Level-one elliptic modular forms 1. Automorphy condition ...

(November 6, 2015)

Level-one elliptic modular forms

Paul Garrett [email protected] http://www.math.umn.edu/ garrett/

[This document is http://www.math.umn.edu/˜garrett/m/mfms/notes 2015-16/10 level one.pdf]

1. Automorphy condition, Fourier expansions, cuspforms2. Explicit example: holomorphic Eisenstein series3. Divisor/dimension formula, applications4. Proof of divisor/dimension formula5. Fourier expansions of holomorphic Eisenstein series6. Automorphic L-functions7. Hecke operators and Euler products8. Poincare series, Petersson inner product

1. Automorphy condition, Fourier expansion, cuspforms

An elliptic (holomorphic) modular form of level one and weight 2k is a holomorphic function f on the upperhalf-plane H meeting the automorphy condition

f(γz) = (cz + d)2k · f(z) (for z ∈ H and γ =

(a bc d

)∈ SL2(Z))

with γz = az+bcz+d , and meeting the growth condition that it is bounded on the closure of the standard

fundamental domainF = {z ∈ H : |z| > 1, |Re(z)| < 1

2}

The function

j : SL2(Z)× H −→ C× by j(

(a bc d

), z) −→ cz + d

is the cocycle. When context makes the details clear, the modifier elliptic is often dropped. [1]

f |2kγ = f(γz) · (cz + d)−2k (with γ =

(a bc d

))

for arbitrary complex-valued functions f on H, allowing the automorphy condition to be rewritten as

f |2kγ = f (for all γ ∈ SL2(Z))

[1.0.1] Note: The holomorphic modular forms of weight 2k for SL2(Z) form a complex vector space undervalue-wise sums. Also, the product of a weight 2k form and a weight 2k′ form gives a weight 2k + 2k′ form.

[1.0.2] Remark: The modifier elliptic modular refers to the fact that the function is on H, as opposedto some other homogeneous space, and is holomorphic, as opposed to meeting some other local analyticcondition. Level one refers to the automorphy requirement for all γ ∈ SL2(Z) rather than some smaller ordifferent subgroup of SL2(R).

[1] Traditional terminology is that f → f |2kγ is the slash operator, although this name fails to suggest any meaning

other than reference to the notation itself. In fact, obviously f(z)→ f(γz)(cz + d)−2k is a left translation operator,

albeit complicated by the automorphy factor. That is, this is a right action of SL2(Z) on functions on H, while the

group action of SL2(Z) on H is written on the left.

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Paul Garrett: Level-one elliptic modular forms (November 6, 2015)

[1.0.3] Remark: Boundedness in the closure of the fundamental domain does not imply boundedness on H,because modular forms are not quite invariant under SL2(Z), but only almost invariant, with the cocyclemaking things more complicated.

[1.1] Fourier expansions The upper-triangular element γ =

(1 10 1

)∈ SL2(Z) sends z → z + 1, and

j(γ, z) = 1, so a level-one modular form f has the property

f(z + 1) = f(γz) = j(γ, z)2k · f(z) = 12k · f(z) = f(z)

That is, modular forms are periodic in x = Re(z), with period 1. Thus, as functions of z, modular formshave Fourier expansions in x, with coefficients depending on y = Im(z):

f(x+ iy) =∑n∈Z

cn(y) e2πinx

Since f is holomorphic, it satisfies the Cauchy-Riemann equation( ∂∂x

+ i∂

∂y

)f(x+ iy) = 0

Differentiating term-wise,

0 =∑n∈Z

( ∂∂x

+ i∂

∂y

)(cn(y)e2πinx

)=∑n∈Z

(2πincn(y)e2πinx + ic′n(y)e2πinx

)By uniqueness of Fourier expansions,

2πincn(y) + ic′n(y) = 0 (for all n ∈ Z)

This is a linear, constant-coefficient differential equation for cn(y):

c′n(y) + 2πncn(y) = 0

Thus,cn(y) = constant × e−2πny

and the Fourier expansion of a (holomorphic) modular form is of the form

f(z) =∑n∈Z

cn e2πinz (constants cn ∈ C)

[1.1.1] Remark: Fourier expansions of modular forms are sometimes called q-expansions, with q = e2πiz.

[1.2] Fourier expansions and growth condition

Use the standard notationAn � Bn

for the assertion that |An| ≤ C ·Bn for some constant C.

[1.2.1] Proposition: A modular form f(z) =∑n∈Z cne

2πinz has cn = 0 for n < 0, and |cn| � e2πn forn ≥ 0, with implied constant depending on f .

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Proof: Let |f(z)| ≤ C for z in the fundamental domain. Then the usual expression for the nth Fouriercomponent gives

|cn|e−2πny =∣∣∣ ∫ 1

2

− 12

e−2πinxf(x+ iy) dx∣∣∣ ≤ ∫ 1

2

− 12

∣∣∣e−2πinxf(x+ iy)∣∣∣ dx ≤ ∫ 1

2

− 12

1 · C dx ≤ C

That is,|cn| ≤ e2πny · C

As y → +∞ with z ∈ F , we find cn = 0 for n < 0. For n ≥ 0, taking y = 1 gives the estimate. ///

[1.2.2] Remark: The estimate |cn| � e2πn is very bad, but useful in preliminaries.

[1.3] Cuspforms A modular form with 0th Fourier coefficient 0 is a cuspform.

This innocent cuspform condition, beyond holomorphy, automorphy, and the growth condition, has importantramifications later.

[1.3.1] Theorem: (Hecke) A weight 2k holomorphic cuspform f has exponential decay

|f(x+ iy)| �f e−2πy (as y → +∞)

with implied constant depending on f . The Fourier coefficients cn of f satisfy

|cn| � nk

Proof: Using the preliminary bound |cn| � e2πny from above,

|f(z)| �∑n≥1

e2πn e−2πny =∑n≥1

e−2πn(y−1) =e−2π(y−1)

1− e−2π(y−1)

by summing the geometric series, giving the exponential decay. Since

|Im(

(a bc d

)z)| =

Im(z)

|cz + d|2

the function yk · |f(z)| is SL2(Z)-invariant, rather than merely satisfying the automorphy condition. Due tothe exponential decay in the fundamental domain, yk · |f(z)| is surely bounded in the fundamental domain.By SL2(Z)-invariance, yk · |f(z)| is bounded on H.

