = cond(τ)+1, we have Okh∨F ⊃ g∨(Okh∨F ) ⊃ πu−1Okh∨F , which implies

that

πu−1Okh∨F ⊂ πu−1g−1∨(Okh∨F

)⊂ Oh∨F .

Therefore πu−1g−1 ∈ Matkh×kh(OF ), and

cond(πu−1g−1) ≤ u− 1.

Since Height(π1−uγ−1) = Height(πu−1g−1), the pair (π1−uγ−1, πu−1g−1) is an equi-height pair.

Now we prove that (3.10) implies (3.11). Recall that (3.10) is ηn+mn (ϕ, τ)[G, ι, α]n+m = [GF/A, id, 0]n.

Composing this identity with βnn−u+1(π1−uγ−1, πu−1g−1), we obtain

βnn−u+1(π1−uγ−1, πu−1g−1) ηn+mn (ϕ, τ)[G, ι, α]n+m

= βnn−u+1(π1−uγ−1, πu−1g−1)[GF/A, id, 0]n

= [GF/A, ψ id πu−1(π1−uγ−1)−1, ψ 0 (π1−ug−1)∨]n−u+1

= [GF/A, id, 0]n−u+1,

(3.13)

23

where ψ : GF/A −→ GF/A is the Serre’s isogeny attached to (π1−uγ−1, π1−ug−1) and the last

equality follows from Lemma 3.8. Moreover, this expression can be calculated in another way as

follows,

βnn−u+1(π1−uγ−1, πu−1g−1) ηn+mn (ϕ, τ)[G, ι, α]n+m

= ηn+mn−u+1(π1−uγ−1ϕ, πu−1g−1τ)[G, ι, α]n+m

= ηn+mn−u+1(π1−uϕ0, π

u−1τ0)[G, ι, α]n+m

= [G, ι ϕ−10 , α πm+u−1τ∨0 ]n−u+1.

(3.14)

Therefore, we have [G, ι ϕ−10 , α πm+u−1τ∨0 ]n−u+1 = [GF/A, id, 0]n−u+1, which implies that two

triples (G, ι ϕ−10 , α πm+u−1τ∨0 ) and (GF/A, id, 0) are equivalent as formal OF -modules. Recall

that G is deformed from GK , and that GF is obtained by only remembering OF -actions of GK .

Therefore, in fact,

[G, ι, α πm+u−1τ∨0 ]n−u+1 = [GK/A, id, 0]n−u+1,

which implies that we have an isomorphism G −→ GK/A of formal OK-modules such that the

following diagrams commute

GKι

⑦⑦⑦⑦⑦⑦⑦⑦ id

!!

G // GK ,

Oh∨Kαπm+u−1τ∨0

⑦⑦⑦⑦⑦⑦⑦⑦ 0

""

G // GK/A.

Letting f = ι−1 α, we have

(3.15) [G, ι, α]n+m = [GK/A, id, f ]n+m.

Therefore by the assumption ηn+mn (ϕ, τ)[G, ι, α]n+m = [GF/A, id, 0]n, we obtain

ηn+mn (ϕ, τ)[GK/A, id, f ]n+m = [GF/A, id, 0]n,

which implies that there is an isogeny ψ : GK/A −→ GF/A such that the following diagrams

commute

GKπmϕ //

id

GF

id

GKψ // GF ,

Oh∨K

f

Okh∨Fτ∨oo

0

GK/A

ψ // GF/A.

Therefore, we have ψ = πmϕ and ψf τ∨ = 0, which implies that πmϕf τ∨ = 0. Combining

this with (3.15), we complete the proof of (3.11).

Proof of statement (2): Before starting the proof, we extend the diagram (3.9) into the following

diagram

GhK [πm3ϕ3τ3]

η∞+m1∞ (ϕ1,τ1) //

GkhF [πm2γg] //

SpecFq

Nn+m1

ηn+m1n (ϕ1,τ1) // Mn

βnn−m2

(γ,g)// Mn−m2

.24

Here the extended right square is constructed in the same way as (3.8), which is a Cartesian dia-

gram. If the left square commutes, then the whole diagram commutes, and therefore the left square

is automatically a Cartesian diagram because the right and the outer squares are Cartesian by state-

ment (1). Therefore, it suffices to show that the left square commutes, that is, for any A ∈ C⊗ Fqand f ∈ GhK [π

m3ϕ3τ3](A), one has

ηn+m1n (ϕ1, τ1)[GK/A, id, f ]n+m1 = [GF/A, id, π

m1ϕ1 f τ∨1 ]n.

Since the outer diagram is commutative, we have

(3.16) βnn−m2(γ, g)

(ηn+m1n (ϕ1, τ1)[GK/A, id, f ]n+m1

)= [GF/A, id, 0]n−m2 .

Since the right square is Cartesian by the first statement of Proposition 3.9, the equation (3.16)

implies that

ηn+m1n (ϕ1, τ1)[GK/A, id, f ]n+m1 = [GF/A, id, g]n

for some g ∈ HomOF(Okh∨F ,GF (A)). Therefore, by Corollary 2.14, there exists an isogeny ψ :

GK/A −→ GF/A such that the following diagrams commute

GKπm1ϕ1 //

id

GF

id

GKψ // GF

Oh∨K

f

Okh∨Fτ∨oo

g

GK/A

ψ // GF/A,

which imply that g = ψ f τ∨ and ψ = πm1ϕ1. Therefore we have g = πm1ϕ1 f τ∨, which

completes the proof.

3.3. CM cycles comparison. Now we define CM cycles on GkhF by similar ways as in Definition

2.16.

Definition 3.10. For any (ϕ, τ) ∈ Equih(K/F ), let Z∞(ϕ, τ) ∈ K(GkhF

)⊗Z Q be the element

obtained by

(3.17) Z∞(ϕ, τ) :=1

deg

(GhK

πm// GhK

)[η∞+m∞ (ϕ, τ)

∗OGh

K

].

Clearly, this definition does not depend on m by Proposition 3.3.

Now we will compare cycles Zn(ϕ, τ) with Z∞(ϕ, τ). Let im∞ be the natural inclusion map

(3.18) im∞ : GkhF [πm] −→ GkhF ,

and for any n ≥ 1 let imm+n be the map

(3.19)

imm+n : GkhF [πm] −→ Mn+m

f 7−→ [GF , id, f ]n+m

as defined in Definition 3.7. The following proposition is the main result of this section.

Proposition 3.11. For any integer m ≥ 0, n > cond(τ) and any (ϕ, τ) ∈ Equih(K/F ), we have

imm+n∗Zn+m(ϕ, τ) = im∞

∗Z∞(ϕ, τ) ∈ K(GkhF [πm]

)⊗Z Q.

25

Proof. By definition of Z∞(ϕ, τ) and Zn+m(ϕ, τ) in (3.17) and (2.18), it suffices to prove

1

deg(Nn+m+w

// Nn+m

)[imm+n

∗ηn+m+wn+m (ϕ, τ)∗ONn+m+w

]

=1

deg

(GhK

πw// GhK

)[im∞

∗η∞+w∞ (ϕ, τ)∗OGh

K

](3.20)

for some w ≥ cond(τ). Since n > 0, we observe that the denominator of two sides are equal

because

deg(Nn+m+w

// Nn+m

)= qkh

2w = deg

(GhK

πw// GhK

).

It therefore suffices to show that

(3.21) imm+n∗ηn+m+wn+m (ϕ, τ)∗ONn+m+w

∼= im∞∗η∞+w

∞ (ϕ, τ)∗OGhK.

Since n > cond(τ), by Proposition 3.9, we obtain the following Cartesian diagram

GhK [πm+wϕτ ]

η∞+w∞ (ϕ,τ)

//

im+wm+w+n(ϕ,τ)

GkhF [πm]

imm+n

Nn+m+w

ηn+m+wn+m (ϕ,τ)

// Mn+m,

where all vertical maps,which are closed embeddings, are defined in the same way as in (3.7) and

labeled as shown in the diagram. We then obtain the following isomorphisms for the left hand side

of (3.21),

imm+n∗ηn+m+wn+m (ϕ, τ)∗ONn+m+w

∼=η∞+w∞ (ϕ, τ)∗i

m+wm+w+n(ϕ, τ)

∗ONn+m+w

∼=η∞+w∞ (ϕ, τ)∗OGh

K [πm+wϕτ ].(3.22)

On the other hand, by using the following Cartesian diagram,

GhK [πm+wϕτ ]

η∞+w∞ (ϕ,τ)

//

im+w∞ (ϕ,τ)

GkhF [πm]

im∞

GhKη∞+w∞ (ϕ,τ)

// GkhF ,

where all vertical maps are canonical closed embeddings and are labeled as shown, we obtain the

following isomorphisms for the right hand side of (3.21)

im∞∗η∞+w

∞ (ϕ, τ)∗OGhK

∼=η∞+w∞ (ϕ, τ)∗i

m+w∞ (ϕ, τ)∗OGh

K

∼=η∞+w∞ (ϕ, τ)∗OGh

K [πm+wϕτ ].(3.23)

Comparing (3.22) with (3.23), we complete the proof that two sides of (3.21) are equal, and hence

complete the proof of this proposition.

26

4. INTERSECTION COMPARISON

From now, we take k = 2, and then K/F is a quadratic extension. In the previous section, we

proved that there are two closed embeddings of formal schemes

(4.1) G2hF G2h

F [πM ]iM∞oo

iMn // Mn,

and that our CM cycles Z∞(ϕ, τ) and Zn(ϕ, τ) satisfy

(4.2) iMn∗Zn(ϕ, τ) = iM∞

∗Z∞(ϕ, τ).

Intuitively, the cycle Z∞(ϕ, τ) is an approximation of Zn(ϕ, τ) for n → ∞. Loosely speaking,

when n is sufficiently large, one can choose a large enough M in the diagram (4.1) such that the

intersection Z∞(ϕ1, τ1) ⊗G2hFZ∞(ϕ2, τ2) is contained completely inside the formal neighborhood

G2hF [πM ]. Then from the diagram (4.1) and the equation (4.2), we identify the intersection numbers

on sufficiently high levels with the one on the infinite level together. The purpose of this section is

to gather commutative algebra arguments to verify this intuition.

From this section onwards, we consider two quadratic extensions K1, K2 of F , which are not

necessarily isomorphic. Our focus in this section is to prove the following key theorem:

Theorem 4.1 (Intersection Comparison). For any (ϕ1, τ1) ∈ Equih(K1/F ) and (ϕ2, τ2) ∈ Equih(K2/F ),if the length of Z∞(ϕ1, τ1)⊗ Z∞(ϕ2, τ2) is finite, then there exists N > 0(see (4.19)) such that for

all n ≥ N , we have

(4.3) χ(Zn(ϕ1, τ1)⊗LMn

Zn(ϕ2, τ2)) = χ(Z∞(ϕ1, τ1)⊗LG2hFZ∞(ϕ2, τ2)).

Remark 4.2. The reader might be interested in the case where the length ofZ∞(ϕ1, τ1)⊗Z∞(ϕ2, τ2)is infinite, in which case, it is easy to show that the length of Z0(ϕ1, τ1) ⊗

LM0

Z0(ϕ2, τ2) is infinite

as well. We omit the proof here since this case is not important for the application of the linear

AFL. However, the author does not know if it is true that the length of Zn(ϕ1, τ1) ⊗LM0

Zn(ϕ2, τ2)is infinite as well for n ≥ 1.

Since we can regard G2hF as a formal scheme over OF by the morphisms

G2hF −→ Spf Fq −→ Spf OF ,

we only need to work in the category of formal schemes over OF , and hence we abbreviate

lengthOF(•) to length(•) for the rest of the article.

4.1. Outline of proof. We will prove Theorem 4.1 in 3 steps.

Step 1: We reduce the calculation of intersection numbers to intersection multiplicities by proving

the following identities:

(4.4) χ(Zn(ϕ1, τ1)⊗LMn

Zn(ϕ2, τ2)) = length(Zn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2)),

(4.5) χ(Z∞(ϕ1, τ1)⊗L

G2hFZ∞(ϕ2, τ2)) = length(Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2))

when the right hand sides of them are finite.

Step 2: By using the following closed embeddings,

(4.6) iMn : G2hF [πM ] −→Mn,

(4.7) iM∞ : G2hF [πM ] −→ G2h

F ,27

we prove that

length(iMn ∗

iMn∗(Zn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2)

))

= length(iM∞∗

iM∞∗(Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2)

))(4.8)

for any n > max (cond(τ1), cond(τ2)) +M by Proposition 3.11.

Step 3: We prove that the approximation of (4.6) and (4.7) will eventually catch the whole inter-

section as M →∞, in other words,

(4.9) iMn ∗iMn

∗(Zn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2)

)= Zn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2),

(4.10) iM∞∗iM∞

∗(Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2)

)= Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2)

for any n > M , where M is a large integer depending only on (ϕ1, τ1) and (ϕ2, τ2). With this

choice of M , and letting N = M + max (cond(τ1), cond(τ2)) + 1, the finiteness assumption for

the length of Zn(ϕ1, τ1) ⊗Mn Zn(ϕ2, τ2) in Step 1 follows from the finiteness assumption for the

length of Z∞(ϕ1, τ1) ⊗G2hFZ∞(ϕ2, τ2), and therefore we will complete the proof of Theorem 4.1

once we finished these three steps.

4.2. Step 1: Reduce to intersection multiplicities. Note that the following identity

(4.11) χ(F ⊗LX G) =

∑

i,j

(−1)i+j lengthOFRjp∗

(ToriX(F,G)

)

holds for any coherent sheaves F and G of OX -modules, whereX is a formal scheme over OF with

structure map p : X −→ Spf OF . Note that Mn and G2nF are affine formal schemes. To prove (4.4)

and (4.5), it suffices to show that

(4.12) ToriMn(Zn(ϕ1, τ1), Zn(ϕ2, τ2)) = 0,

(4.13) ToriG2hF(Z∞(ϕ1, τ1), Z∞(ϕ2, τ2)) = 0

for any i > 0, and that higher direct images vanish for Mn −→ Spf OF and G2hF −→ Spf OF .

However, since the vanishing of higher direct images follows directly from the fact that G2hF and

Mn are affine formal schemes, the results (4.4) and (4.5) follow directly from (4.12) and (4.13).

We prove (4.12) and (4.13) in the following lemmas.

Lemma 4.3 (Acyclicity Lemma). [Sta17, Tag 00N0] Let A be a Noetherian local ring, and let

M• = 0→ Mh → · · · →M0

be a complex of A-modules such that depthA(Mi) ≥ i. If depthA(H

i(M•)) = 0 for any i, then

M• is exact.

Proof. See Stacks Project [Sta17, Tag 00N0].

The following lemma gives some condition for the vanishing of higher torsion groups.

Lemma 4.4. [Sta17, Tag 0B01] Suppose A → B1 and A → B2 are homomorphisms of regular

local rings such that

(1) dim(A) = 2h, dim(B1) = dim(B2) = h;

(2) depthA(B1) = depthA(B2) = h;

(3) lengthA(B1 ⊗A B2) <∞.28

Then the higher torsion groups vanish, that is,

ToriA(B1, B2) = 0

for any i > 0.

Proof. 1 Since A is a regular local ring, we have depthA(A) = 2h. Therefore, there is a finite free

A-module resolution F• → B1 of length

depthA(A)− depthA(B1) = 2h− h = h.