For any y > 0,

|cn · e−2πny| ≤∫ 1

2

− 12

∣∣∣e−2πinx f(x+ iy)∣∣∣ dx �f y−k

That is, |cn| �f y−ke2πny. The bounding expression blows up as y → 0+ and as y → +∞, but we can find

its minimum: solve

0 =∂

∂y

(y−ke2πny

)= −ky−k−1e2πny + 2πny−ke2πny = (−k + 2πny)y−k−1e2πny

The minimizing y = k/2πn gives

|cn| �( k

2πn

)−ke2πn· k

2πn = nk ·(2πe

k

)kgiving the asserted bound. ///

[1.3.2] Remark: [Hecke 1937]’s bound given above was improved by [Rankin 1939] and [Selberg 1940].

[Ramanujan 1916]’s and [Petersson 1930]’s conjecture that |cp| ≤ 2pk−12 for prime p and weight 2k cuspforms,

was proven by [Deligne 1974] as application of his completion of proof of the Weil conjectures.

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2. Explicit example: holomorphic Eisenstein series

One normalization of (holomorphic) Eisenstein series is

E2k(z) = 12

∑coprime c,d

1

(cz + d)2k

Legitimate analogues of an integral test show that this is absolutely convergent, and uniformly so for z incompacts, for 2k ≥ 4. Thus, E2k is holomorphic. [2]

[2.0.1] Remark: Unless 2k is an integer, there are serious problems with the definition of the 2kth powers.When 2k ≥ 3 is an odd integer, the pairs (c, d) and (−c,−d) produce terms that cancel each other, and theexpression is identically 0.

As earlier, direct computation shows that

E2k(γz) = (cz + d)2kE2k(z) (with γ =

(a bc d

))

Namely, with γ =

(A BC D

),

E2k(γz) = 12

∑coprime C,D

1

(C az+bcz+d +D)2k

= (cz + d)2k∑

coprime C,D

1

(C(az + b) +D(cz + d))2k

= (cz + d)2k∑

coprime C,D

1((Ca+Dc)z + (Cb+Dd))2k

and (A BC D

)(a bc d

)=

(∗ ∗

Ca+Dc Cb+Dd

)Thus, the map (C,D) → (Ca + Dc,Cb + Dd) is a bijection on the set of coprime integers, and we have

(cz + d)2kE2k(z). [3]

The leading fraction and the coprimality condition are elementary shadows of a more meaningful expression,

E2k(z) =∑

γ∈Γ∞\Γ

1

(cz + d)2k(with γ =

(a bc d

))

[2] An infinite sum∑n≥1 fn of holomorphic functions, if uniformly absolutely convergent on compacts, is again

holomorphic. This follows from Morera’s theorem, that a function f is holomorphic if its integrals over small

triangles are 0. Namely, any given triangular path γ traces out a compact set, so, given ε > 0, there is N such

that∑n≥N |fn(z)| < ε for all z on γ, and the integral of this tail over γ is at most ε times the length of γ. Since the

finite sum∑n≤N fn is holomorphic, its integral over γ is 0. Thus, the integral over every triangle is smaller than

every positive real, so is 0.

[3] The same computation demonstrates the cocycle relation j(gh, z) = j(g, hz)j(h, z) for g, h ∈ SL2(R) and z ∈ H.

This certifies that the action f → f |2kγ has the associativity

(f |2kγ)|2kδ = f |2k(γδ)

necessary for this to be a legitimate right action.

4


where Γ = SL2(Z), Γ∞ = {(∗ ∗0 ∗

)∈ Γ}. Indeed, for integers c, d to be the lower row of an element γ ∈ Γ,

necessarily c, d are coprime. With even integer 2k, changing c, d for −c,−d does not change (cz+d)2k. And,

given

(a bc d

)and

(a′ b′

c d

)in Γ,

(a′ b′

c d

)(a bc d

)−1

=

(a′ b′

c d

)(d −b−c a

)=

(∗ ∗

cd− dc ∗

)=

(∗ ∗0 ∗

)∈ Γ∞

proving the bijection.

So E2k(z) satisfies the automorphy condition.

Thus, E2k(z) meets the holomorphy condition and the automorphy condition. Demonstration that it isbounded in the closure of the standard fundamental domain would complete proof that it is an ellipticmodular form.

This demonstration is postponed till after computation of the Fourier coefficients of holomorphic Eisensteinseries below.

3. Divisor/dimension formula, applications

A useful relation on the orders of vanishing of an elliptic modular form f of weight 2k for SL2(Z) is producedvia the argument principle, by path-integration of f ′(z)/f(z) around the boundary of a height-T truncation

FT = {|z| ≥ 1, |Re(z)| ≤ 12 , Im(z) ≤ T}

of the standard fundamental domain F .

The divisor of a function is the set of it zeros, counting order-of-vanishing, that is, counting multiplicities.[4] Less usually, the order of vanishing at i∞, νf (i∞), of f(z) =

∑n cne

2πinz is the smallest no such thatcn = 0 for n < no. Still, this is consistent with the usual notion by viewing the Fourier expansion as a powerseries in q = e2πiz.

[3.0.1] Theorem: Let νf (z) be the order of vanishing of not-identically-zero f at z ∈ H. Including only anirredundant collection of representatives for SL2(Z)\H,

νf (i)

2+

νf (ρ)

3+ νf (i∞) +

∑other z

νf (z) =2k

12

where ρ is a cube root of unity in H and f is weight 2k. (Proof in following section.)

This divisor relation yields important corollaries.

[3.1] The first cuspform A small further preparation: Ramanujan’s ∆(z)-function is a non-zero constantmultiple of E3

4 − E26 , which the proof of the following shows to be not identically zero. The choice of the

[4] As usual in complex analysis, at a point zo ∈ H, the order of vanishing νf (zo) of a holomorphic function f is the

smallest no so that the ntho power series coefficient of f at zo is non-zero. That is, with

f(z) =

∞∑n=0

cn (z − zo)n

the order (of vanishing) of f at zo is the smallest no such that cno 6= 0.

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multiplying constant is usually to make ∆(z) have Fourier expansion (with vanishing 0th Fourier coefficient,and) 1st Fourier coefficient 1:

∆(z) = 1 · e2πiz +∑n≥2

τ(n) e2πinz

The higher Fourier coefficients are sometimes denoted τ(n) for reasons of tradition. When we compute theFourier coefficients of E2k, we will see that they are of the form

E2k(z) = 1 · e2πi·0·z +∑n≥1

cn e2πinz

Granting this, since there are no negative-index Fourier components,

E4(z)3 − E6(z)2 = (1 + higher)3 − (1 + higher)2 = (1 + higher)− (1 + higher)

= vanishing 0th Fourier component + higher Fourier components

Thus, granting this feature of the Fourier expension of Eisenstein series, the constant multiple ∆(z) ofE4(z)3 − E6(z)2 is indeed a cuspform.

[3.1.1] Corollary: The spaces M2k of modular forms of weight 2k for SL2(Z) are {0} for 2k < 0 or 2k anodd integer. In small non-negative weights: M0 = C and M2 = {0}, while for even integer weights 2k ≥ 4,

M2k = C · E2k ⊕ ∆ ·M2k−12

That is, for weights up through 22,

M0 = CM2 = {0}M4 = C · E4

M6 = C · E6

M8 = C · E8

M10 = C · E10

M12 = C · E12 ⊕ C ·∆M14 = C · E14

M16 = C · E16 ⊕ C ·∆E4

M18 = C · E18 ⊕ C ·∆E6

M20 = C · E20 ⊕ C ·∆E8

M22 = C · E22 ⊕ C ·∆E10

Proof: For odd integers 2k (momentarily resisting the suggestion of the notation that it’s an even integer),and f ∈M2k,

f(z) = f(−z + 0

0 · z − 1) = f(

(−1 00 −1

)z) = (0 · z − 1)2k · f(z) = (−1) · f(z)

so f(z) = 0.

For even integer 2k, the point is that, for small non-negative even integers 2k, it is not easy to meet thecondition

ni2

+nρ3

+ ni∞ +∑

other z

nz =2k

12

with non-negative integers n∗.

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To begin the more serious discussion, for 2k = 0, all orders of vanishing must be 0, since they are non-negative integers. Constants are obviously in M0. The trick is that, for a holomorphic modular form f ofweight 0, f(z) − f(zo) vanishes at zo for every zo ∈ H. Thus, f(z) is identically equal to f(zo), that is, isconstant.

For 2k = 2, there is no collection of orders of vanishing combining to give the required 2k/12 = 1/6, soM2 = {0}.

For 2k = 4, on one hand, the only way to get 4/12 = 1/3 is

0

2︸︷︷︸at i

+1

3︸︷︷︸at ρ

+ 0︸︷︷︸at i∞

+∑

other z

0 =4

12

On the other hand, we are granting ourselves that the holomorphic Eisenstein series E4 is in M4, so evidentlyE4(ρ) = 0, and the vanishing is just first-order. Given f ∈ M4, take zo ∈ H not in the Γ-orbit of ρ, andconsider

f2 = f − f(zo)

E4(zo)· E4

By design, f2 vanishes at zo:

f2(zo) = f(zo) −f(zo)

E4(zo)· E4(zo) = 0

Such vanishing can occur only for f2 identically zero, so f is a constant multiple of E4.

Similarly, for 2k = 6, 8, 10, there is only one way to satisfy the divisor relation:

1

2︸︷︷︸at i

+0

3︸︷︷︸at ρ

+ 0︸︷︷︸at i∞

+∑

other z 0 =6

12

0

2︸︷︷︸at i

+2

3︸︷︷︸at ρ

+ 0︸︷︷︸at i∞

+∑

other z 0 =8

12

1

2︸︷︷︸at i

+2

3︸︷︷︸at ρ

+ 0︸︷︷︸at i∞

+∑

other z 0 =10

12

and E2k ∈ M2k. The same argument as for M4 shows that every element of M6,M8,M10 is a constantmultiple of E6, E8, E10.

Things change at M12, since 12/12 = 1: there is no numerical obstacle to vanishing at i∞ and other points,in addition to the special points i and ρ. Still, E12 is present, and we are granting in advance that its Fourierexpansion is of the form

E12(z) = 1 · e2πi·0·z +∑n≥1

an e2πinz

Given f ∈M12 with Fourier expansion

f(z) =∑n≥0

bn e2πinz

substract a multiple of E12 to make the 0th Fourier coefficient 0: consider

f2(z) = f(z) − b0 · E12

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Thus, νf2(i∞) = 1, and f2 is a cuspform, by definition. The divisor relation shows that f2 has no otherzeros, unless by mischance f2 is identically 0.

To prove existence of a not-identically-zero cuspform of weight 12, note that E34 − E2

6 is weight 12, and has0th Fourier coefficient 0, so is a candidate. To show that E3

4 −E26 is not identically 0, recall from above that

E4(ρ) = 0 and does not vanish otherwise, while E6(i) = 0 and does not vanish otherwise. Thus, E34 − E2

6

cannot vanish at either ρ or i, so is not identically 0. Up to normalizing constant, ∆ = E34 − E2

6 .

By the divisor relation, ∆ only vanishes at i∞, and there to order 1. Now we will see that M12 = CE12 +C∆.Given f ∈M12, as before, subtract a multiple E12 to make the 0th Fourier coefficient of f2 = f − cE12 be 0.Then divide f2 by ∆, taking advantage of the fact that ∆ does not vanish in H, and vanishes only to firstorder at i∞. Thus, f2/∆ is in M0 = C, proving that f2 is a multiple of ∆, and M12 = CE12 + C∆.

Similarly, now that the non-zero cuspform ∆ is identified, a similar argument gives the structure of M2k, for2k ≥ 4 so that Eisenstein series converge. Namely, given f ∈ M2k, subtract a multiple of E2k to obtain acuspform of weight 2k, and then divide by ∆ to obtain a modular form of weight 2k − 12. This shows thatM2k = CE2k + ∆M2k−12, as claimed. ///

For present purposes, an isobaric polynomial P (X,Y ) ∈ C[X,Y ] (with weights 4, 6) is a polynomial withthe property that there is an integer 2k such that every monomial XaY b appearing has the property that4a+ 6b = 2k. This has the effect that P (E4, E6) is a modular form of weight 12.

[3.1.2] Corollary: Every holomorphic modular form for SL2(Z) is an isobaric polynomial in E4, E6.

Proof: The assertion is vacuously true for weight 0 since holomorphic modular forms of weight 0 areconstants. Holomorphic modular forms of weight 2 are all identically 0. At weights 4 and 6, all modularforms are multiples of the respective Eisenstein series.

At weight 8, the only modular form is E8, but also E24 has weight 8. Both have 0th Fourier coefficient 1, so

E8 = E24 . Similarly, E10 = E4 · E6.

We already showed that ∆ is a constant multiple of the isobaric polynomial E34 − E2

6 . Since E12 − E34 is a

cuspform of weight 12, it is a multiple of ∆, proving that E12 has an isobaric polynomial expression in termsof E4 and E6.