Note that ToriA(B1, B2) is the i’th cohomology group of the complex F• ⊗A B2 → B1 ⊗A B2.

Using Lemma 4.3, it suffices to prove the following claims:

• depthA(ToriA(B1, B2)) = 0,

• depthA(Fi ⊗A B2) = depthA(B2) = h ≥ i.

Note that each cohomology group of the complex F• ⊗A B2 → B1 ⊗A B2 is a finite module over

the Artinian ring B1 ⊗A B2. Hence depthA(ToriA(B1, B2)) = 0, which verifies the first claim. As

each Fi is a free A-module, we have depthA(Fi ⊗A B2) = depthA(B2) = h ≥ i, which verifies

the second claim and therefore completes the proof.

In fact, we are in a special situation where the depth of our ring is the same as its dimension,

which we will describe in the following lemma.

Lemma 4.5. SupposeA,B are regular local rings with residue field Fq. Let f : Spf B −→ Spf A

be a map such that f−1(SpecFq) is an Artinian scheme. We have

depthA(B) = dim(B).

Proof. Let J = f1, f2, · · · fn ⊂ mA be a subset that generates the maximal ideal mA of A. The

scheme f−1(SpecFq) is the spectrum of the ring

B/(f1, f2, · · · , fn) = B/mAB.

Consider the subset J ′ := fii∈J ⊂ J , where

J = i : dim(B/(f1, · · · , fi−1)) > dim(B/(f1, · · · , fi−1, fi)) .

Note that depthA(B) is the number of elements in a regular sequence of B. It suffices to show

that J ′ is a regular sequence of length dim(B). Indeed, since f−1(SpecFq) is Artinian, we have

dim(B/(f1, · · · , fn)) = 0. Since dim(B/(f1, · · · , fi−1)) − dim(B/(f1, · · · , fi−1, fi)) equals ei-

ther 0 or 1, we have

depthA(B) = #J = dim(B),

which completes the proof.

Now we are ready to complete the proof of Step 1.

Proof of Step 1: (4.12) and (4.13). We prove these two identities together. For any

m ≥ max cond(τ1), cond(τ2) ,

we put

A = OMn , B1 = ηn+mn (ϕ1, τ1)∗ONn+m , B2 = ηn+mn (ϕ2, τ2)∗ONn+m

1This proof is written according to [Sta17, Tag 0B01].

29

for the proof of (4.12), and put

A = OG2hF, B1 = η∞+m

∞ (ϕ1, τ1)∗OGhK, B2 = η∞+m

∞ (ϕ2, τ2)∗OGhK

for the proof of (4.13). Note that Proposition 3.9 implies that

η∞+m∞ (ϕi, τi)

−1(SpecFq

)= GhK [π

mϕiτi] ∼= ηn+mn (ϕi, τi)−1(SpecFq

)

for i = 1, 2. Since GhK [πmϕτ ] is an Artinian scheme, we have depthA(B1) = dim(B1) and

depthA(B2) = dim(B2) by Lemma 4.5. By Lemma 4.4, it then suffices to show that

• dim(A) = 2h, dim(B1) = dim(B2) = h,

• lengthA(B1 ⊗A B2) <∞,

which are clearly true. This completes the proof of Step 1.

4.3. Step 2: Multiplicities inside the thickening.

Proof of Step 2: In this step, we prove (4.8), which states that the following elements

iMn ∗iMn

∗(Zn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2)

), iM∞∗

iM∞∗(Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2)

)

have the same length. Abbreviate Zi,n := Zn(ϕi, τi) and Zi,∞ := Z∞(ϕi, τi) for i = 1, 2. Note that

length(iMn ∗

iMn∗(Z1,n ⊗Mn Z2,n)

)= length

(iMn

∗Z1,n ⊗G2h

F [πM ] iMn

∗Z2,n

)

and

length(iM∞∗

iM∞∗(Z1,∞ ⊗G2h

FZ2,∞

))= length

(iM∞

∗Z1,∞ ⊗G2h

F [πM ] iM∞

∗Z2,∞

).

It suffices to prove for i = 1, 2,

iMn∗Zn(ϕi, τi) = iM∞

∗Z∞(ϕi, τi),

which follows directly from Proposition 3.11 when n > max(cond(τ1), cond(τ2)) + M . This

completes the proof of Step 2.

4.4. Step 3: Actual Multiplicity. In this step, our goal is to show that

iMn ∗iMn

∗(Zn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2)

)= Zn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2),

iM∞∗iM∞

∗(Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2)

)= Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2).

Lemma 4.6. Let A, B be Noetherian local rings and let mA, mB be their maximal ideals respec-

tively. Let F be a coherent sheaf of OSpf A-module. Suppose there is a closed embedding

s : Spf B/mMB −→ Spf A

such that lengthA(s∗s∗F) < M . If either of the following conditions hold

• dim(A) = dim(B) and both A and B are complete regular local rings with the same

residue field,

• A = B and s is a morphism over Spf A,

then we have

s∗s∗F ∼= F.

30

Proof. Let E = F(Spf A) be the A-module of global sections of F, and we denote the induced

map of s by

S : A −→ B/mMB .

Since s is a closed embedding, the homomorphism S is surjective. Since Spf A and Spf B are

affine formal schemes, it suffices to prove that

E ⊗A B/mMB = E,

which is equivalent to

(4.14) S−1(mMB )E = 0.

The key step of the proof is showing that S−1(mMB ) = mM

A , which is clearly true when A = B with

s a morphism over Spf A. Therefore, it remains to prove for the other case of complete regular

local rings of the same dimension. Since A and B are complete local rings, we have S(mA) ⊂ mB

and S−1(mB) ⊂ mA. Therefore the surjective ring homomorphism S gives rise to a surjective

graded ring homomorphism

S :M−1⊕

n=0

mnA/m

n+1A −→

M−1⊕

n=0

mnB/m

n+1B ,

where we denote m0A := A and m0

B := B. Letting d := dim(A) = dim(B) and k := A/mA∼=

B/mB , we have

dimk

(mnA/m

n+1A

)=

(n+ d− 2)!

(d− 1)!(n− 1)!= dimk

(mnB/m

n+1B

),

which implies that S is an isomorphism. Therefore in both cases, we obtain the key identity

(4.15) S−1(mMB ) = m

MA .

Since lengthA(E) < M , the descending chain

E ⊃ mAE ⊃ · · · ⊃ mMA E

must have the property that miAE = m

i+1A E for some i ≤ M − 1, which implies m

i+1A E = 0

by Nakayama lemma. Hence mMA E = 0, which together with (4.15) prove (4.14), and therefore

complete the proof of this proposition.

Next, we finish the proof of Step 3.

Proof of Step 3. Finally, we prove (4.9) and (4.10). Let OG2hF

be the ring of global sections of the

structure sheaf of G2hF , and let mG2h

Fbe the corresponding maximal ideal. Since

deg( GFπM

// GF ) = q2hM ,

one sees that SpecOG2hF/mq2hM

G2hK

is a closed subscheme of

ker

(G2hF

πM// G2hF

)= G2h

F [πM ],

and we denote the corresponding closed embedding by

jM : SpecOG2hF/mq2hM

G2hK

−→ G2hF [πM ].

31

Consider the following closed embeddings

(4.16) Spf OG2hF/mq2hM

G2hF

jM // G2hF [πM ]

iM∞ // G2h

F ,

(4.17) Spf OG2hF/mq2hM

G2hF

jM // G2hF [πM ]

iMn // Mn .

Let m be an integer such that m ≥ maxcond(τ1), cond(τ2), and denote by N1,m+n, N2,m+n the

corresponding πm+n-level Lubin Tate spaces associated to the pair (ϕ1, τ1) and (ϕ2, τ2). Put

Fn := ηn+mn (ϕ1, τ1)∗ON1,n+m ⊗Mn ηn+mn (ϕ2, τ2)∗ON2,n+m

and

F∞ := η∞+m∞ (ϕ1, τ1)∗OGh

K1⊗G2h

Fη∞+m∞ (ϕ2, τ2)∗OGh

K2.

Note that n ≥ 1, and therefore deg(GhKi

πm

−−→ GhKi

)= q2hm = deg

(Ni,n+m

// Ni,n

)for

i = 1, 2, which implies that

[F∞] = q4hmZ∞(ϕ1, τ1)⊗G2hFZ∞(ϕ2, τ2), [Fn] = q4hmZn(ϕ1, τ1)⊗Mn Zn(ϕ2, τ2).

Therefore (4.8) implies length(iMn∗Fn) = length(iM∞

∗F∞). It then suffices to prove iMn ∗

iMn∗Fn =

Fn and iM∞∗iM∞

∗F∞ = F∞. Let M be any integer such that

M >1

2hlogq length(F∞).(4.18)

Note that we have a canonical isomorphism

OG2hF/mq2hM

G2hF

∼= OG2hF [πM ]/m

q2hM

G2hF [πM ]

,

and that length(iMn∗Fn) = length(iM∞

∗F∞) < q2hM . Applying Lemma 4.6 to the map jM :

Spf OG2hF [πM ]/m

q2hM

G2hF [πM ]

→ G2hF [πM ], we have

jM∗j∗M

(iM∞

∗F∞

)= iM∞

∗F∞, jM∗j

∗M

(iMn

∗Fn)= iMn

∗Fn.

Applying Lemma 4.6 again to the map iM∞ jM : Spf OG2hF/mq2hM

G2hF→ G2h

F , and to the map iMn jM :

Spf OG2hF/mq2hM

G2hF→Mn, we have

(iM∞ jM)∗(iM∞ jM)∗F∞ = F∞, (iMn jM)∗(i

Mn jM)∗Fn = Fn.

Therefore iM∞∗iM∞

∗F∞ = F∞ and iMn ∗

iMn∗Fn = Fn as desired.

Proof of Theorem 4.1. The theorem follows from combining all steps. Note that Step 2 works for

N ≥ M + max(cond(τ1), cond(τ2)) + 1, and that Step 3 works for such M and m that satisfies

(4.18) and m ≥ max(cond(τ1), cond(τ2)). Therefore, any choice of N such that

(4.19) N ≥1

2hlogq length(Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2)) + 3max(cond(τ1), cond(τ2)) + 1

completes the proof of Theorem 4.1.

32

5. COMPUTATION OF INTERSECTION NUMBERS ON THE INFINITE LEVEL

In this section, we give an explicit formula for

(5.1) length(Z∞(ϕ1, τ1)⊗G2hFZ∞(ϕ2, τ2)).

We keep the same notation as in Section 4. We recall some elementary linear algebra facts for

formal modules, which will be used to obtain our main results in Theorem 5.12, which applies

generally whether K1∼= K2 or K1 6∼= K2. Then we simplify the result to Theorem 5.20 for the

case of K1∼= K2. Let K be either K1 or K2.

5.1. Some linear algebra for formal modules. We briefly review some elementary linear algebra

concepts that will be used in our calculation.

5.1.1. Formal matrix multiplication. For R = F,K,OF and OK , we fix canonical isomorphisms

Rn ∼= Matn×1(R) and Rn∨ ∼= Mat1×n(R).

The paring Rn∨ ⊗R Rn −→ R is given by the matrix multiplication in an obvious way.

Definition 5.1 (Formal vector). When R = F or K, for any formal OR-module G over Fq, and for

any A ∈ C⊗ Fq, we represent each element f ∈ Gn(A) as the n× 1 matrix

f =

f1...

fn

,

such that fi = f(ei) ∈ G(A) for any 1 ≤ i ≤ n, where ei ∈ O∨R is the 1×n matrix (0 · · · 1 · · · 0)

with i’th entry 1 and 0 elsewhere.

Definition 5.2 (Formal matrix multiplication). When R = F or K, for any formal OR-module G

over Fq, and any matrix M = (mij)1≤j≤b1≤i≤a ∈ Mata×b(End(G)), we define the formal left multipli-

cation map GbM // Ga , functorially in A ∈ C⊗ Fq, by

M : Gb(A) −→ Ga(A)

f1...

fb

7−→

[m11]G(f1)[+]G · · · [+]G[m1b]G(fb)

...

[ma1]G(f1)[+]G · · · [+]G[mab]G(fb)

·

Remark 5.3. Under the identification Gn(A) ∼= HomOR(On∨R ,G(A)) for any A ∈ C ⊗ Fq, one

sees that for any f ∈ Gn(A), after expressing f as an n × 1 matrix as in Definition 5.1, the map

f : On∨R → G(A) is obtained from the formal matrix multiplication v 7→ v · f for any v ∈ On∨R .

Let ϕ′ : G1 −→ G2 be a quasi-isogeny of two formal groups. By abuse of notation, we also use

the same symbol ϕ′ : Gn1 −→ Gn2 for the map obtained by applying ϕ′ componentwisely on Gn1 for

any n ≥ 0.

In this section, we work with two quadratic extensions K1/F and K2/F . Let GK1 = GF (resp.

GK2 = GF ) be the formal OF -module with an extra OK1- (resp. OK2) -action induced from a fixed

embedding K1 (resp. K2) ⊂ End(GF ) ⊗OFF . We fix these embeddings throughout the section.

Since GK1 = GK2 = GF as formal OF -modules, there are canonical isomorphisms

HomOF(GK1,GF )

∼= EndOF(GF ) ∼= HomOF

(GK2 ,GF ),33

which identifies the quasi-isogenies ϕ1 : G2hK1−→ G2h

F and ϕ2 : G2hK2−→ G2h

F with scalar matrices

ϕ1I2h, ϕ2I2h ∈ Mat2h×2h(DF ) through DF = EndOF(GF )⊗OF

F .

Let a, b be two non-negative integers. For any matrix M ∈ Mata×b(Ki), we denote by MGKi

its corresponding image in Mata×b(DF ) through the embedding Ki ⊂ DF for any i = 1, 2. Note

that even if K := K1∼= K2, the embeddings K1 ⊂ DF and K2 ⊂ DF may have different images,

where one might have MGK16= MGK2

for some M ∈ Mata×b(K). However, if M ∈ Mata×b(F ),then we view M as a matrix in Mata×b(DF ) naturally through the embedding of the center F →DF .

For ease of exposition, if a construction is made for both GK1 and GK2 , then we will omit sub-

scripts. The map η∞+m∞ (ϕ, τ) : GhK −→ G2h

F in Definition 3.1 can be represented as a matrix in

Mat2h×h(DF ), which we describe now explicitly.

Definition 5.4. Let Mτ ∈ Mat2h×h(K) be such that for any 1 × 2h matrix (a1 · · · a2h) ∈ F2h∨,

we have

τ∨(a1 · · · a2h) = (a1 · · · a2h)Mτ ∈ Kh∨.

Proposition 5.5. For any (ϕ, τ) ∈ Equih(K/F ), and any integerm ≥ cond(τ), the map η∞+m∞ (ϕ, τ) :

GhK −→ G2hF can be decomposed as described by the following commutative diagram

GhK

η∞+m∞ (ϕ,τ) &&

(Mτ )GK // G2hK

πmϕxx♣♣♣

♣♣♣♣♣♣♣♣♣♣

G2hF .