Given 12 < 2k ∈ 2Z, find non-negative integers a, b such that 4a+ 6b = 2k. Then E2k−Ea4Eb6 is a cuspform,and

E2k − Ea4Eb6∆

∈ M2k−12

By induction, E2k is an isobaric polynomial in E4, E6. Given f ∈M2k, subtract a multiple of E2k to producea cuspform f2, allowing division by ∆ to put f2/∆ in M2k−12, completing the induction. ///

[3.1.3] Corollary: For every weight 2k, the space of holomorphic cuspforms is finite-dimensional.

Proof: The space of cuspforms of weight 2k is ∆ ·M2k−12, and M2k−12 is cuspforms together with multiplesof E2k−12, for 2k − 12 ≥ 4. ///

[3.1.4] Remark: [Ramanujan 1916] conjectured that the nth Fourier coefficient τ(n) of ∆ satisfies

|τ(p)| ≤ 2p112 (for prime p)

andτ(mn) = τ(m) · τ(n) (for coprime m,n)

andτ(pn+1) = τ(p)τ(pn)− p11τ(pn−1) (for prime p)

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[Mordell 1917] prove the latter two properties, using operators systematically investigated in [Hecke 1937],

nowadays called Hecke operators. As noted earlier, [Deligne 1974] proved |cp| ≤ 2pk−12 for prime p and

weight 2k cuspforms, as consequence of his completion of proof of the Weil conjectures.

[3.1.5] Remark: In this context, the modular function or modular invariant j(z) is defined to be a constantmultiple of E3

4/∆, since now we know that ∆ 6= 0 in H, and that ∆ has a simple pole at i∞ and vanishes atρ since E4(ρ) = 0.

[3.1.6] Remark: Yes, there is some conflict with the notation that j can refer to the cocycle, as well as tothe modular function, but context usually clarifies.

4. Proof of divisor/dimension formula

This proof ofni2

+nρ3

+ ni∞ +∑

other z

nz =2k

12

is an application of the argument principle, exploiting the near-invariance of modular forms.

Proof: Let f be a not-identically-zero holomorphic modular form of weight 2k. Let

FT = {|z| ≥ 1, |Re(z)| ≤ 12 , Im(z) ≤ T}

be the truncation at height T of the standard fundamental domain F , and γ a path tracing its boundary.

On one hand, by the argument principle,∫γ

f ′(z)

f(z)dz = 2πi

∑z inside FT

νf (z)

In fact, points on the boundary itself require special treatment, especially the points i and ρ. Treatment ofthis is postponed to the end of the proof.

On the other hand, the individual pieces of the path integral nearly cancel each other out, except for somemanageable pieces, as follows.

The easiest part is that the integrals along the upward path along Re(z) = + 12 and downward path along

Re(z) = − 12 cancel each other, because f(z + 1) = f(z).

Let f(z) =∑n≥no

cn e2πinz, with cno

6= 0. That is, νi∞(f) = no. The path-integral along the top of ∂FT ,

from 12 + iT to − 1

2 + iT is an integral in the coordinate q = e2πinz around a circle: letting g(q) = f(z),∫ − 12

12

f ′(x+ iT )

f(x+ iT )dx =

∫ − 12 +iT

12 +iT

g′(q) · dqdzg(q)

dz =

∫C

g′(q)

g(q)dq

with C a circle of radius e−2πT at 0, traced clockwise. The Fourier expansion of f in z is a power seriesexpansion in q, so by the argument principle, and by the convention about νf (i∞),∫ − 1

2

12

f ′(x+ iT )

f(x+ iT )dx = −2πi · νf (i∞)− 2πi

∑z:Im(z)>T

νf (z)

The path from the cube-root of unity ρ to i is mapped by z → −1/z to that running backward from thesixth root of unity to i, but these do not quite cancel each other, because f is not invariant under z → −1/z.Rather, differentiating f(−1/z) = z2k · f(z) gives

f ′(−1/z) · 1

z2= 2kz2k−1f(z) + z2kf ′(z)

9


sof ′(−1/z) = 2kz2k+1f(z) + z2k+2f ′(z)

andf ′(−1/z)

f(−1/z)d(−1/z) =

2kz2k+1f(z) + z2k+2f ′(z)

z2kf(z)

dz

z2=

2k

z+f ′(z)

f(z)

Thus, the integral from the cube root of 1 to the sixth root of 1 cancel except for the −2k/z. Letting z = eit

as t goes from 23π to 1

2π,∫ 12π

23π

(f ′(z)f(z)

dz − f ′(−1/z)

f(−1/z)

)d(−1/z) =

∫ 12π

23π

−2k

e−itd(eit) =

∫ 12π

23π

−2ik dt = 2ik · π6

= 2πi · 2k

12

Thus, if there were no vanishing on the boundary, evaluating the integral around the truncated fundamentaldomain in two ways gives ∑

z Im(z)<T

νf (z) = −νf (i∞)−∑

z:Im(z)>T

νf (z) +2k

12

or

νf (i∞) +∑z∈F

νf (z) =2k

12

Now we consider points on the boundary of FT . Any vanishing on the top edge Im(z) = T can be avoidedby adjusting T slightly. Any vanishing on the vertical edges Re(z) = ± 1

2 can be easily accommodated byslightly deforming the contour γ inward on the left side Re(z) = − 1

2 to exclude a point zo with f(zo) = 0,and deforming the contour slightly outward on the right side Re(z) = 1

2 to include zo + 1. Similarly, for anypoint on the bottom part of the boundary, except for i and ρ, at which f vanishes, the left half of that arccan be deformed slightly inward, and the right half outward, to avoid the points. [5] Thus, the ordinaryargument principle is sufficient for these cases.

[4.1] Points i, ρ on the boundary

Unfortunately, there is no deformation of the contour to avoid the points i, ρ while counting order-of-vanishing. We first consider the situation at i.

To simplify the discussion, use the Cayley map z → z−i−iz+1 to convert the arc along |z| = 1 to a straight

line segment σ along the real axis, and replace f by its composition g with the inverse z → z+iiz+1 to the

Cayley map. This does not alter order-of-vanishing. In these coordinates modify σ traversing the interval[−a, a] left-to-right to include a small semi-circular detour along |z| = ε in the upper half-plane. That is, themodified path σε goes along the interval [−a,−ε] left-to-right, along the arc clockwise from −ε to +ε, andleft-to-right along the interval [ε, a].

For g(0) = 0, the logarithmic derivative g′/g has a simple pole at 0, with Laurent expansion

g′(z)

g(z)=

ν0(g)

z+ (holomorphic near 0)

By continuity, the limit as ε → 0+ of the integral of a holomorphic function along the modified paths σε isjust the integral along the segment σ. This leaves us explicit computation of∫

σε

dz

z=

∫ −ε−a

dt

t+

∫ 0

−π

d(εeit)

eit

∫ a

ε

dt

t= −(log a− log ε)− πi+ (log a− log ε) = −πi

[5] One might reasonably worry that there might be infinitely-many points near FT where f vanishes. However, the

compactness of any slightly larger region containing FT , and the holomorphy of f , assures that this cannot happen.

10


That is, the limit of the integrals over paths σε excluding 0 produces 12 · 2πi · νg(0). Thus, the corresponding

modification of the path around the boundary of FT gives − 12 · 2πi · νf (i).

The point ρ is treated similarly, with slight further complications. One way to describe the outcome is totreat ρ and ρ+ 1 separately, as follows. Here, unlike at i, we cannot completely convert the path near ρ intostraight line segments. Nevertheless, there is a well-defined angle to the boundary of F at ρ, namely, π/3.Modifying the path-integral along the boundary by indenting upward along a small arc of radius ε > 0, andtaking a limit as ε→ 0+, produces − 1

6 · 2πi · νf (ρ), rather than the full −2πi · νf (ρ). Similarly, the limit ofslightly-indented paths around ρ+ 1 produces another − 1

6 · 2πi · νf (ρ), noting that νf (ρ+ 1) = νf (ρ).

Thus, by integrating over the boundary of FT modified by indentations of radius ε at i and ρ, and takingthe limit as ε→ 0+, we obtain

νf (i∞) +∑z∈F

νf (z) = −νf (i)

2− νf (ρ)

3+

2k

12

Moving the suitably weighted orders of vanishing at i, ρ to the left-hand side gives the divisor/dimensionformula. ///

[4.1.1] Remark: The idea that path integrals essentially running directly through a simple pole can beconstrued as giving half the residue, or half the negative, depending on the direction of indentation, can belegitimized as in the discussion of i above. The further idea, applied above to ρ and ρ+1, that path integralsalong paths having a corner with angle θ at a simple pole, can be construed as producing − θ

2π of the residue,can likewise be legitimized. In all these cases, the underlying mechanism is that∫ θ2

θ1

d(εeit)

εeit=

∫ θ2

θ1

i dt = (θ2 − θ1)i (independent of ε > 0)

5. Fourier expansions of holomorphic Eisenstein series

[5.0.1] Theorem: For weight 2k ≥ 4, the holomorphic Eisenstein series

E2k(z) =∑

coprime c,d

1

cz + d

2k

has Fourier expansion

E2k(z) = 1 +(−2πi)2k

(2k − 1)! ζ(2k)

∑n≥1

σ2k−1(n) e2πinz

Before the important computation that determines the Fourier coefficients, two corollaries:

[5.0.2] Corollary: Given a modular form f(z) = co+∑n≥1 cn e

2πinz, the difference f−co ·E2k is a cuspform.

Proof: The leading Fourier coefficient of the Eisenstein series is 1, so the indicated subtraction exactlyannihilates the leading Fourier coefficient. ///

[5.0.3] Corollary: For weight 2k ≥ 4, the holomorphic Eisenstein series E2k(z) is bounded in the standardfundamental domain, so is a elliptic modular form in the strongest sense.

Proof: The absence of negative-index Fourier terms, and an easy estimate

σ2k−1(n) ≤∑

1≤`≤n

`2k−1 ≤ (n+ 1)2k � e2πn (as n→ +∞)

11


give

|E2k(z) � 1 +∑n≥1

e2πn e−2πny ≤ 1 +e−2πy

1− e−2πy

which is bounded for y ≥√

32 . ///

Proof: We directly compute the Fourier coefficients

cn = e2πny ·∫ 1

2

− 12

e−2πinx E∗2k(x+ iy) dx

of the renormalized Eisenstein series

E∗2k(z) = ζ(2k) · E2k(z) =∑

(c,d) 6=(0,0)

1

(cz + d)2k

First, the subsum over d 6= 0 with c = 0 is literally 2ζ(2k), and this is translation-invariant, so is part of the0th Fourier coefficient 0

Each subsum over d ∈ Z for fixed c 6= 0 is invariant under z → z + 1, so has a Fourier expansion, with nth

coefficient

e2πny ·∫ 1

2

− 12

e−2πinx∑d

1

(cz + d)2kdx

The integral is∫ 12

− 12

e−2πinx∑d

1

(cx+ d+ ciy)2kdx = c−2k

∫ 12

− 12

e−2πinx∑d

1

(x+ dc + iy)2k

dx

Aiming to unwind the sum-and-integral to have a simpler sum and an integral over R, rewrite∫ 12

− 12

e−2πinx∑d

1

(x+ dc + iy)2k

dx =

∫ 12

− 12

e−2πinx∑`∈Z

∑d mod c

1

(x+ `+ dc + iy)2k

dx

and replace x by x− `, to obtain

∑`∈Z

∫ 12 +`

− 12 +`

e−2πinx∑

d mod c

1

(x+ dc + iy)2k

dx =

∫Re−2πinx

∑d mod c

1

(x+ dc + iy)2k

dx

=∑

d mod c

∫Re−2πinx 1

(x+ dc + iy)2k

dx =∑

d mod c

e2πind/c

∫Re−2πinx 1

(x+ iy)2kdx

by replacing x by x − dc in each integral. Now neither c nor d appears inside the integral, while neither x

nor y appear in the sum.

The integral can be evaluated by residues, treating x itself as a complex variable, as follows. Fix y, theimaginary part of the original z. For n ≤ 0, the function e2πinx is rapidly decreasing as x moves intothe upper half-plane, so the indicated integral is the limit as R → +∞ of an integral left-to-right along[−R,R] and then along an arc of a circle of radius R in the upper half-plane. This picks up residues ofx→ e−2πinx/(x+ iy)2k in the upper half-plane: there are none, so these Fourier coefficients are 0.