Proof. For any A ∈ C⊗ Fq, and any

f =

f1...

fh

∈ G

hK(A),

let f ′ = η∞+m∞ (ϕ, τ)f = πmϕ f τ∨. For any a = (a1 · · · a2h) ∈ O2h∨

F , we have

f ′(a) = πmϕ f τ∨(a)

= πmϕ f ((a1 · · · a2h)Mτ )

= πmϕ ((a1 · · · a2h)GK· (Mτ )GK

· f) .

Since πmϕ : GK −→ GF is an isogeny of formal OF -modules, we have πmϕ [x]GK= [x]GF

πmϕfor any x ∈ OF . Therefore the preceding results imply

f ′(a) = (a1 · · · a2h) · πmϕ ((Mτ )GK

· f) .

Hence

f ′ = πmϕ((Mτ )GK· f),

which completes the proof.

Remark 5.6. Using a similar method, for any (γ, g) ∈ Equi2h(F/F ), and any integer m ≥cond(g), the map β∞+m

∞ (γ, g) : G2hF −→ G2h

F can be decomposed as described by the following34

commutative diagram

G2hF

β∞+m∞ (γ,g) &&

g // G2hF

πmγxx♣♣♣

♣♣♣♣♣♣♣♣♣♣

G2hF .

Note that gMτ =Mgτ for any g ∈ GL2h(F ) and any τ∨ ∈ HomF (F2h∨, Kh∨).

5.1.2. Reduced norm and degree. Note that dimF (DF ) = 4h2. For any n ≥ 0, the algebra

Matn×n(DF ) is a (2h)2 × n2-dimensional F -central simple algebra, for which Matn×n(DF ) ⊗FF alg is isomorphic to Mat2hn×2hn(F

alg) over the algebraic closure F alg of F . The reduced norm

of an element g ∈ Matn×n(DF ) is simply its determinant as an element of Mat2hn×2hn(Falg). In

particular, in cases of n = 2h, n = h, and n = 1, we denote by

NRD(g), Nrd(g′), and nrd(g′′)

the reduced norms of elements g ∈ Mat2h×2h(DF ), g′ ∈ Math×h(DF ) and g′′ ∈ DF respectively.

We will show that the absolute value of the reduced norm for an invertible element is related to the

degree of the quasi-isogeny induced by it. From now on, we denote ODF:= EndOF

(GF ), which is

the maximal order of DF . We start with the following basic case.

Lemma 5.7. For any γ ∈ ODF∩D×

F , we have

deg

(GF

γ // GF

)= |nrd(γ)|−1

F .

Proof. Let GL be the formal OL-module over Fq obtained from GF by choosing a maximal subfield

L ⊂ DF such that γ ∈ L, where the tangent space Lie(GL) = Lie(GF ) is equipped with an

OL-algebra structure via the action of OL on GF. Note that the height of GL is 1. Let L be the

completion of the maximal unramified extension of L. By Lubin and Tate [Lub66], there is a

deformation GL of GL over OL with its Tate-module isomorphic to OL. It suffices to prove this

lemma for GL. Note that the degree of the induced map

deg

(GL

γ // GL

)

equals the number of elements in the cokernel of the induced map for their Tate modules

#coker

(OL

γ // OL

),

which equals |γ|−1L = |nrd(γ)|−1

F . This completes the proof of the Lemma.

More generally, we have the following.

Lemma 5.8. For any g ∈ Math×h(ODF), we have

deg

(GhF

g // GhF

)=∣∣Nrd(g)

∣∣−1

F.

35

Proof. By Cartan decomposition of matrix algebras over division algebras, there exists u1, u2 ∈GLh(ODF

) and a diagonal matrix t = diag(t1, t2, · · · , th) such that

g = u1tu2,

which decomposes the map g : GhF −→ GhF into the following homomorphisms

GhFu2 // GhF

t // GhFu1 // GhF .

This implies that the degree of g : GhF → GhF equals the degree of t : GhF → GhF since u1 : GhF → GhF

and u2 : GhF → GhF are isomorphisms. Finally, note that

deg(GhF

t // GhF

)=

h∏

i=1

deg

(GF

[ti]GF // GF

)=

h∏

i=1

∣∣nrd(ti)∣∣−1

F=∣∣Nrd(t)

∣∣−1

F=∣∣Nrd(g)

∣∣−1

F,

which completes the proof.

5.1.3. Associated quasi-isogeny. Let σ be the non-trivial element in Gal(K/F ), and let Mστ be

the matrix obtained by applying σ entrywise to Mτ . The following construction is the key to our

calculation.

Definition 5.9. We associate the pair (ϕ, τ) to the following matrix

(5.2) ∆(ϕ, τ) := ϕ ·(Mτ Mσ

τ

)GK∈ Mat2h×2h(DF ),

which represents the composition of the following quasi-homomorphisms

G2hK

(

Mτ Mστ

)

GK // G2hK

ϕ // G2hF .

Let |DiscK/F |F be the norm of the relative discriminant of K/F . We show that ∆(ϕ, τ) is in

fact a quasi-isogeny.

Lemma 5.10. For any (ϕ, τ) ∈ Equih(K/F ), we have

|NRD(∆(ϕ, τ))|F = |DiscK/F |h2

F 6= 0.

In particular, ∆(ϕ, τ) ∈ GL2h(DF ).

Proof. Let µ ∈ OK be a generator such that OK = OF [µ]. Let

M0 =

(IhµIh

).

Since right multiplication by M0 induces an isomorphism of OF -modules O2h∨F → Oh∨K , there

exsits g ∈ GL2h(F ) such that Mτ = gM0. Then we have

Vol(τ∨(O2h∨F )) = Vol(O2h∨

F Mτ ) = Vol(O2h∨F gM0) = Vol(O2h∨

F g).

Hence

| det(g)|F =Vol(O2h∨F g

)

Vol(O2h∨F

) =Vol(τ∨(O2h∨

F ))

Vol(O2h∨F

) = qHeight(τ),

which implies that |NRD(g)|F = | det(g)2h|F = q2hHeight(τ).36

On the other hand, note that the quasi-isogeny ϕ : GK → GF is of degree |nrd(ϕ)|−1F . By the

definition of Height(ϕ), we have

|nrd(ϕ)|F = q−Height(ϕ),

which implies |NRD(ϕI2h)|F = |nrd(ϕ)2h|F = q−2hHeight(ϕ). Finally,

|NRD((M0 Mσ0 )GK

) |F =

∣∣∣∣NmK/F

(detK

(Ih IhµIh µσIh

))∣∣∣∣h

F

= |DiscK/F |h2

F .

Note that ∆(ϕ, τ) = ϕ · g · (M0 Mσ0 )GK

, and Height(τ) = Height(ϕ). Combining the preceding

equations, we complete the proof of this lemma.

5.2. Decomposition of Z∞(ϕ, τ). We fix the following notation throughout the proof of Lemma

5.11 and Theorem 5.12. Abbreviate [OX ] to [X ] for any formal scheme X . For any τ∨ ∈HomF (F

2h∨, Kh∨) such that τ∨ is an isomorphism, consider the Iwasawa decomposition of the

matrix (Mτ Mστ ) ∈ GL2h(K)

(Mτ Mστ ) = Γτ ·

(Pτ ∗

Qτ

)∈ GL2h(K),

where Qτ , Pτ ∈ GL2h(K) and Γτ ∈ GL2h(OK). We also fix an integer m such that πmϕMτ ∈Mat2h×h(ODF

). Therefore πmϕPτ , πmϕQτ ∈ Math×h(ODF

)∩GLh(DF ), which implies that they

are corresponding to actual isogenies. Furthermore, let iϕ,τ and pϕ,τ be the maps defined by

iϕ,τ : GhF

ϕ·(Γτ )GK ·ϕ−1

Ih0h

// G2hF ,

and

pϕ,τ : G2hF

(

0h Ih)

ϕ·(Γ−1τ )

GK·ϕ−1

// GhF .

Note that iϕ,τ is a closed embedding, and that these maps depend on the choice of Γτ . The following

lemma gives a decomposition of the cycle Z∞(ϕ, τ).

Lemma 5.11. We have

(5.3) Z∞(ϕ, τ) = |Nrd (ϕ · (Pτ )GK)|−1F

[Im

(GhF

iϕ,τ // G2hF

)],

(5.4) Z∞(ϕ, τ) = |Nrd (ϕ · (Pτ )GK)|−1F

[Ker

(G2hF

pϕ,τ // GhF

)].

Proof. Since

πmϕMτ = πmϕ (Mτ Mστ )

(Ih0

)= πmϕΓτ

(Pτ ∗

Qτ

)(Ih0

)= ϕΓτϕ

−1 ·

(Ih0

)· πmϕPτ ,

the map η∞+m∞ (ϕ, τ) : GhK

πm·(Mτ )GK // G2hF can be decomposed into

(5.5) GhK

πmϕ·(Pτ )GK // GhF

Ih0

// G2hF

ϕ·(Γτ )GK ·ϕ−1

// G2hF .

37

Note that πmϕ·(Pτ)GK: GhK −→ GhF is an isogeny, that

(Ih0

): GhF −→ G2h

F is a closed embedding,

and that ϕ · (Γτ )GK· ϕ−1 : G2h

F −→ G2hF is an isomorphism. This implies that

(5.6) η∞+m∞ (ϕ, τ)

∗[GhK ] = deg

(GhK

πm·ϕ·(Pτ )GK // GhK

)[Im

(GhF

iϕ,τ // G2hF

)].

By Definition 3.10, we have

η∞+m∞ (ϕ, τ)

∗[GhK ] = deg

(GhK

πm// GhK

)Z∞(ϕ, τ).

Comparing this identity with (5.6), we complete the proof of (5.3). Moreover, considering the

following exact sequence

(5.7) 0 // GhF

Ih0

// G2hF

(

0 Ih)

// GhF// 0 ,

and twisting it by the isomorphism ϕ−1 · (Γτ )GK· ϕ : G2h

F −→ G2hF , we obtain an exact sequence

(5.8) 0 // GhF

ϕ−1·(Γτ )GK ·ϕ·

Ih0

// G2hF

(

0 Ih)

·ϕ−1·(Γ−1τ )GK ·ϕ

// GhF// 0 ,

which implies that

(5.9) Im

(GhF

iϕ,τ // G2hF

)= Ker

(G2hF

pϕ,τ // GhF

).

Comparing this identity with (5.3), we finish the proof of (5.4), and hence complete the proof of

the lemma.

5.3. Computation of the intersection number. In this part, we give a formula for the intersection

number χ(Z∞(ϕ1, τ1) ⊗L

G2hFZ∞(ϕ2, τ2)). We give the most general version of the intersection

formula by the following theorem.

Theorem 5.12. Suppose (ϕi, τi) ∈ Equih(Ki/F ) for i = 1, 2. If the following expression

(5.10) |DiscK1/F |−h2

F ·

∣∣∣∣Nrd((

0h Ih)·∆(ϕ1, τ1)

−1 ·∆(ϕ2, τ2) ·

(Ih0h

))∣∣∣∣−1

F

is finite, then it equals to the intersection number χ(Z∞(ϕ1, τ1)⊗L

G2hFZ∞(ϕ2, τ2)).

Proof. By (4.5), we have

χ(Z∞(ϕ1, τ1)⊗L

G2hFZ∞(ϕ2, τ2)) = length(Z∞(ϕ1, τ1)⊗G2h

FZ∞(ϕ2, τ2))

when the right hand side is finite, which, by Proposition 5.11, equals

(5.11)∣∣Nrd(ϕ1·(Pτ1)GK1

·ϕ2·(Pτ2)GK2)∣∣−1

Flength

([Im

(GhF

iϕ2,τ2 // G2hF

)×G2h

FKer

(G2hF

pϕ1,τ1 // GhF

)]).

38

The map iϕ2,τ2 : GhF → G2h

F induces a natrual map

Ker

(GhF

pϕ1,τ1iϕ2,τ2 // GhF

)−→ Im

(GhF

iϕ2,τ2 // G2hF

)×G2h

FKer

(G2hF

pϕ1,τ1 // GhF

),

which is an isomorphism since iϕ2,τ2 is a closed embedding. Therefore

(5.12)

length

([Im

(GhF

iϕ2,τ2 // G2hF

)×G2h

FKer

(G2hF

pϕ1,τ1 // GhF

)])= |Nrd(pϕ1,τ1 · iϕ2,τ2)|

−1F .

Note that

pϕ1,τ1 · iϕ2,τ2 =(0h Ih

)· ϕ1 ·

(Γ−1τ1

)GK1

· ϕ−11 · ϕ2 · (Γτ1)GK2

· ϕ−12 ·

(Ih0h

),

ϕ2 · (Γτ1)GK2· ϕ−1

2 ·

(Ih0h

)= ∆(ϕ2, τ2) ·

(Ih0h

)·(P−1τ2

)GK2

· ϕ−12 ,

and that(0h Ih

)· ϕ1 ·

(Γ−1τ1

)GK1

· ϕ−1 = ϕ1 · (Qτ1)GK1·(0h Ih

)·∆(ϕ1, τ1)

−1.

Therefore,

(5.13) pϕ1,τ1 · iϕ2,τ2 = ϕ1 · (Qτ1)GK1·(0h Ih

)·∆(ϕ1, τ1)

−1 ·∆(ϕ2, τ2) ·

(Ih0

)·(P−1τ2

)GK2

·ϕ−12 .

Combining the preceding equations (5.11),(5.12) and (5.13), the intersection number equals

(5.14)∣∣Nrd

(ϕ21(Pτ1)GK1

(Qτ1)GK1

)∣∣−1

F

∣∣∣Nrd((

0 Ih)·∆(ϕ1, τ1)

−1 ·∆(ϕ2, τ2) ·

(Ih0

)) ∣∣∣−1

F.

Note that

∣∣Nrd(ϕ21(Pτ1)GK1

(Qτ1)GK1

)∣∣F=

∣∣∣∣∣NRD((

ϕ1Ihϕ1Ih

)(Pτ1 ∗0 Qτ1

)

GK1

)∣∣∣∣∣F

.

Since | det Γτ1 |F = 1, the above equation can be simplifeid to

(5.15) |NRD(∆(ϕ1, τ1))|F = |DiscK/F |h2

F

by Lemma 5.10. Combining (5.14) and (5.15), we complete the proof of this theorem.

5.4. The invariant polynomial and the resultant formula. In this section, we restrict ourselves

to the case where the two fixed embeddings K1 ⊂ DF and K2 ⊂ DF are the same. In particular,

we have K1∼= K2 and GK1 = GK2 . We abbreviate them to K ⊂ DF and GK respectively.

We introduce the invariant polynomials to simplify the formula (5.10) further. We abbreviate

ϕ := ϕ1,τ := τ1, and note that there exists elements g ∈ GL2h(F ) and γ ∈ D×F describing the

relative position of ϕ1, ϕ2, τ1 and τ2 by

(5.16) ϕ2 = γ ϕ1, Mτ2 = gMτ1 ,

where in this case, we have

(5.17) ∆(ϕ1, τ1)−1 ·∆(ϕ2, τ2) = ∆(ϕ, τ)−1 · γ · g ·∆(ϕ, τ).

39

Therefore, to calculate χ(Z∞(ϕ, τ)⊗LG2hFZ∞(γϕ, gτ)), it suffices to simplify the following expres-

sion

(5.18)

∣∣∣∣Nrd((

0 Ih)·∆(ϕ, τ)−1 · γ · g ·∆(ϕ, τ) ·

(Ih0

))∣∣∣∣−1

F

.