For n > 0, the integral can be evaluated by residues, using an arc of a circle in the lower half-plane, pickingup −2πi times the residue of x→ e−2πinx/(x+ iy)2k at −iy, namely,

−2πi

(2k − 1)!·( ∂∂x

)2k−1

e−2πinx∣∣∣x=−iy

=−2πi

(2k − 1)!· (−2πin)2k−1 · e−2πny =

(2πi)2k

(2k − 1)!n2k−1 e−2πny

12


That is, ∫Re−2πinx 1

(x+ iy)2kdx =

(2πi)2k

(2k − 1)!n2k−1 e−2πny (for n ≥ 1)

0 (for n ≤ 0)

The sum over d mod c is a sum of the character d → e2πind/c over the finite abelian group Z/c. Thecancellation lemma says this sum is 0 unless the character is trivial, in which case it is the cardinality of thegroup, namely, |c|. The character is trivial if and only c|n. Thus,

∑d mod c

e2πind/c =

|c| (for c|n)

0 (otherwise)

In summary, the 0th Fourier coefficient is 2ζ(2k), the negative-index Fourier coefficients are 0, and for n > 1the Fourier coefficient is

∑c|n

1

c2k· |c| × (2πi)2k

(2k − 1)!n2k−1 (for n > 1)

As c runs over positive and negative divisors of n, so does n/c, and the last expression can be simplifiedsomewhat by doing so:

∑c|n

c2k

n2k

∣∣∣nc

∣∣∣× (2πi)2k

(2k − 1)!n2k−1 =

2(2πi)2k

(2k − 1)!

∑0<c|n

c2k−1

Often the sum of `th powers of positive divisors of an integer n is denoted σ`(n), so the Fourier expansion ofthe Eisenstein series can be written

2ζ(2k) · E2k(z) = 2ζ(2k) +2(2πi)2k

(2k − 1)!

∑n≥1

σ2k−1(n) e2πinz

and

E2k(z) = 1 +(2πi)2k

(2k − 1)! ζ(2k)

∑n≥1

σ2k−1(n) e2πinz

as claimed. ///

[5.0.4] Corollary: E24 = E8, E4E6 = E10, and E4E10 = E6E8 = E14.

Proof: In dimensions 8, 10, 14 there are no holomorphic modular forms other than the correspondingEisenstein series, and the leading Fourier coefficients are always 1. ///

[5.0.5] Corollary: Granting that ζ(2k) is a rational multiple of π2k, the Fourier coefficients of Eisensteinseries are rational numbers. ///

[5.0.6] Remark: The rationality of the Fourier coefficients of holomorphic Eisenstein series is significantin later developments. The following corollaries are slightly frivolous examples of proving number-theoreticidentities by relations among automorphic forms. Nevertheless, more serious results do use the same proofmechanism of which these simple examples are prototypes.

13


[5.0.7] Corollary: For positive integers N ,

σ7(N) = 2 · 7! ζ(8)

3! (2πi)4 ζ(4)σ3(N) +

7! ζ(8)

(3!)2 ζ(4)2

∑m+n=N

σ3(m)σ3(n) (with m,n ≥ 1)

σ9(N) =9! ζ(10)

3! (2πi)6 ζ(4)σ3(N) +

9! ζ(10)

5! (2πi)4 ζ(6)σ5(N) +

9! ζ(10)

3! 5! ζ(4) ζ(6)

∑m+n=N

σ3(m)σ5(n) (m,n ≥ 1)

Proof: The first identity comes from equating the Fourier coefficients of E24 = E8. A similar one arises from

E4E6 = E10. Fourier expansions without negative-index terms multiply as∑m≥0

am e2πimz ·

∑n≥0

bm e2πinz =

∑N≥0

( ∑m+n=N

am · bn)e2πiNz

From E24 = E8, noting that the 0th Fourier coefficients do not quite fit into the general pattern, for N ≥ 1,

equating the N th coefficients of E24 and E8 gives

(2πi)8

7! ζ(8)σ7(N) = 2 · (2πi)4

3! ζ(4)σ3(N) +

( (2πi)4

3! ζ(4)

)2 ∑m+n=N

σ3(m)σ3(n)

Rearranging,

σ7(N) = 2 · 7! ζ(8)

3! (2πi)4 ζ(4)σ3(N) +

7! ζ(8)

(3!)2 ζ(4)2

∑m+n=N

σ3(m)σ3(n)

The second computation is entirely analogous. ///

[5.0.8] Remark: Also, these frivolous relations completely determine ζ(4), ζ(6), ζ(8), and ζ(10), by lookingat the relations for N = 1, 2. And since there are no cuspforms of weight 14, also ζ(14) is determined.

More generally, from [Gunning 1959/62] p. 55, Ramanujan proved the following, but with a worse error term,since Hecke’s estimate on Fourier coefficients of cuspforms was not available. That is, in general, E2k · E2`

is probably not exactly E2k+2`, but it misses only by a cuspform:

[5.0.9] Corollary: For 2k ≥ 4 and 2` ≥ 4 and N ≥ 1,

σ2k+2`−1(N) =(2k + 2`− 1)! ζ(2k + 2`)

(2πi)2` (2k − 1)! ζ(2k − 1)σ2k−1(N) +

(2k + 2`− 1)! ζ(2k + 2`)

(2πi)2k (2`− 1)! ζ(2`)σ2`−1(N)

+(2k + 2`− 1)! ζ(2k + 2`)

(2k − 1)! (2`− 1)! ζ(2k) ζ(2`)

∑m+n=N

σ2k−1(m) · σ2`−1(m) + O(n2k+2`

2 ) (with m,n ≥ 1)

Proof: Up to a cuspform, E2k ·E2` = E2k+2`. Equating the N th Fourier coefficients and multiplying throughby (2k+ 2`− 1)! ζ(2k+ 2`)/(2πi)2k+2` gives the identity, with the big-O term arising from Hecke’s estimateon the Fourier coefficients of the cuspform = E2k+2` − E2k · E2`. ///

[5.0.10] Remark: Of course, for weights 2k + 2` among 8, 10, 14, there are no cuspforms, and the errorterm is exactly 0.

14


6. Automorphic L-functions

[6.1] Euler product attached to ∆(z) A little later, we will prove two of the conjectures of Ramanujanproven by Mordell, in a form applicable to all holomorphic cuspforms of for SL2(Z). First, we examine theimplications for Dirichlet series.

With ∆(z) = 1 · e2πinz +∑n≥1 τ(n) e2πinz the unique cuspform of weight 12 for SL2(Z), the associated

Dirichlet series is

L(s,∆) =∑n≥1

τ(n)

ns

The Hecke estimate |τ(n)| � n122 shows that the series for L(s,∆) is absolutely convergent for Re(s) > 12

2 +1.

The weak multiplicativity τ(mn) = τ(m) · τ(n) for coprime m,n is equivalent to an Euler factorization ofL(s,∆):

L(s,∆) =∑n≥1

τ(n)

ns=

∏p prime

(1 +

τ(p)

ps+τ(p2)

p2s+τ(p3)

p3s+ . . .

)The more peculiar relation

τ(pn+1) = τ(p)τ(pn)− p11τ(pn−1) (for prime p, for n ≥ 1)

gives a recursion for the τ(pn): to simplify notation, let X = p−s, observe that powers of p−s do multiplylike powers of X, and

1 · τ(pn+1)Xn+1 − τ(p)X · τ(pn)Xn + p11X2 · τ(pn−1)Xn−1 = 0 (for n ≥ 1)

For n ≥ 1, the left-hand side of the last equality is the Xn+1th term in(1− τ(p)X + p11X2

)(1 + τ(p)X + τ(p2)X2 + τ(p3)X3 + . . .

)The constant component of the latter product is 1. That is,(

1− τ(p)X + p11X2)(

1 + τ(p)X + τ(p2)X2 + τ(p3)X3 + . . .)

= 1

That is, (1− τ(p)

ps+p11

p2s

)(1 +

τ(p)

ps+τ(p2)

p2s+τ(p3)

p3s+ . . .

)= 1

and

1 +τ(p)

ps+τ(p2)

p2s+τ(p3)

p3s+ . . . =

1

1− τ(p)

ps+p11

p2s

Thus, ∑n

τ(n)

ns=∏p

1

1− τ(p)

ps+p11

p2s

This Euler product factorization partly justifies calling∑nτ(n)ns an automorphic L-function.

Further, the discriminant of the quadratic equation

X2 − τ(p)X + p11 = 0

15


is τ(p)2 − 4p11. From the expression of ∆ as a real constant multiple of E64 − E2

6 , τ(p) ∈ R. Thus, theroots occur in complex conjugate pairs exactly when Ramanujan’s conjectured, Deligne’s proven, inequality|τ(p)| < 2p

112 holds.

[6.1.1] Remark: We have given Hecke’s proof of |τ(p)| � p122 , but will not attempt to follow [Deligne 1974]

to prove |τ(p)| < 2p112 .

[6.1.2] Remark: We will show below that the space of weight 2k holomorphic cuspforms for SL2(Z) has abasis of cuspforms f(z) =

∑n≥1 cn e

2πinz with cn = 1 and whose associated Dirichlet series

L(s, f) =∑n≥1

cnns

have Euler product factorizations

L(s, f) =∑n≥1

cnns

=∏p

1

1− cpps

+p2k−1

p2s

Having an Euler product partly justifies calling L(s, f) an automorphic L-function attached to f . The Hecke

estimate cn � n2k2 proves absolute convergence of L(s, f) for Re(s) > 2k

2 + 1.

[6.2] Analytic continuation and functional equation A holomorphic cuspform f(z) =∑n≥1 cn e

2πinz

of weight 2k for SL2(Z) has associated Dirichlet series

L(s, f) =∑n≥1

cnns

whether or not this has an Euler product.

[6.2.1] Remark: Merely copying Fourier coefficients to coefficients of a Dirichlet series accomplishes little,without further analytic features.

We do know that f is rapidly decreasing as y → +∞, and that y2k2 · |f(z)| is bounded on H, so |f(z)| � y−k

as y → 0+. Thus, for Re(s) > k we have absolute convergence of the Mellin transform∫ ∞0

ys f(iy)dy

y

In that range, ∫ ∞0

ys f(iy)dy

y=

∫ ∞0

ys∑n

cn e−2πny dy

y=∑n

cn

∫ ∞0

ys e−2πny dy

y

=∑n

cn(2πn)s

·∫ ∞

0

ys e−ydy

y= (2π)−s Γ(s)

∑n

cnns

= (2π)−s Γ(s)L(s, f)

[6.2.2] Claim: (2π)−s Γ(s)L(s, f) has an analytic continuation to an entire function, satisfying

(2π)−(2k−s) Γ(2k − s)L(2k − s, f) = (−1)2k2 · (2π)−s Γ(s)L(s, f)

[6.2.3] Remark: This integral representation of L(s, f), with Gamma-factor (2π)−s Γ(s) to complete it,plays the role for L(s, f) as did the integral representation of the completed ζ(s) in terms of θ(z).

16


[6.2.4] Remark: With hindsight, seeing that the functional equation is with respect to s ↔ 2k − s, acontemporary choice would be to renormalize to have a functional equation s↔ 1− s, as we describe below.The latter convention is not universal.

Proof: The rapid decay of a cuspform f(x+ iy) as y → +∞ assures that part of the integral is entire:∫ ∞1

ys f(iy)dy

y= entire

Meanwhile, using the automorphy condition with z → −1/z,∫ 1

0

ys f(iy)dy

y=

∫ 1

0

ys (iy)−2k · f(−1/iy)dy

y= (−1)

2k2

∫ 1

0

ys−2k · f(−1/iy)dy

y

= (−1)2k2

∫ ∞1

y2k−s · f(iy)dy

y= entire

Thus,

(2π)−s Γ(s)L(s, f) =

∫ ∞1

ys f(iy)dy

y+ (−1)

2k2

∫ ∞1

y2k−s f(iy)dy

y= entire

and the behavior under s↔ 2k − s is clear. ///

[6.2.5] Remark: To translate so that the functional equation is s↔ 1− s, instead of the natural but naivenormalization above, put

L(s, f) =∑n

cn/n2k−1

2

ns=∑n

cn

ns+2k−1

2

The corresponding integral representation becomes

(2π)−s−2k−1

2 Γ(s+2k − 1

2)L(s, f) =

∫ ∞0

ys−12

(f(iy) · y 2k

2

) dyy

Then one might further divide through by a constant so that the extra constant power of π disappears,giving functional equation

(2π)−(1−s) Γ(1− s+2k − 1

2)L(1− s, f) = (−1)k · (2π)−s Γ(s+

2k − 1

2)L(s, f)

[6.2.6] Remark: Thus, we have shown that automorphic L-functions L(f, s) arising from holomorphiccuspforms for SL2(Z) have analytic continuations and functional equations. Euler product factorizations areproven below.

17


7. Hecke operators and Euler products

Following [Hecke 1937]’s general application of ideas of [Mordell 1917], we will eventually prove that thereis a basis for the space of holomorphic cuspforms of weight 2k for SL2(Z) whose associated Dirichlet serieshave Euler products of the same shape as that attached to Ramanujan’s ∆.

This will be accomplished using the operators on modular forms introduced in this section, the Heckeoperators.

The viewpoint here is the simplest possible treatment of this example of Hecke operators, but has manydeficiencies, among them the difficulty of comparing this naive notion of Hecke operators to other objects instandard mathematics. Hecke operators will be reconsidered later in a broader, more explanatory context.For the moment, the treatment is ad hoc, but as simple as possible.

One vague point that will be made more clearly later is that Hecke operators are analogues of analyticconditions such as holomorphy.