Letting DK ⊂ DF be the centralizer of K, we note that ODK:= DK ∩ ODF

is the endomorphism

ring of GK as a formal OK-module. Note that the maps ϕ : GK → GF and τ∨ : F 2h∨ → Kh∨

induce F -algebra embeddings

Math×h(K)→ Mat2h×2h(F ) and DK → DF

by x 7→ g and y 7→ γ such that Mτx = gMτ and ϕ y = γ ϕ respectively. Denote their images

by Mat2h×2h(F )+ and DF+. Note that

Mat2h×2h(F )+ ∼= Math×h(K), DF+ = ϕ ·DK · ϕ−1.

Let K ′ ⊂ Mat2h×2h(F )+ be the center of Mat2h×2h(F )+. Obviously K ′ ∼= K. Define

Mat2h×2h(F )− := g ∈ Mat2h×2h(F ) : gk = kσg for any k ∈ K ′,

and

DF− := γ ∈ DF : γk = kσγ for any k ∈ ϕKϕ−1.

Then we have

(5.19) Mat2h×2h(F ) = Mat2h×2h(F )+ ⊕Mat2h×2h(F )−, DF = DF+ ⊕DF−

because they are the eigenspace decompositions of DF and Mat2h×2h(F ) as left K-linear spaces

under the right K-multiplications. With respect to this decomposition, for any γ ∈ DF and any

g ∈ Mat2h×2h(F ), we write

γ = γ+ + γ−, g = g+ + g−.

When γ (resp. g) is invertible, conjugating it by a trace 0 element µ ∈ ϕ−1 ·K× · ϕ ⊂ D×F (resp.

µ ∈ K ′× ⊂ GL2h(F )), we have

(5.20) µ(γ+ + γ−)µ−1 = (γ+ − γ−), resp. µ(g+ + g−)µ

−1 = (g+ − g−),

which implies that γ+ − γ−(resp. g+ − g−) is also invertible.

Definition 5.13. For any γ ∈ D×F and g ∈ GL2h(F ), we define

γ′ϕ := (γ+ − γ−)−1γ+(γ+ + γ−)

−1γ+ ∈ DF ,

g′τ := (g+ − g−)−1g+(g+ + g−)

−1g+ ∈ Mat2h×2h(F ).(5.21)

Remark 5.14. When γ+ (resp. g+) is invertible, we can write

γ′ϕ = (γ+ − γ−)−1γ+(γ+ + γ−)

−1γ+ = (1− γ−1+ γ−γ

−1+ γ−)

−1,

g′τ = (g+ − g−)−1g+(g+ + g−)

−1g+ = (1− g−1+ g−g

−1+ g−)

−1.

Lemma 5.15. We have γ′ϕ ∈ DF+ = ϕ ·DK · ϕ−1 and g′τ ∈ Mat2h×2h(F )+.40

Proof. Note that the assignment g 7→ g′ϕ induces a morphism GL2h(F ) → Mat2h×2h(F ) of al-

gebraic varieties over F . Since Mat2h×2h(F )+ is a Zariski closed subset, it suffices to prove this

lemma for the following Zariski-dense subset

g ∈ GL2h(F ) : g+ ∈ GLh(K).

By Remark 5.14, we can write g′τ = (1− (g−1+ g−)

2)−1. Note that for any k ∈ K ′, we have

g−1+ g−g

−1+ g−k = g−1

+ g−kσg−1

+ g− = kg−1+ g−g

−1+ g−,

which implies (g−1+ g−)

2 ∈ Mat2h×2h(F )+, and therefore g′τ ∈ Mat2h×2h(F )+. A similar argument

shows that γ′ϕ ∈ DF+, which completes the proof.

Definition 5.16. We define the invariant polynomial of γ ∈ D×F (resp. g ∈ GL2h(F )) with respect

to ϕ (resp. τ ) as the K-coefficient reduced characteristic polynomial of γ′ϕ as an element in the

K-central simple algebra DF+ (resp. Mat2h×2h(F )+). We denote this polynomial by P ϕγ (resp.

P τg ). We also abbreviate them to Pγ and Pg if the choice of ϕ and τ is clear from the context.

From the definition, we know that these polynomials are of degree h.

Corollary 5.17. For any x ∈ F×, we have Pxγ = Pγ and Pxg = Pg for any γ ∈ D×F and

g ∈ GL2h(F ).

Proof. This is obvious since (xγ)+ = xγ+, (xγ)− = xγ−, (xg)+ = xg+ and (xg)− = xg−.

Proposition 5.18. The coefficients of the polynomials P ϕγ and P τ

g are in F .

Proof. It suffices to prove this proposition for a Zariski-dense subset consisting of γ ∈ D×F such

that γ+, γ− ∈ D×F . Let p(x) = xn +

∑n−1i=0 aix

i be the minimal polynomial of γ′ϕ with coefficients

ai ∈ ϕKϕ−1 ⊂ DF . Since γ−1

+ γ−ai = aσi γ−1+ γ−, and γ−1

+ γ− commutes with γ′ϕ, we have pσ(γ′ϕ) =

γ−1+ γ−p(γ

′ϕ)γ

−1− γ+ = 0, which implies that pσ(x) is also the minimal polynomial of γ′ϕ. Therefore

p(x) has coefficients in F , which implies P ϕγ (x) ∈ F [x]. A similar method will show that P τ

g (x) ∈F [x]. We complete the proof.

Definition 5.19. For any γ ∈ DF and g ∈ GL2h(F ), we define the relative resultant

(5.22) Res(P ϕγ , P

τg ) :=

∏

1≤i,j≤h

(xi − yj)

as the resultant of polynomials P ϕγ and P τ

g . Here xi and yi are roots of P ϕγ and P τ

g for 1 ≤ i ≤ h,

with multiplicities counted.

Using (5.18), we will prove the following theorem.

Theorem 5.20. Suppose (ϕ, τ) ∈ Equih(K/F ) and (γ, g) ∈ Equi2h(F/F ). We have

χ(Z∞(ϕ, τ)⊗LG2hFZ∞(γϕ, gτ)) = |DiscK/F |

−h2

F ·∣∣Res(P ϕ

γ , Pτg )∣∣−1

F

when the right hand side is finite.

This theorem is a direct consequence of the following proposition.

Proposition 5.21. For any γ ∈ D×F , g ∈ GL2h(F ) and (ϕ, τ) ∈ Equih(K/F ), we have

∣∣Nrd((

0 Ih)·∆(ϕ, τ)−1 · γ · g · ϕ ·Mτ

)∣∣−1

F=∣∣Res(P ϕ

γ , Pτg )∣∣−1

F| det g · nrd(γ)|−hF .

41

Proof. It suffices to prove this proposition for those γ such that γ+ is invertible. Since

(Ih 0h

)·∆(ϕ, τ)−1 ·∆(ϕ, τ) ·

(Ih0h

)=(0h Ih

)·∆(ϕ, τ)−1 ·∆(ϕ, τ) ·

(0hIh

)= Ih,

(0h Ih

)·∆(ϕ, τ)−1 ·∆(ϕ, τ) ·

(Ih0h

)=(Ih 0h

)·∆(ϕ, τ)−1 ·∆(ϕ, τ) ·

(0hIh

)= 0h,

and ∆(ϕ, τ) = ϕ · (Mτ Mστ ), we have

(5.23)(Ih 0h

)·∆(ϕ, τ)−1 · ϕ ·Mτ =

(0h Ih

)·∆(ϕ, τ)−1 · ϕ ·Mσ

τ = Ih,

(5.24)(Ih 0h

)·∆(ϕ, τ)−1 · ϕ ·Mσ

τ =(0h Ih

)·∆(ϕ, τ)−1 · ϕ ·Mτ = 0h.

Let

X :=(0 Ih

)·∆(ϕ, τ)−1 · γ · g · ϕ ·Mτ .

It suffices to prove Nrd (X)4 = nrd(γ)4h det(g)4hResϕ,τ(γ, g)4 for those γ, g such that γ+, g+ are

invertible. By constructions of g+ and g−, there exists h× h matrices x+, x− ∈ Math×h(K) such

that

(5.25) g+Mτ =Mτx+, g−Mτ =Mστ x−.

Similarly for γ+ and γ−, we have

(5.26) (ϕ−1 · γ+ · ϕ) ·Mτ =Mτ · (ϕ−1 · γ+ · ϕ), (ϕ−1 · γ− · ϕ) ·Mτ =Mσ

τ · (ϕ−1 · γ− · ϕ).

Using preceding equations (5.24),(5.25),(5.26), we have

(5.27)(0h Ih

)·∆(ϕ, τ)−1 · γ+ · ϕ · g+ ·Mτ = 0h,

(5.28)(0h Ih

)·∆(ϕ, τ)−1 · γ− · ϕ · g− ·Mτ = 0h,

(5.29)(Ih 0h

)·∆(ϕ, τ)−1 · γ+ · ϕ · g− ·Mτ = 0h,

(5.30)(Ih 0h

)·∆(ϕ, τ)−1 · γ− · ϕ · g+ ·Mτ = 0h.

Note that γ = γ+ + γ−, that g = g+ + g−, and that g+, g− commute with γ+, γ− and ϕ. Using

equations (5.27) and (5.28), we have

(5.31) X =(0h Ih

)·∆(ϕ, τ)−1 · (γ+ · g− + γ− · g+) · ϕ ·Mτ .

By (5.26), we have (ϕ−1 · γ− · γ−1+ ϕ) ·Mτ =Mσ

τ (ϕ−1 · γ− · γ

−1+ ϕ), which implies

γ− · γ−1+ ·∆(ϕ, τ) = ∆(ϕ, τ)

(Ih

Ih

)· (ϕ−1 · γ− · γ

−1+ · ϕ).

Letting

X ′ := (ϕ−1 · γ− · γ−1+ · ϕ)

−1 ·X · (ϕ−1 · γ−1+ · γ− · ϕ),

we have

(5.32) X ′ =(Ih 0h

)·∆(ϕ, τ)−1 · (γ+ · g− + γ− · g+) · ϕ ·M

στ .

Combining (5.31) and (5.32), we have

∆(ϕ, τ)−1 · (γ+ · g− + γ− · g+) ·∆(ϕ, τ) =

(0h X ′

X 0h

).

42

Since Nrd(X) = Nrd(X ′), we have

(−1)hNrd(X)2 = NRD

(0h X ′

X 0h

)= NRD(γ+g− + γ−g+) = NRD(γ+(γ

−1+ γ− + g−g

−1+ )g+).

Letting µ ∈ ϕ ·K× · ϕ−1 be such that µσ = −µ, we have

µ(γ−1+ γ− + g−g

−1+ )µ−1 = −γ−1

+ γ− + g−g−1+ ,

which implies NRD(γ−1+ γ− + g−g

−1+ ) = NRD(−γ−1

+ γ− + g−g−1+ ). Therefore,

(5.33)

Nrd(X)4 = NRD(γ2+g2+)NRD((g−g

−1+ )2 − (γ−1

+ γ−)2) = NRD(γ2+g

2+)NRD((γ

′ϕ)

−1 − (g′τ )−1).

By (5.20), we have NRD(g+ + g−) = NRD(g+ − g−) and NRD(γ+ + γ−) = NRD(γ+ − γ−),which implies

(5.34) NRD(γ′ϕ) = NRD(γ2+)NRD(γ−2), NRD(g′τ ) = NRD(g2+)NRD(g

−2).

Note that, as elements of K-central simple algebras, the characteristic polynomials of γ′ϕ ∈ DF+

and g′τ ∈ Mat2h×2h(F )+ are given by P ϕγ and P τ

g . Therefore, as elements inDF and Mat2h×2h(F ),

their characteristic polynomials are (P ϕγ )

2 and (P τg )

2. Let L ⊂ DF+ be a maximal subfield con-

taining γ′ϕ and ϕ ·K · ϕ−1. Since γ′ϕ commutes with g′τ , we have

γ′ϕ − g′τ ∈ Mat2h×2h(L) ⊂ Mat2h×2h(DF ).

Therefore detL(γ′ϕ − g

′τ ) = (P τ

g (γ′))2, and we have

(5.35) NRD(γ′ϕ − g′τ ) = NmL/F (P

τg (γ

′))2 = Res((P τg )

2, (P ϕγ )

2) = Res(P τg , P

ϕγ )

4.

Combining preceding equations (5.33),(5.34),(5.35), we have

Nrd(X)4 = NRD(γ2+g2+)NRD((γ

′ϕ)

−1(g′τ )−1)NRD(γ′ϕ − g

′τ ) = NRD(γ2 · g2)Res(P τ

g , Pϕγ )

4.

Noting that NRD(γ) = nrd(γ)2h and NRD(g) = det(g)2h, we complete the proof.

Proof of the Theorem 5.20. By (5.17) and Theorem 5.12, we know that

χ(Z∞(ϕ, τ)⊗LG2hFZ∞(γ ·ϕ, g ·τ)) = |DiscK/F |

−h2

F

∣∣Nrd((

0 Ih)·∆(ϕ, τ)−1 · γ · g · ϕ ·Mτ

)∣∣−1

F.

Since (γ, g) is an equi-height pair, we have | det(g)nrd(γ)|F = 1. the theorem follows from the

Proposition 5.21.

Finally we prove the following result, which will be used in the rest of the article.

Proposition 5.22. For any γ ∈ D×F and g ∈ GL2h(F ), if P ϕ

γ is an irreducible polynomial and

P ϕγ (0)P

ϕγ (1) 6= 0, then Resϕ,τ (γ, g) 6= 0.

Proof. Since P ϕγ and P τ

g are monic polynomials of degree h, it suffices to show that P ϕγ 6= P τ

g .

Suppose for the sake of contradiction that P ϕγ = P τ

g . Since these polynomials are irreducible and

P ϕγ (0) 6= 0, we have nrd(γ′ϕ) 6= 0 and det(g′τ ) 6= 0, which implies that γ+ and g+ are invertible.

In this case, by Remark 5.14, the characteristic polynomial of (γ−1+ γ−)

2 and (g−1+ g−)

2 will be

the same. Since P ϕγ is irreducible and P ϕ

γ (1) 6= 0, we have γ−1+ γ− 6= 0. Note that there exists

matrices x+, x− ∈ Math×h(K) such that g+Mτ = Mτx+ and g−Mτ = Mστ x−. This implies

that (g−1+ g−)

2Mτ = Mτxxσ where x = x−1

+ x−. So P τg is the characteristic polynomial of xxσ .

Let L = F [(γ−1+ γ−)

2], and let DL ⊂ DF be the centralizer of L. Since [L : F ] = h, one sees

that DL is a quaternion algebra over L. For any k ∈ K, since both γ−1+ γ− and k commute with

43

(γ−1+ γ−)

2, we have γ−1+ γ− ∈ DL and K ⊂ DL. Let λ = (γ−1

+ γ−)2. Since λ can be identified with

an eigenvalue of xxσ , there exists y ∈ LK such that λ = yyσL where σL is the non-trivial element

in Gal(LK/L). Since γ−1+ γ−y = yσLγ−1

+ γ−, we have

0 = (γ−1+ γ−)

2 − yyσL = (γ−1+ γ− + y)(γ−1

+ γ− − yσL),

which implies either γ−1+ γ− = −y or γ−1

+ γ− = yσL. This contradicts to the assumption that γ−1+ γ−

does not commute with some elements in KL.

6. PROOF OF MAIN THEOREMS

Combining Theorems 4.1 and 5.12 in previous sections, we have essentially calculated the in-

tersection number χ(Zn(ϕ1, τ1)⊗LMnZn(ϕ2, τ2)) for sufficiently large n, where the minimal choice

of n depends on the pairs ϕ1, ϕ2 and τ1, τ2. This illustrates the fact that the intersection number

induced by two fixed pairs will eventually be stable as the level increases. In this section, we elim-

inate the restriction on n by proving a formula relating intersection numbers on higher and lower

levels. Then we combine all formulas together to give a proof of the main theorems.

6.1. Intersection number formula for low levels. Let

Rn := ker

(GL2h(OF )

g 7→g // GL2h(OF/πn)

).

That is, R0 = GL2h(OF ), and Rn = I2h + πnMat2h×2h(OF ) for any n ≥ 1. By associating each

g ∈ GL2h(OF ) to the morphism

βnn(id, g) : Mn −→Mn

as defined in (2.16), the group GL2h(OF ) naturally acts on Mn for any n ≥ 0. This action is the

natural action of GL2h(OF ) on πn-level structures of Mn. Note that Rn ⊂ GL2h(OF ) is the maxi-

mal subgroup that acts trivially on Mn, and that the quotient group Rn/Rn+m is the automorphism

group of Mn+m as a formal scheme over Mn via the transition map βn+mn : Mn+m −→ Mn for

any n,m ≥ 0. We put additional subscripts and write N1,n, N2,n for πn-level Lubin–Tate spaces

associated to GK1 and GK2 respectively. Moreover, for each i = 1, 2, n ≥ 0, and m ≥ 0, we simply

denote by Ni,n+m → Ni,n the corresponding transition map.

We state the key theorem in this subsection as follows. The reader willing to accept this theorem

may skip this subsection.

Theorem 6.1. Suppose (ϕi, τi) ∈ Equih(Ki/F ) for i = 1, 2. For any n,m ≥ 0, we have

(6.1) χ(Zn(ϕ1, τ1)⊗LMn

Zn(ϕ2, τ2)) =

∑k∈Rn/Rn+m

χ(Zn+m(ϕ1, τ1)⊗LMn+m

Zn+m(ϕ2, kτ2))

deg(N1,m+n → N1,n)deg(N2,m+n → N2,n).

Intuitively, up to some multiplicities, the cycles Zn+m(ϕ2, kτ2) run over all possible cycles that

map to Zn(ϕ2, τ2) under the transition map Mn+m →Mn when k runs through Rn/Rn+m. Before

proceeding to the proof, we fix some notations. Let w1,w2 be integers such that w1 ≥ cond(τ1)and w2 ≥ cond(τ2). Let

(6.2) F1 := ηm+n+w1m+n (ϕ1, τ1)∗ON1,m+n+w1

, F2 := ηm+n+w2m+n (ϕ2, τ2)∗ON2,m+n+w2

be coherent sheaves of OMm+n-modules, and let βn+mn : Mm+n −→ Mn be the transition map.

Note that by Definition 2.16, we have

[Fi] = deg (Ni,m+n+wi→ Ni,m+n)Zm+n(ϕi, τi)44

for i = 1, 2. We use the following two lemmas to write both sides of the equation (6.1) in terms of

F1 and F2.

Lemma 6.2. For the left hand side of (6.1), we have

(6.3) χ(Zn(ϕ1, τ1)⊗LMn

Zn(ϕ2, τ2)) =χ(F1 ⊗

LMm+n

βn+mn∗βn+mn ∗

F2)

deg(N1,m+n+w1 → N1,n) deg(N2,m+n+w2 → N2,n).

Proof. Using the projection formula, we have

χ(F1 ⊗LMm+n

βn+mn∗βn+mn ∗

F2) = χ(βn+mn ∗F1 ⊗

LMn

βn+mn ∗F2).

Note that βn+mn ∗Zm+n(ϕi, τi) = deg (Ni,m+n → Ni,n)Zn(ϕi, τi) for i = 1, 2, which implies that[βn+mn ∗

Fi]= deg(Ni,m+n+wi

→ Ni,n)Zn(ϕi, τi) for i = 1, 2.

Combining all preceding equations together, we complete the proof of the lemma.

Lemma 6.3. For each summand of the right hand side of (6.1), we have

(6.4)

χ(Zn+m(ϕ1, τ1)⊗LMn+m

Zn+m(ϕ2, kτ2)) =χ(F1 ⊗Mm+n k

−1∗F2)

deg(N1,m+n+w1 → N1,m+n) deg(N2,m+n+w2 → N2,m+n).

Proof. This lemma follows directly from the observation that

[k−1∗Fi] = [k∗Fi] = deg(Ni,m+n+wi→ Ni,m+n)Zn+m(ϕi, kτi)

for i = 1, 2 and any k ∈ GL2h(OF ).

Combining (6.3) and (6.4) together, it suffices to prove that

(6.5) χ(F1 ⊗LMn+m

βn+mn∗βn+mn ∗

F2) = χ

F1 ⊗

LMm+n

⊕

k∈Rn/Rn+m

k∗F2

.

Therefore, it is important to compare βn+mn∗βn+mn ∗

F2 with ⊕k∈Rn/Rn+mk∗F2. The following

lemma is useful to construct a comparison morphism between them.

Lemma 6.4. There is an injective canonical morphism of sheaves of OMm+n-modules

(6.6) βn+mn∗βn+mn ∗

OMm+n//⊕

k∈Rn/Rn+mk∗OMm+n

such that its induced map

(6.7) βn+mn∗βn+mn ∗

OMm+n

[1π

] ∼= //⊕

k∈Rn/Rn+mk∗OMm+n

[1π

]

is an isomorphism.

Proof. Since Mm+n is affine, it suffices to describe a coherent sheaf F by taking its global sections.

In this proof, by abuse of notation, we will use the same symbol F to denote the set of global

sections F(Mm+n). For any s ∈ OMm+n , we denote k∗s by sk. Note that by definition, we have the

following isomorphisms of OMm+n-modules

βn+mn∗βn+mn ∗

F2∼= OMm+n ⊗Mn F2, k∗F2

∼= OMm+n ⊗Mm+n,k F2.45

Here the twisted tensor product OMm+n ⊗Mm+n,k F2 is defined so that ak ⊗ v = 1 ⊗ av for all

a ∈ OMm+n and v ∈ F2. The assignment a⊗ b 7→ a⊗ b defines a canonical morphism

OMm+n ⊗Mn OMm+n −→ OMm+n ⊗Mm+n,k OMm+n

of sheaves of OMm+n-modules for any k ∈ Rn/Rn+m, which gives rise to a canonical morphism

βn+mn∗βn+mn ∗

OMm+n//⊕

k∈Rn/Rn+mk∗OMm+n

of sheaves of OMm+n-modules as desired. By inverting π, we obtain the rigid generic fiber Mn of

Mn for any n ≥ 0. It is well known that the transition maps Mn+m −→ Mn

of the rigid generic

fibers of Lubin–Tate spaces are etale for any n,m ≥ 0. Since Rn/Rn+m acts on the etale cover

Mn+m −→Mn

faithfully, and

#(Rn/Rn+m) = deg(Mn+m −→Mn

),

the induced map (6.7) is an isomorphism. Since Mm+n is flat over Spf OF , the natural map

βn+mn∗βn+mn ∗

OMm+n −→ βn+mn∗βn+mn ∗

OMm+n

[1

π

]

is injective, which implies that the map (6.6) is also injective.

In the following two lemmas, we review some sufficient conditions for the vanishing of Euler

characteristics.

Lemma 6.5 (Serre’s multiplicity vanishing theorem). Let R be a regular local ring, and let p, q be

primes of R such that R/p⊗R R/q is an Artinian ring. If dim(R/p) + dim(R/q) < dim(R), then

χ(R/p⊗LR R/q) = 0.

Proof. This was proven in 1985 by Paul C. Roberts [Rob85].

Lemma 6.6. LetM,N be finite modules over a regular Noetherian local ringA such thatM⊗ANis of finite length. If dim(Supp(M)) + dim(Supp(N)) < dim(A), then

χ(M ⊗LA N) = 0.

Proof. Consider a filtration

0 =Mn ⊂ · · · ⊂M0 =M

such that Mi/Mi+1∼= A/pi for some associated primes pi ∈ Ass(M) for any 0 ≤ i < n. Similarly

for N , let

0 = Nr ⊂ · · · ⊂ N0 = N

be a filtration such that Nj/Nj+1∼= A/qj for some qj ∈ Ass(N) for any 0 ≤ j < r. Then,

χ(M ⊗LA N) =

∑

1≤i<n1≤j<r

χ(A/pi ⊗LA A/qj).

However, the fact that

dim(A/pi) + dim(A/qj) ≤ dim(Supp(M)) + dim(Supp(N)) < dim(A)

implies that each summand χ(A/pi ⊗LA A/qj) = 0 for any 1 ≤ i < n, 1 ≤ j < r by Serre’s

multiplicity vanishing theorem. Hence χ(M ⊗LA N) = 0 as desired.

46

Combining all previous lemmas, we prove the following proposition, which implies (6.8) and

therefore proves Theorem 6.1. Actually, Proposition 6.7 below proves something stronger than

(6.8); this is necessary for our application in the next subsection.

Proposition 6.7. Suppose (ϕ2, τ2) ∈ Equih(K2/F ). For any integer m,n ≥ 0, and any coherent

sheaf F of OMm+n-modules such that dim(Supp(F)) = h, we have

(6.8) χ(F ⊗LMn+m

βn+mn∗βn+mn ∗

Zn+m(ϕ2, τ2)) =∑

k∈Rn/Rn+m

χ(F ⊗L

Mm+nZn+m(ϕ2, kτ2)

).

Proof. Recall F2 as defined in (6.2). It suffices to prove that

χ(F ⊗LMn+m

βn+mn∗βn+mn ∗

F2) = χ

F ⊗L

Mm+n

⊕

k∈Rn/Rn+m

k∗F2

.

Let J be the cokernel of the map in (6.6). By Lemma 6.4, there is an exact sequence

(6.9) 0 // βn+mn∗βn+mn ∗

OMm+n//⊕

k∈Rn/Rn+mk∗OMm+n

// J // 0 .

Since k∗OMm+n is free, we have Tor1Mm+n(k∗OMm+n ,F2) = 0. Tensoring the sequence (6.9) by F2,

we have an exact sequence

0 // Tor1Mm+n(J,F2) // βn+mn

∗βn+mn ∗

F2//⊕

k∈Rn/Rn+mk∗F2

// J⊗Mm+n F2// 0,

where Tor1Mm+n(J,F2) is the sheaf of OMm+n-modules associated to Tor1Mm+n

(J,F2). Then it

suffices to prove that

χ(F ⊗L

Mm+n(J⊗Mm+n F2)

)= 0; χ

(F ⊗L

Mm+nTor1Mm+n

(J,F2))= 0.

By Lemma 6.4, tensoring the sequence (6.9) by OMm+n

[1π

], we obtain an isomorphism

βn+mn∗βn+mn ∗

OMm+n

[ 1π

]−→

⊕

k∈Rn/Rn+m

k∗OMm+n

[ 1π

],

which implies that

(6.10) J⊗Mm+n OMm+n

[ 1π

]= 0.

Let V (π) ⊂ Mm+n be the vanishing locus of π. Therefore, we have Supp(J) ⊂ V (π), which

implies that

Supp(J⊗Mm+n F2) ⊂ V (π) ∩ Supp(F2),

and that

Supp(Tor1Mm+n(J,F2)) ⊂ V (π) ∩ Supp(F2).

It then suffices to show that for any coherent sheaf M of OMm+n-modules with Supp(M ) ⊂V (π) ∩ Supp(F2), one has

χ(F ⊗LMm+n

M ) = 0.

Indeed, since π is not a zero-divisor for F2, we have

dim(V (π) ∩ Supp(F2)) ≤ dim(Supp(F2))− 1 = h− 1.47

However, by what given, we have

dim(Supp(F)) = h,

which implies that

dim(V (π) ∩ Supp(F2)) + dim(Supp(F)) ≤ 2h− 1 < dim(Mm+n).

Hence χ(F ⊗LMm+n

M ) = 0 by Lemma 6.6, which completes the proof.

Proof of Theorem 6.1. By Lemma 6.2 and Lemma 6.3, it suffices to prove (6.5), which is a special

case when F = F1 in Proposition 6.7.

6.2. A generalization of the intersection number formula for correspondences. Note that the

groupD×F ×GL2h(OF/π

n) acts on Mn∼, and therefore also acts on K (Mn

∼)⊗ZQ. Here the group

K (Mn∼)⊗Z Q is defined in §2.5. In fact, we can consider a more general action on K (Mn

∼)⊗Z

Q. Let H (Rn\GL2h(F )/Rn) be the set of compactly supported double Rn-invariant Q-valued

functions. For some particular elements

(γ, f) ∈ D×F ×H (Rn\GL2h(F )/Rn),

we define their actions on K (Mn∼) ⊗Z Q via certain correspondences. Our main result in this

subsection is Theorem 6.8, which is a key theorem to generalize our intersection formula for those

correspondences.

Let g0 ∈ GL2h(F ), and let n,m be arbitrary non-negative integers such that m ≥ cond(g0).Consider a function

(6.11) fRng0Rn,m(x) :=

C x ∈ Rng0Rn;

0 x /∈ Rng0R0,

where C = deg(Mm+n −→Mn)/Vol(Rn · g0 · Rn) is the constant such that∫

GL2h(F )

fRng0Rn,m(x)dx = deg(Mm+n −→Mn).

Let γ ∈ D×F be an element such that Height(γ) = Height(g0). To the pair (γ, fRng0Rn,m) we attach

the following correspondence

βnn(γ, fRng0Rn,m) : Mn Mn+mβn+mnoo

βn+mn (γ,g0) // Mn .

Recall that two correspondences X ← Z → Y and X ← Z ′ → Y are the same if and only if there

exists an isomorphism Z → Z ′ such that the following diagram commutes

Z //

X

Y Z ′.oo

OO

Therefore replacing g0 in the preceding definition by any element in Rng0Rn gives the same cor-

respondence, which implies that our notation βnn(γ, fRng0Rn,m) is well-defined.

For any coherent sheaf of OMn-modules F on Mn, the definition

(6.12) βnn(γ, fRng0Rn,m)∗F := βn+mn ∗

βn+mn (γ, g0)∗F

also defines an action of the pair (γ, fRng0Rn,m) on K (Mn) ⊗Z Q. Note that when (γ, g0) is not

an equi-height pair, the same construction gives the action of (γ, fRng0Rn,m) on K (Mn∼) ⊗Z Q,

48

but in that case the subset K (Mn) ⊗Z Q ⊂ K (Mn∼) ⊗Z Q is not invariant. Our main result of

this subsection is the following theorem. The reader willing to accept this theorem may skip this

subsection.

Theorem 6.8. Suppose (ϕi, τi) ∈ Equih(Ki/F ) for i = 1, 2 and (γ, g0) ∈ Equi2h(F/F ). Let

f = fRng0Rn,m. We have

χ(Zn(ϕ1, τ1)⊗

LMn

βnn(γ, f)∗Zn(ϕ2, τ2)

)=

∑

x∈Rn/Rn+m

χ(Zn(γϕ1, g0xτ1)⊗

LMn

Zn(ϕ2, τ2)).

Proof. By definition in (6.12), we have

χ(Zn(ϕ1, τ1)⊗

LMn

βnn(γ, f)∗Zn(ϕ2, τ2)

)= χ

(Zn(ϕ1, τ1)⊗

LMn

βn+mn ∗βn+mn (γ, g0)

∗Zn(ϕ2, τ2)),

which, by using the projection formula, equals

χ(βn+mn

∗Zn(ϕ1, τ1)⊗

LMn+m

βn+mn (γ, g0)∗Zn(ϕ2, τ2)

).

Note that deg(N1,m+n

// N1,n

)Zn(ϕ1, τ1) = βn+mn ∗

Zn+m(ϕ1, τ1). We have

deg(N1,m+n

// N1,n

)χ(βn+mn

∗Zn(ϕ1, τ1)⊗

LMn+m

βn+mn (γ, g0)∗Zn(ϕ2, τ2)

)

= χ(βn+mn

∗βn+mn ∗

Zn+m(ϕ1, τ1)⊗LMn+m

βn+mn (γ, g0)∗Zn(ϕ2, τ2)

),

which, by Proposition 6.7, equals

(6.13)∑

x∈Rn/Rn+m

χ(Zn+m(ϕ1, xτ1)⊗LMm+n

βn+mn (γ, g0)∗Zn(ϕ2, τ2)).

Using the projection formula again, the preceding equation can be simplified to

(6.14)∑

x∈Rn/Rn+m

χ(βn+mn (γ, g0)∗Zn+m(ϕ1, xτ1)⊗

LMn

Zn(ϕ2, τ2)).

Note that, by Proposition 2.22, we have

βn+mn (γ, g0)∗Zn+m(ϕ1, xτ1) = deg(N1,m+n

// N1,n

)Zn(γϕ1, g0xτ1).

Plugging this equation back to (6.14), we complete the proof of this theorem.

6.3. Proof of the main theorem. In this subsection, we combine all results of previous sec-

tions together to prove the main theorem. Fix three equi-height pairs (ϕ1, τ1) ∈ Equih(K1/F ),(ϕ2, τ2) ∈ Equih(K2/F ) and (g0, γ) ∈ Equi2h(F/F ). Recall the definition of fRng0Rn,m in (6.11),

and the definition of ∆(ϕi, τi) ∈ Mat2h×2h(DF ) in (5.2) for i = 1, 2. We denote

R(g) := Nrd

((0h Ih

)·∆(ϕ1, τ1)

−1 · γ−1 · g−1 ·∆(ϕ2, τ2) ·

(Ih0h

))∈ F.

Recall that |DiscK1/F |−h2

F |R(g)|−1F is the intersection number χ(Z∞(γ ·ϕ1, g · τ1)⊗

LG2hFZ∞(ϕ2, τ2))

by Theorem 5.12. Note that our intersection formula in Theorem 5.12 only holds for

R(g) 6= 0.

Intuitively, this condition is describing that the cycle Z∞(γ · ϕ1, g · τ1) intersects properly with

the cycle Z∞(ϕ2, τ2).When K1∼= K2, the non-degeneracy condition holds automatically for any

g ∈ GL2h(F ) when P ϕγ is irreducible and P ϕ

γ (0)Pϕγ (1) 6= 0 by Proposition 5.22. When K1 6∼= K2,

49

there is a similar criterion based on constructions of invariant polynomials for bi-quadratic cases,

which was discussed in detail in [HL20]. When L = F , K1, or K2, we define their local zeta

functions by ζL(x) := (1− q−xL )−1, where qL is the cardinality of the residue field of OL. We state

our main theorem as follows.

Theorem 6.9. Assume the non-degeneracy condition

(6.15) R(g) 6= 0

for any g ∈ Rng0Rn. Suppose (ϕi, τi) ∈ Equih(Ki/F ) for i = 1, 2, (γ, g0) ∈ Equi2h(F/F ),n ≥ 0 and m ≥ cond(g0). Let f := fRng0Rn,m. Then the intersection number

χ(Zn(ϕ1, τ1)⊗

LMn

βnn(γ, f)∗Zn(ϕ2, τ2)

)

is finite and we have the following formula

χ(Zn(ϕ1, τ1)⊗

LMn

βnn(γ, f)∗Zn(ϕ2, τ2)

)= Cn · |DiscK1/F |

−h2

F

∫

GL2h(F )

f(x)|R(x)|−1F dx,

where Cn = 1 for any n > 0, and

(6.16) C0 =

∏hi=1 ζK1(i)ζK2(i)∏2h

i=1 ζF (i).

Before proceeding to the proof, we state a lemma for simplifying integrals.

Lemma 6.10. For any real-valued function X(x) on GL2h(F ) and any g ∈ GL2h(F ), we have∫

Rn

∫

Rn

X(ygx)dydx =Vol(Rn)

2

Vol(Rn · g ·Rn)

∫

Rn·g·Rn

X(t)dt.

Proof. Applying a substitution y → yx−1g−1, we have

(6.17)

∫

Rn

∫

Rn

X(ygx)dy dx =

∫

Rn

∫

Rn·gx

X(y)dy dx.

Note that

x ∈ Rn and y ∈ Rn · gx ⇐⇒ y ∈ Rn · g · Rn and x ∈ Rn ∩ g−1 · Rn · y.

By changing the order of the integration, we have∫

Rn

∫

Rn·gx

X(y)dy dx =

∫

Rn·g·Rn

∫

Rn∩g−1·Rn·y

X(y)dx dy,

which equals

(6.18)

∫

Rn·g·Rn

Vol(Rn ∩ g

−1 · Rn · y)X(y)dy.

Note that for any t ∈ Rn, we have(Rn ∩ g

−1 ·Rn · y)· t = Rn ∩ g

−1 · Rn · yt,

and

Rn ∩ g−1 · Rn · ty = Rn ∩ g

−1 · Rn · y,50

which imply that the value Vol (Rn ∩ g−1 · Rn · y) does not depend on y for all y ∈ Rn · g · Rn.

Letting X be a function that constantly equals 1, we obtain∫

Rn

∫

Rn

dy dx =

∫

Rn·g·Rn

Vol(Rn ∩ g

−1 · Rn · y)dy,

which implies that

Vol(Rn ∩ g

−1 · Rn · y)=

Vol(Rn)2

Vol(Rn · g ·Rn).

Combining this equation with (6.18), we complete the proof of this lemma.

Proof of Theorem 6.9. By Theorem 6.8, we have

χ(Zn(ϕ1, τ1)⊗

LMn

βnn(γ, f)∗Zn(ϕ2, τ2)

)=

∑

x∈Rn/Rn+m

χ(Zn(γϕ1, g0xτ1)⊗

LMn

Zn(ϕ2, τ2)),

which, by Theorem 6.1, can be written further into

(6.19)

∑x∈Rn/Rn+m

∑y∈Rn/Rn′

χ(Zn′(γϕ1, yg0xτ1)⊗

LMn′

Zn′(ϕ2, τ2))

deg (N1,n′ → N1,n) deg (N2,n′ → N2,n)

for any n′ ≥ n. Let U = Rng0Rn be the support of f . Note that the non-degeneracy condition

(6.15) implies that |R(g)|−1F is a continuous function on U . Therefore there is a uniform bound

|R(g)|−1F < M for any g ∈ U since U is compact. Using (4.19), we can choose a large enough

bound N such that the Theorem 4.1 holds uniformly for (γϕ1, kτ1) and (ϕ2, τ2) whenever k ∈ U .

Letting n′ be a value such that n′ > N , by Theorem 4.1, each summand of (6.19) simplifies to

(6.20) χ(Zn′(γϕ1, yg0xτ1)⊗

LMn′

Zn′(ϕ2, τ2))= χ

(Z∞(γϕ1, yg0xτ1)⊗

LG2hFZ∞(ϕ2, τ2)

),

which, by Theorem 5.12, equals

|DiscK1/F |−h2

F |R(yg0x)|−1F .

Since the action of βn′

n′ (id, t) on Mn′ is trivial for any t ∈ Rn′ , we have

Zn′(γϕ1, tyg0xτ1) = βn′

n′ (id, t)∗Zn′(γϕ1, yg0xτ1) = Zn′(γϕ1, yg0xτ1).

Therefore |R(tyg0x)|F = |R(yg0x)|F for any t ∈ Rn′ , which implies that

|R(yg0x)|−1F =

∫Rn′ ·y|R(tg0x)|

−1F dt

∫Rn′

dt.

Then, summing up the simplified summands over Rn/Rn′ , we have

(6.21)∑

y∈Rn/Rn′

χ(Zn′(γϕ1, yg0xτ1)⊗

LMn′

Zn′(ϕ2, τ2))=|DiscK1/F |

−h2

F

∫Rn|R(tg0x)|

−1F dt∫

Rn′

dt.

Combining preceding equations (6.19),(6.20), and (6.21), we obtain

χ(Zn(ϕ1, τ1)⊗

LMn

βnn(γ, f)∗Zn(ϕ2, τ2)

)

=|DiscK1/F |

−h2

F

∑x∈Rn/Rm+n

∫Rn|R(tg0x)|

−1F dt

deg (N1,n′ → N1,n) deg (N2,n′ → N2,n)∫Rn′

dt.

(6.22)

51

Since cond(g0) ≤ m, we have g∨0 (O2h∨F ) ⊃ πmO2h∨

F , which implies that

g0 · Rn+m · g−10 ⊂ Rn.

Therefore for any t ∈ Rn+m, we have yg0tx ∈ y · Rn · g0x = Rn · yg0x, which implies that the

integral∫Rn|R(yg0x)|

−1F dy is invariant when replacing x 7→ tx. Then we have

(6.23)∑

x∈Rn/Rm+n

∫

Rn

|R(yg0x)|−1F dy =

∫Rn

∫Rn|R(yg0x)|

−1F dydx∫

Rm+ndx

,

which, by Lemma 6.10, equals

Vol(Rn)2

Vol(Rn · g0 · Rn)Vol(Rm+n)

∫

Rn·g0·Rn

|R(g)|−1F dg.

Moreover, note that by definition of f in (6.11), we have

(6.24)deg(Mm+n →Mn)

Vol(Rn · g0 ·Rn)

∫

Rn·g0·Rn

|R(g)|−1F dg =

∫

GL2h(F )

f(g)|R(g)|−1F dg.

Note that deg(Mm+n →Mn) = Vol(Rn)/Vol(Rm+n). Combining preceding equations (6.22),(6.23),

and (6.24), the intersection number χ(Zn(ϕ1, τ1)⊗

LMn

βnn(γ, f)∗Zn(ϕ2, τ2)

)equals

(6.25)|DiscK1/F |

−h2

F Vol(Rn)

deg (N1,n′ → N1,n) deg (N2,n′ → N2,n)Vol(Rn′)

∫

GL2h(F )

f(g)|R(g)|−1F dg.

To simplify each leading factors, note that for any integer r and i = 1, 2, we have

(6.26) deg (Ni,r → Ni,0) = #GLh(OKi/πr) = q2rh

2h∏

j=1

(ζKi(j))−1 ,

(6.27) deg (Ni,n′ → Ni,n) =deg (Ni,n′ → Ni,0)

deg (Ni,n → Ni,0),

(6.28) Vol(R0/Rr) = #GL2h(OF/πr) = q4rh

22h∏

j=1

(ζF (j))−1 ,

and

(6.29) Vol(Rn)/Vol(R′n) =

Vol(R0/Rn′)

Vol(R0/Rn).

If n = 0, then putting r = n′ in (6.26) and (6.28), we have

Vol(R0)/Vol(Rn′)

deg (N1,n′ → N1,0) deg (N2,n′ → N2,0)=

q4n′h2∏2h

j=1 ζF (j)−1

(∏hj=1 ζK1(j)

−1)q2n′h2

(∏hj=1 ζK2(j)

−1)q2n′h2

,

where one sees that the term q4n′h2 cancels out. So we complete the proof for n = 0. When

n > 0, using (6.27),(6.29) together with (6.26), (6.28), we have Vol(Rn)/Vol(Rn′) = q4(n′−n)h2

and deg (N1,n′ → N1,n) = deg (N2,n′ → N2,n) = q2(n′−n)h2 , which completes the proof of the

theorem for all n.

52

Proof of main Theorems 1.5,1.4,1.3 and 1.2. The Theorem 1.5 is identical with Theorem 6.9. Let

∆i = ∆(ϕi, τi) for i = 1, 2. The Theorem 1.4 is a special case of Theorem 6.9 when K1∼= K2,

ϕ2 = γϕ1, and τ2 = gτ1. In this case we have ∆2 = γg∆1. Since (γ, g) is an equi-height pair, we

have, by Proposition 5.21,∣∣∣∣Nrd

((0h Ih

)·∆−1

1 · γ−10 · x

−1 · g · γ ·∆1 ·

(Ih0h

))∣∣∣∣F

=∣∣∣Resϕ1,τ1

(P ϕ1

γ−10 γ

, P τ1x−1g

)∣∣∣F,

which is non-zero everywhere for all g ∈ GL2h(F )when P ϕ1

γ−10 γ

is irreducible and P ϕ1

γ−10 γ

(0)P ϕ1

γ−10 γ

(1) 6=

0 thanks to Proposition 5.22. Therefore Theorem 6.9 implies Theorem 1.4. The Theorems 1.3 and

1.2 are special cases of Theorem 1.4 when m = 0, γ = id, and g0 = id.

6.4. Generalization of the theorem to non-equi-height pairs. When defining CM cyclesZ•(ϕ, τ)on M•

∼ in Definition 2.16, we do not put any restrictions for the height of (ϕ, τ). In this subsection,

we emphasize the fact that the intersection number involving non-equi-height pairs is essentially

the same as the one corresponding to equi-height pairs.

Let K be a uniformizer ofDK , and let vD(•) be the normalized discrete valuation ofDF . Note

that vD(K) = 2 if K/F is unramified and vD(K) = 1 if K/F is ramified.

For any pairs (ϕ, τ) as in (2.3), it is clear from the definition that Z∼n (ϕ, τ) = Z∼

n (ϕ ·mK , τ) for

any m. Therefore, if m = Height(τ)−Height(ϕ)vD(K)

is an integer, then (ϕ · mK , τ) is an equi-height pair,

and Z∼n (ϕ, τ) = Z∼

n (ϕ · mK , τ). Therefore, after such a replacement, we could apply the main

theorem to these cycles.

It remains to consider the pairs (ϕ, τ) such thatK/F is unramified and Height(τ)−Height(ϕ) ∈2Z+ 1, which we call odd pairs. We call it an even pair if Height(τ)−Height(ϕ) ∈ 2Z. If (ϕ, τ)

is an odd pair, we have Z(2j)n (ϕ, τ) = 0 and Z

(2j+1)n (ϕ, τ) 6= 0 for any j ∈ Z. Therefore, the cycle

Zn(ϕ, τ)∼ can not intersect with other cycles on Mn

(0). Instead, one needs to consider Mn(1). Let

F be a uniformizer of DF . Using an isomorphism βnn(F , id) : Mn(1) −→Mn

(0), we have

βnn(F , id)∗

(Z(1)n (ϕ1, τ1)⊗

L

Mn(1) Z

(1)n (ϕ2, τ2)

)= Z(0)

n (F · ϕ1, τ1)⊗L

Mn(0) Z

(0)n (F · ϕ2, τ2),

which essentially translates the intersection problems from Mn(1) to Mn

(0). If neither (F ·ϕ1, τ1)nor (F · ϕ2, τ2) is an odd pair, we can reduce this to the equi-height pairs case and our main

theorem applies. The only exception happens if (ϕ1, τ1) is an odd pair but (ϕ2, τ2) is an even pair

whenK1∼= K2 are unramified extensions over F . However, these two cycles do not intersect since

they are located on different components. Therefore, we can reduce all non-trivial intersection

problems of CM cycles to the equi-height pairs case.

7. PROOF OF THE LINEAR AFL FOR GL2(F )

In this section, we prove the linear AFL for the case of h = 1. Assume K/F is an unramified

quadratic extension of non-archimedean local fields of odd residue characteristic.

7.1. Notation. Let H ′ := GL1(F ) × GL1(F ) = F× × F× and G′ := GL2(F ). We identify H ′

with a subgroup of G′ by the blockwise diagonal embedding. Let ηK/F be the quadratic character

associated toK/F . We lift the characters |•|F and ηK/F to characters onH ′ by defining ηK/F (x) =ηK/F (x

−11 x2) and |x|F = |x−1

1 x2|F for any x = (x1, x2) ∈ F× × F×. For any g ∈ G′ and any53

real-valued test function f on G′, we define the orbital integral

(7.1) OrbF (g, f, s) :=

∫

H′×H′

I(g)

f(h−11 gh2

)ηK/F (h2)|h1h2|

sFdh1dh2

as a complex-valued function with s ∈ C, where I(g) := (h1, h2) ∈ H′ ×H ′ : h1g = gh2.

Fix an embedding K× ⊂ D×F and an element γ ∈ D×

F . Denote the invariant polynomial of

γ ∈ D×F by Pγ . Let g(γ) ∈ G′ be an element having the same invariant polynomial with γ. We

assume that Pγ is irreducible and Pγ(0)Pγ(1) 6= 0.

Let H := GL1(K) = K× and G := GL2(F ). Let H ⊂ G be an embedding which restricts

to O×K ⊂ GL2(OF ). Denote by Pg the corresponding invariant polynomial for any g ∈ G. We

fix the non-trivial element σ ∈ Gal(K/F ), and denote by kσ the Galois conjugate of k. For any

m ≥ n ≥ 0, we denote

gm,n :=

(πm

πn

).

Let M be an integer such that M > cond(gm,n).

7.2. The linear AFL. Let Z∼0 = Z∼

0 (id, id) be CM cycles on M0∼ as in Definition 2.16. Recall

thatZ∼0 is a collection of cycles on each M0

(n) for any integer n. We are interested in its intersection

with cycles

Z ′0∼:= βM0∗β

M0 (γ, gm,n)

∗Z∼0 (id, id)

as constructed in Section 6.2. In this expression, the symbolZ ′0∼

is an abbreviation since it depends

on the choice of γ, gm,n and M . Recall that we have defined the function

(7.2) fR0gm,nR0,M(g) =

deg(MM→M0)Vol(R0gm,nR0)

if g ∈ R0gm,nR0

0 otherwise

in (6.11). For h = 1, the linear AFL is the following theorem, which we will prove in this section.

Theorem 7.1. Let f = fR0gm,nR0,M be the function as described in (7.2), we have

χ(Z

(0)0 ⊗M0 Z

′0(0))= ±(2 ln q)−1 d

ds

∣∣∣∣s=0

OrbF (g(γ), f, s).

Here the sign is chosen so that the right hand side is non-negative.

Note that the pair (γ, gm,n) is not necessarily an equi-height pair. We will show that the conjec-

ture essentially reduces to the case of equi-height pairs with n = 0. Let

~ := Height(γ)− Height(gm,n)

be the difference of the height of the two elements. Recall the map βM0 (γ, gm,n) : MM∼ → M0

∼

restricts to

βM0 (γ, gm,n) : MM(0) −→M0

(−~)

as defined in (2.16), and recall that by using the superscript (−~) on Z(−~)0 we mean the part of Z∼

0

that is located on M0(−~) as described in (2.18).

54

Lemma 7.2. If ~ is an odd integer, then

Z ′0(0)

= 0.

If ~ is an even integer, then (π− ~

2γ, gm,n) is an equi-height pair, and

Z ′0(0)

= βM0∗βM0 (π− ~

2γ, gm,n)∗Z

(0)0 .

Proof. Note that

Z ′0(0)

= βM0∗βM0 (γ, gm,n)

∗Z(−~)0 .

By definition, Z(2i)0 6= 0 and Z

(2i+1)0 = 0 for all i ∈ Z. If ~ is an odd integer, then we have

Z(−~)0 = 0,

which implies that Z ′0(0) = 0.

Suppose ~ is an even number. Since

Height(gm,n) = −~+Height(γ) = Height(π−~

2 ) + Height(γ) = Height(π−~

2γ),

we know that (π− ~

2γ, gm,n) is an equi-height pair. Furthermore, since

βM0 (γ, gm,n) = β00(π

~

2 , id) βM0 (π− ~

2γ, gm,n)

and β00(π

~

2 , id)∗Z(−~)0 = Z

(0)0 , the cycle Z ′

0(0)

can be written as

βM0∗βM0 (γ, gm,n)

∗Z(−~)0 = βM0∗β

M0 (π− ~

2γ, gm,n)∗β0∗

0 (π~

2 , id)Z(−~)0 = βM0∗β

M0 (π− ~

2γ, gm,n)∗Z

(0)0

as desired.

Since we are in the equi-height situation now, we are able to apply our intersection formula to

χ(Z

(0)0 ⊗M0 Z

′0(0))

.

Lemma 7.3. If Height(γ)−m ∈ 2Z, we have

χ(Z

(0)0 ⊗M0 Z

′0(0))=

ζK(1)2

ζF (1)ζF (2)

∫

GL2(F )

fR0gm,nR0,M(g)∣∣∣Res(Pγ, Pg)

∣∣∣−1

Fdg.

If Height(γ)−m ∈ 2Z+ 1, we have

χ(Z

(0)0 ⊗M0 Z

′0(0))= 0.

Proof. If Height(γ)− Height(gm,n) is odd then Z ′0(0) = 0. Therefore χ

(Z

(0)0 ⊗M0 Z

′0(0))= 0.

If ~ = Height(γ)−Height(gm,n) is even, then by Lemma 7.2 we have

χ(Z

(0)0 ⊗M0 Z

′0(0))= χ

(Z

(0)0 ⊗M0 β

M0∗β

M0 (π− ~

2γ, gm,n)∗Z

(0)0

).

By Theorem 6.9, we have

χ(Z

(0)0 ⊗M0 Z

′0(0))=

ζK(1)2

ζF (1)ζF (2)

∫

GL2(F )

fR0gm,nR0,M(g)∣∣∣Res(P

π−~2 γ, Pg)

∣∣∣−1

Fdg.

Therefore the lemma follows from the fact that Pπ−

~2 γ

= Pγ by Corollary 5.17.

55

Furthermore, it is easy to see that

fR0gm,nR0,M(g) = fR0gm−n,0R0,M(π−ng)

by (6.11). Since Pπ−ng = Pg by Corolary 5.17, we have∫

GL2(F )

fR0gm,nR0,M(g)∣∣∣Res(Pγ, Pg)

∣∣∣−1

Fdg =

∫

GL2(F )

fR0gm−n,0R0,M(π−ng)∣∣∣Res(Pγ, Pπ−ng)

∣∣∣−1

Fdg.

By replacing g 7→ πng, the integral can be simplified to∫

GL2(F )

fR0gm−n,0R0,M(g)∣∣∣Res(Pγ, Pg)

∣∣∣−1

Fdg,

which implies that the intersection number only depends on m− n.

The similar phenomenon also happens on the orbital integral side. Since |πnh′|F = |(πnx1)−1(πnx2)|F =

|x−11 x2| = |h

′|F for any h′ = (x1, x2) ∈ H′, we have

∫

H′×H′

I(g)

fR0gm,nR0,M

(h−11 gh2

)ηK/F (h2)|h1h2|

sFdh1dh2

=

∫

H′×H′

I(g)

fR0gm−n,0R0,M

((πmh1)

−1gh2)ηK/F (h2)|(π

mh1)h2|sFdh1dh2

=

∫

H′×H′

I(g)

fR0gm−n,0R0,M

(h−11 gh2

)ηK/F (h2)|h1h2|

sFdh1dh2.

This implies that we only need to prove the linear AFL for f(x) = fR0gm,0R0,M . Since Height(gm,0) =m, we have

~ = Height(γ)− Height(gm,0) = Height(γ)−m.

The intersection number is given by Lemma 7.3 when ~ is even. The intersection number is 0when ~ is odd by Lemma 7.2. To simplify our notations, let

Tm,0(g) :=

1 if g ∈ R0gm,0R0;

0 otherwise.

Since fR0gm,0R0,M is a scalar multiple of Tm,0, the theorem reduces to the following theorem.

Theorem 7.4. Suppose Pγ(x) is the invariant polynomial of g(γ) and Pγ(0)Pγ(1) 6= 0. For any

m, if Height(γ)−m ∈ 2Z, then we have

(7.3) ±(2 ln q)−1 d

ds

∣∣∣∣s=0

OrbF (g(γ), Tm,0, s) =ζK(1)

2

ζF (1)ζF (2)

∫

GL2(F )

Tm,0(g)∣∣∣Res(Pγ, Pg)

∣∣∣−1

Fdg.

Otherwise, if Height(γ)−m ∈ 2Z+1, then the left hand side of the above equation is 0. Here the

sign is chosen so that the left hand side is non-negative.

Call the left hand side of this identity the analytic side, and the right hand side the geometric

side. For any two sets A and B, by A−B we mean the subset of A consisting of all elements that

is not in B. Note that

(7.4) Tm,0(g) = 1 ⇐⇒ det g ∈ πmO×F and g ∈ M2(OF )− πM2(OF ).

56

7.3. Matching orbits. Fix the decomposition γ = γ+ + γ− according to the embedding K ⊂ DF

such that γ+k = kγ+ and γ−k = kσγ− for all k ∈ K. Note that when h = 1, DF is a quaternion

algebra over F . Let vD(•), vK(•) and vF (•) be normalized valuations on DF , K and F . It is

well-known that vD(x) = 2vF (x) = 2vK(x) for any x ∈ F ⊂ K ⊂ DF . Since Pγ is irreducible

and Pγ(1)Pγ(0) 6= 0, there exists an element c ∈ F such that c 6= 0, 1 and Pγ(X) = X− (1−c)−1.

Note that Pγ is the characteristic polynomial of γ+(γ++ γ−)−1γ+(γ+−γ−)

−1 as an element in K.

Therefore,

γ+(γ+ + γ−)−1γ+(γ+ − γ−)

−1 = (1− c)−1 ∈ F×,

in particular, we have γ+ 6= 0. Note that this element can also be written as

γ+(γ+ + γ−)−1γ+(γ+ − γ−)

−1 = (1− γ−1+ γ−γ

−1+ γ−)

−1,

which implies that γ−1+ γ−γ

−1+ γ− = c.

Lemma 7.5. If γ ∈ D×F such that γx = xσγ for any x ∈ K, then vD(γ) is an odd integer.

Proof. Since γx = xσγ for any x ∈ K, one sees that γ2 commutes with all elements in K,

which implies γ2 ∈ K. Since γx = xσγ for any x ∈ K, one sees that γ(γ2) = (γ2)σγ, which

implies (γ2) = (γ2)σ and therefore γ2 ∈ F . Assume, for the sake of contradiction, that vD(γ)is even. Then vF (γ

2) = 12vD(γ

2) = vD(γ) is also an even integer. Since the norm subgroup of

the unramified extension K/F is π2Z · O×K , there exists some element y ∈ K such that γ2 = yyσ.

Since γy = yσγ, we have

(γ + yσ)(γ − y) = γ2 − yyσ = 0.

SinceDF is a division algebra, we have either γ = y or γ = −yσ, which contradicts the assumption

that γx = xσγ for any x ∈ K.

Note that c = (γ−1+ γ−)

2, and that γ−1+ γ−k = kσγ−1

+ γ− for any k ∈ K. We have vF (c) ∈ 2Z+1.

Similarly, we have vD(γ−) ∈ 2Z + 1. Hence vD(γ+) ∈ 2Z. Therefore vD(γ) = vD(γ+ + γ−) =minvD(γ+), vD(γ−). Moreover,

vD(γ) ∈ 2Z ⇐⇒ vD(γ+) > vD(γ−) ⇐⇒ vF (c) > 0,

vD(γ) ∈ 2Z+ 1 ⇐⇒ vD(γ+) < vD(γ−) ⇐⇒ vF (c) < 0.(7.5)

7.4. The calculation for the analytic side. Recall that H ′ ⊂ G′ ∼= GL2(F ) is the subgroup of

diagonal matrices. For any g ∈ G′, the decomposition g = g++ g− gives a diagonal matrix g+ and

a matrix g− with zero entries on the diagonal.

Note that g(γ) is an element with invariant polynomial Pγ(X) = X − (1 − c)−1. Since c 6= 1or 0, we know (g(γ)−1

+ g(γ)−)2 = c. Since the diagonal entries of g(γ)−1

+ g(γ)− are 0, there exists

a diagonal matrix h such that

h−1g(γ)−1+ g(γ)−h =

(0 c1 0

).

Therefore (h−1g(γ)−1

+

)· g(γ) · h =

(1 c1 1

).

Since g(γ)+ ∈ H′ and h ∈ H ′, we have h−1g(γ)−1

+ ∈ H′ and h ∈ H ′. Since replacing g(γ) by

h′g(γ)h′′ does not change the absolute value of the orbital integral (7.1) for any h′, h′′ ∈ H ′, we57

can assume

g(γ) :=

(1 c1 1

)∈ G′.

Let k be the integer such that vF (c) = 2k+1. Indeed, the invariant polynomial of g(γ) is Pγ(X) =X − (1− c)−1.

Proposition 7.6. Let M = m−min0, vF (c). We have

(1 + q2s)OrbF (g(γ), Tm,0, s) =

0 if M ∈ 2Z+ 1;

−(1− q−2s)(q−Ms + q(M+4k−2)s

)if M ∈ 2Z r 0;

−q2vF (c)s + q−2s if M = 0.

Proof. Recall the definition of OrbF (g(γ), Tm,0, s) in (7.1). Note that I(g) ∼= F× is the set of

scalar matrices, which embeds diagonally in H ′ × H ′ ∼= (F× × F×) × (F× × F×). Therefore

H ′×H ′/I(g) ∼= F××F××F×, with its Haar measure normalized by O×F ×O×

F ×O×F . Then, the

orbital integral OrbF (g(γ), Tm,0, s) equals

(7.6)

∫

F××F××F×

Tm,0

((a

b

)(1 c1 1

)(d

1

))ηK/F (d)|a

−1bd|−sF da×db×dd×.

Letting x := vF (a), y := vF (b) and z := vF (d), we rewrite the above integral into

OrbF (g(γ), Tm,0, s) =

∫

F××F××F×

Tm,0

(ad acbd b

)(−1)zq(y+z−x)sda×db×dd×,

which equals

∑

x,y,z∈Z

∫

πxO×

F×πyO×

F×πzO×

F

Tm,0

(ad acbd b

)(−1)zq(y+z−x)sda×db×dd×.

Furthermore let w := vF (det(g(γ))) = vF (1 − c). Since vF (c) = 2k + 1 6= 0, we have w =min0, 2k+ 1. Then M = m−w. Note that each summand is either (−1)zq(y+z−x)s or 0, where

it is non-zero if and only if

Tm,0

(ad acbd b

)= 1,

which, by (7.4), is equivalent to

x+ y + z =M and minx+ z, x+ (2k + 1), y + z, y = 0.

Replacing z by M − x− y, this restriction can be simplified to

(7.7) z =M − x− y and minM − y, x+ (2k + 1),M − x, y = 0.

Let SM,k = (x, y) : minM − y, x+ (2k + 1),M − x, y = 0. Then we have

(7.8) OrbF (g(γ), Tm,0, s) =∑

(x,y)∈SM,k

(−1)M−x−yq(M−2x)s.

From now on, we will discuss the orbital integral in two cases depending on the parity of M .

Let us start with the case of M ∈ 2Z + 1. Since the set SM,k is invariant under the replacement

y 7→M − y, we have∑

(x,y)∈SM,k

(−1)M−x−yq(M−2x)s =∑

(x,y)∈SM,k

(−1)y−xq(M−2x)s = −∑

(x,y)∈SM,k

(−1)M−x−yq(M−2x)s,

58

which implies OrbF (g(γ), Tm,0, s) = 0.

In the case of M ∈ 2Z, we decompose SM,k into the following subsets

SM,k = (−(2k + 1), y) : 0 ≤ y < M ∪ (M, y) : 0 < y ≤M

∪ (x, 0) : −(2k + 1) < x ≤M ∪ (x,M) : −(2k + 1) ≤ x < M.

It suffices to calculate (7.8) by each component. For the first two components (−(2k + 1), y) :0 ≤ y < M and (M, y) : 0 < y ≤ M, since M is even, we have

M∑

y=1

(−1)M−x−yq(M−2x)s = 0 =

M−1∑

y=0

(−1)M−x−yq(M−2x)s.

Therefore,

OrbF (g(γ), Tm,0, s) =

M∑

x=−2k

(−1)M−xq(M−2x)s +

M−1∑

x=−2k−1

(−1)−xq(M−2x)s,

which equals

q(M+4k)s + q(−M−2)s

1 + q−2s+−q(M+4k+2)s − q−Ms

1 + q−2s= −

(1− q−2s)(q−Ms + q(M+4k+2)s

)

1 + q−2s.

Suppose M = 0. We have

S0,k = (x, 0) : −(2k + 1) ≤ x ≤ 0.

Therefore

OrbF (g(γ), Tm,0, s) =

0∑

x=−(2k+1)

(−1)−xq−2xs =−q(4k+2)s + q−2s

1 + q2s.

Combining all previous cases, we complete the proof of the proposition.

Proof of Theorem 7.4 in the case that m− Height(γ) ∈ 2Z+ 1. Note that Height(γ) = vD(γ).By (7.5), vD(γ)−minvF (c), 0 ∈ 2Z. This implies that

m−min0, vF (c) ∈ 2Z+ 1 ⇐⇒ m− Height(γ) ∈ 2Z+ 1.

In this case, by Proposition 7.6, the orbital integral is constantly zero as desired.

We consider the remaining cases. Note that m− Height(γ) ∈ 2Z implies that

m−min0, vF (c) ∈ 2Z ⇐⇒ m ∈ 2Z and vF (c) > 0, or m ∈ 2Z+ 1 and vF (c) < 0.

Therefore, when m ∈ 2Z r 0 and vF (c) > 0, or when m ∈ 2Z+ 1 and vF (c) < 0, we have

(7.9) −1

2 ln q

d

ds

∣∣∣∣s=0

OrbF (g(γ), Tm,0, s) = 1;

when m = 0 and vF (c) > 0, we have

(7.10) −1

2 ln q

d

ds

∣∣∣∣s=0

OrbF (g(γ), Tm,0, s) =vF (c) + 1

2.

59

7.5. The calculation for the geometric side. In this subsection, we calculate the geometric side

of (7.1) in cases of (7.9) and (7.10) respectively.

Recall that H := GL1(K) = K× and G := GL2(F ) are algebraic groups over F . Fix an

embedding H ⊂ G that restricts to O×K ⊂ GL2(OF ). Let 02 be the zero matrix in Mat2×2(F ).

Note that

(H ∩Mat2×2(OF )) ∪ 02 ∼= OK .

By abuse of notation, we also denote (H ∩Mat2×2(OF )) ∪ 02 by OK if no confusion can arise.

Fix the decomposition g = g+ + g− according to the embedding H ⊂ G such that g+k = kg+ and

g−k = kσg− for all k ∈ H ⊂ G. Let

Pg(X) = X − b

be the invariant polynomial of g. Note that b ∈ F and b = g+(g+ − g−)−1g+(g+ + g−)

−1.

Therefore as an element in GL2(F ) we have detF (b) = b2. On the other hand, by (5.20), we have

det(b) = det(g+)2 det(g)−2, which implies that

(7.11) b = g+(g+ − g−)−1g+(g+ + g−)

−1 = ±(g+gσ+) det(g)

−1 ∈ F.

Note that ζK(s) = (1 − q−2s)−1 and ζF (s) = (1 − q−s)−1. Recall that Pγ(X) = X − (1 − c)−1.

We have Res(Pγ , Pg) = (1− c)−1 − b. Therefore the geometric side of (7.3) equals

(7.12)1

1 + q−1

∫

GL2(F )

Tm,0(g)∣∣(1− c)−1 − b

∣∣−1

Fdg.

Lemma 7.7. If Tm,0(g) = 1, then g+, g− ∈ Mat2×2(OF ). Furthermore, if m > 0, then g+ ∈ O×K .

Proof. By definition (7.4), Tm,0(g) = 1 implies that g ∈ Mat2×2(F ) and g 6≡ 0 modulo π. Let

µ ∈ O×K be an element such that µσ = −µ. We have 2µ · g+ = µg + gµ and 2µ · g− = µg − gµ,

which implies g+, g− ∈ Mat2×2(OF ). Suppose m > 0, and suppose for the sake of contradiction

that g+ ∈ πOK . Then g+ ≡ 0 modulo π, which implies that g− = g − g+ 6≡ 0 modulo π. Note

that there is an element σ ∈ GL2(OF ) such that σkσ−1 = kσ. Then σ · g− ∈ K, which implies

σ · g− ∈ O×K ⊂ GL2(OF ). Therefore g = g+ + g− ∈ GL2(OF ), which contradicts to Tm,0(g) = 1

since m > 0.

7.5.1. Case 1: m > 0. By Lemma 7.7 and (7.11), one sees that Tm,0(g) = 1 implies |b|F =| deg(g)−1|F = qm > 1. Recall that vF (c) is an odd integer, and, in particular, vF (c) 6= 0, which

implies that (1 − c)−1 ∈ OF . Therefore |(1 − c)−1 − b|F = qm, which implies that the equation

(7.12) equals

1

1 + q−1

∫

GL2(F )

Tm,0(g)q−mdg.

Note that under the Haar measure dg normalized by GL2(OF ), we have

Vol

(GL2(OF )

(1

πm

)GL2(OF )

)= (1 + q−1)qm,

which implies that the equation (7.12) equals 1 as desired. Comparing this result with (7.9), we

complete the proof of the linear AFL for the case of m > 0.60

7.5.2. Case 2: m = 0. In this case, we only need to prove the formula for vF (c) > 0. Since the

volume of the subset g ∈ G : g+ = 0 is zero, we may assume that g+ is invertible for integration

purposes. Recall Pγ(X) = X − (1− c)−1, and Pg(X) = X − (1− g−1+ g−g

−1+ g−)

−1. We have

Res(Pγ, Pg) =c− g−1

+ g−g−1+ g−

(1− c)(1− g−1+ g−g

−1+ g−)

.

If g ∈ GL2(OF ), by (7.11), we have |(1−g−1+ g−g

−1+ g−)

−1|F = |g+gσ+|F . Since g−g

−1+ = (g−1

+ )σg−,

we have g−1+ g−g

−1+ g− = g−1

+ (g−1+ )σg2−. Note that vF (c) > 0 implies |1− c|F = 1. Therefore,

(7.13) |Res(Pγ, Pg)|F = |g+gσ+c− g

2−|F .

Note that the embedding O×K ⊂ GL2(OF ) gives rise to an OK-action on O2

F , which induces an

isomorphism O2F → OK . The non-trivial Galois conjugation of OK corresponds to an element

σ ∈ GL2(OF ) such that σk = kσσ for any k ∈ H . Therefore, for any g ∈ GL2(OF ), there exists

k1, k2 ∈ H such that

g+ = k1, g− = k2σ.

For any n ≥ 1, let

(7.14) Ωn := g = k1 + k2σ ∈ GL2(OF ) : k2 ∈ πnOK.

Recall the decomposition (5.19). In particular, we have πnMat2×2(OF ) = πnOK⊕πnσOK for any

n ≥ 1, which implies that

Ωn = O×K + πnMat2×2(OF ).

Under the normalized Haar measure of GL2(OF ), the volume of Ωn is

(7.15) Vol(Ωn) =#(OK/π

n)×

#(GL2(OF/πn))=

q−2n

1− q−1.

Therefore, note that g+gσ+c− g

2− ∈ O×

F for any g /∈ Ω1, we have∫

GL2(OF )−Ω1

|Res(Pγ, Pg)|−1F dg =

∫

GL2(OF )−Ω1

dg = Vol(GL2(OF ))−Vol(Ω1) = 1−q−2

1− q−1.

For any 1 ≤ 2n < vF (c), we have |g+gσ+c− g

2−|F = |g2−|F = q−2n for any g ∈ Ωn − Ωn+1, which

implies that∫

Ωn−Ωn+1

|Res(Pγ, Pg)|−1F dg =

∫

Ωn−Ωn+1

|g2−|−1F dg = q2n (Vol(Ωn)−Vol(Ωn+1)) = 1 + q−1.

For 2n = vF (c) + 1, we have |g+gσ+c− g

2−|F = |c|F = q−vF (c) for any g ∈ Ωn, which implies that

∫

Ωn

|Res(Pγ, Pg)|−1F dg =

∫

Ωn

|c|−1F = qvF (c)Vol(Ωn) =

qvF (c)−2n

1− q−1=

q−1

1− q−1.

Combining preceding equations, we have

∫

GL2(OF )

|Res(Pγ, Pg)|−1F dg = 1−

q−2

1− q−1+

vF−1

2∑

n=1

(1 + q−1) +q−1

1− q−1=

(1 + q−1)(vF (c) + 1)

2,

which proves that1

1 + q−1

∫

GL2(OF )

|Res(Pγ, Pg)|−1F dg =

vF (c) + 1

261

as desired. Therefore we proved the linear AFL for the case of m = 0. Combining all cases, we

complete the proof of the linear AFL for h = 1.

REFERENCES

[Dri74] Vladimir Drinfel’d, Elliptic modules, Mathematics of the USSR-Sbornik 23 (1974), no. 4, 561.

[GGP12] Wee Teck Gan, Benedict Gross, and Dipendra Prasad, Symplectic local root numbers, central critical

L-values, and restriction problems in the representation theory of classical groups, Asterisque 346 (2012),

no. 1, 1–109.

[Gro86] Benedict Gross, On canonical and quasi-canonical liftings, Inventiones mathematicae 84 (1986), no. 2, 321–

326.

[GZ86] Benedict Gross and Don Zagier, Heegner points and derivatives of L-series, Inventiones mathematicae 84

(1986), no. 2, 225–320.

[HL20] Benjamin Howard and Qirui Li, Intersections in Lubin-Tate space and biquadratic fundamental lemmas, arXiv

preprint arXiv:2010.07365 (2020).

[HS19] Benjamin Howard and Ari Shnidman, A Gross-Kohnen-Zagier formula for Heegner-Drinfeld cycles, Ad-

vances in Mathematics 351 (2019), 117–194.

[KR14] Stephen Kudla and Michael Rapoport, Special cycles on unitary Shimura varieties II: Global theory, Journal

fur die reine und angewandte Mathematik 2014 (2014), no. 697, 91–157.

[Li19] Qirui Li, On Gross-Keating’s result of lifting endomorphisms for formal modules, arXiv preprint

arXiv:1902.10789 (2019).

[Li20] , A proof of the Linear Arithmetic Fundamental Lemma for GL4 , Canadian Journal of Mathematics

(2020), 1–47 https://doi.org/10.4153/s0008414x20000814.

[Lub66] John Tate Lubin, Formal moduli for one-parameter formal Lie groups, Bulletin de la SMF 94 (1966), 49–59.

[Rob85] Paul Roberts, The vanishing of intersection multiplicities of perfect complexes, Bulletin of the American

Mathematical Society 13 (1985), no. 2, 127–130.

[RSZ17] Michael Rapoport, Brian Smithling, and Wei Zhang, On the arithmetic transfer conjecture for exotic smooth

formal moduli spaces, Duke Mathematical Journal 166 (2017), no. 12, 2183–2336.

[RSZ18] , Regular formal moduli spaces and arithmetic transfer conjectures, Mathematische Annalen 370

(2018), no. 3-4, 1079–1175.

[Sta17] The Stacks Project Authors, stacks project, http://stacks.math.columbia.edu, 2017.

[SW13] Peter Scholze and Jared Weinstein, Moduli of p-divisible groups, Cambridge Journal of Mathematics 1

(2013), no. 2, 145–237.

[Wei16] Jared Weinstein, Semistable models for modular curves of arbitrary level, Inventiones mathematicae 205

(2016), no. 2, 459–526.

[XZ20] Jingwei Xiao and Wei Zhang, Periods and heights for unitary symmetric pairs, In preparation, 2020.

[YZ17] Zhiwei Yun and Wei Zhang, Shtukas and the Taylor expansion ofL-functions, Annals of Mathematics (2017),

767–911.

[YZZ13] Xinyi Yuan, Shouwu Zhang, and Wei Zhang, The Gross-Zagier formula on Shimura curves, no. 184, Prince-

ton University Press, 2013.

[Zha12] Wei Zhang, On Arithmetic Fundamental Lemmas, Inventiones mathematicae 188 (2012), no. 1, 197–252.

[Zha17a] Wei Zhang, In preparation, 2017.

[Zha17b] , A conjectural Linear Arithmetic Fundamental Lemma for Lubin-Tate space, Unpublished, 2017.

[Zha18] Wei Zhang, Periods, cycles, and L-functions: A relative trace formula approach, Proceedings of the Interna-

tional Congress of Mathematicians, vol. 1, World Scientific, 2018, pp. 485–520.

[Zha21] , Weil representation and Arithmetic Fundamental Lemma, Annals of Mathematics 193 (2021), no. 3,

863–978.

62