[7.1] Sublattices of Zz + Z and Hecke operators For a positive integer n, by the structure theorem

for submodules of finitely-generated free modules over principal ideal domains [6] such as Z, the index-msub-lattices Λ′ of Λ = Zz + Z have bases which are linear combinations az + b, cz + d of z and 1, with(

a bc d

)(z1

)=

(az + bcz + d

)(with integers a, b, c, d and ad− bc = m)

Let

Hm = {(a bc d

): (integers a, b, c, d and ad− bc = m)}

Although the matrices in Hm producing index-m sublattices are not in SL2(R), they are still in the positive-determinant component

GL+2 (R) = {real

(a bc d

): det

(a bc d

)> 0}

of the general linear group

GL2(R) = {real

(a bc d

): det

(a bc d

)6= 0}

The linear fractional transformation action of GL+2 (R) still preserves H, so the images γz corresponding to

index-m sublattices of Zz + Z are in H.

In a naive normalization, the mth Hecke operator on holomorphic modular forms of weight 2k is

f(z) −→∑

γ∈SL2(Z)\Hm

f(γz) · (cz + d)−2k (where each γ is

(a bc d

))

The effect of the Hecke operators on Fourier coefficients, computed below, gives motivation to normalize bya constant, setting

Tmf(z) = m2k−1 ·∑

γ∈SL2(Z)\Hm

f(γz) · (cz + d)−2k

[6] The structure theorem for submodules N of finitely-generated free modules M over a principal ideal domain

shows that there is an R-basis m1, . . . ,mr for M and elementary divisors d1| . . . |dr such that d1m1, . . . , drmr is an

R-basis for N , dropping any dimi where di = 0.

18


To check that Tm does stabilize the space of weight 2k modular forms for SL2(Z), we must verify holomorphy,the automorphy condition, and the growth condition. We first check that the sum is well-defined, and is finite,by identifying convenient representatives.

[7.1.1] Remark: There is not universal agreement about normalization of the Hecke operators Tm byconstants depending on m. The immediate benefit of the second normalization here is insufficient to declareit optimal in the long run.

[7.2] Standard representatives for SL2(Z)\Hm Certainly Hm is stable under left and rightmultiplications by Γ = SL2(Z).

Given

(a bc d

)∈ Hm with gcd(a, c) = g, there are coprime s, t ∈ Z such that sa+ tc = g. The coprime s, t

admit u, v ∈ Z such that su− tv = gcd(s, t) = 1, so γ =

(s tv u

)is in Γ, and

γ

(a bc d

)=

(s tv u

)(a bc d

)=

(sa+ tc ∗∗ ∗

)=

(g b′

c′ d′

)(for some b′, c′, d′ ∈ Z)

Left multiplication by γ does not alter the gcd of the left column, so this gcd is still g, and g|c′. Thus,(1 0

−c′/g 1

)is in Γ, and

(1 0

−c′/g 1

)γ

(a bc d

)=

(1 0

−c′/g 1

)(g b′

c′ d′

)=

(g b′′

0 d′′

)The determinant has also not been altered by multiplications in Γ, so g · d′′ = m. Left multiplications by(

1 r0 1

)with r ∈ Z put b′′ into any chosen set of representatives for Z mod d′′. Thus, Γ\Hm has standard

representatives

Γ\Hm ≈ {(a b0 d

): a · d = m, a, d > 0, b mod d}

The irredundancy of these representatives follows from the fact that, given δ ∈ Hm, the upper-left entry ofthe corresponding standard representative is the gcd of the left column of δ. Then the lower right entry iscompletely determined. Further, if

γ

(a b0 d

)=

(a b′

0 d

)for γ ∈ Γ, then

γ =

(a b′

0 d

)(a b0 d

)−1

=

(a b′

0 d

)(1/a −b/ad0 1

d

)=

(1 (b′ − b)/d0 1

)For this to be in Γ requires that d|(b− b′), so b = b′ mod d, proving irredundancy.

[7.3] Hecke operators map modular forms to modular forms Before anything else, observe thatextending the automorphy factor

j(

(a bc d

), z) = cz + d (for

(a bc d

)∈ GL+

2 (R) and z ∈ H)

even on the larger group GL+2 (R) does not disturb the cocycle relation

j(gh, z) = j(g, hz) j(h, z)

19


This is easily checked: with g =

(a bc d

)and h =

(A BC D

),

j(g, hz) j(h, z) = (cAz +B

Cz +D+d)(Cz+D) = c(Az+B)+d(Cz+D) = (cA+dC)z+(cB+dD) = j(gh, z)

Thus, the associativity of the weight 2k action on GL+2 (R) on functions on H still holds:(

f |2kg)|2kh = f |2k(gh) (for g, h ∈ GL+

2 (R))

This gives associativity of the weight 2k action of GL+2 (R) on functions on H:(

f |2k(gh))

(z) = f((gh)z) j(gh, z)−2k = f(g(hz)) j(g, hz)−2k j(h, z)−2k

=(f |2kg(hz)

)j(h, z)−2k =

(f |2kg)|2kh

)(z)

Thus, Tmf is well-defined, since changing representatives for Γ\Hm does not change the summands: givenrepresentatives δj for Γ\Hm, and any γj ∈ Γ,

m1−2kTmf =∑j

f |2kδj =∑j

(f |2kγj)|2kδj =∑j

f |2k(γjδj)

since f |2kγj = f is the automorphy condition on f .

Since Γ\Hm is finite, the image Tmf of a holomorphic modular form f of weight 2k for SL2(Z) is a finitesum of holomorphic functions, so is holomorphic.

To see that Tmf still satisfies the automorphy condition,

m1−2k(Tmf)|2kγ =∑

δ∈Γ\Hm

(f |2kδ)|2kγ =∑

δ∈Γ\Hm

f |2k(δγ)

Right multiplication of Hm by Γ stabilizes Hm, and merely changes one set of representatives of Γ\Hm foranother. Thus, ∑

δ∈Γ\Hm

f |2k(δγ) =∑

δ∈Γ\Hm

f |2kδ = Tmf

[7.3.1] Remark: The individual summands entering the expression for the Hecke operator are not level onemodular forms, but are modular forms for congruence subgroups.

[7.4] Effect of Hecke operators on Fourier coefficients Now we find the motivation, such as it is, for

the normalizing constant m2k−1 in the definition of Tm. [7]

Hecke operators have an understandable effect on Fourier coefficients:

[7.4.1] Theorem:

Tmf(z) =∑N

( ∑0<a|m, a|N

a2k−1 cmNa2

)e2πiNz

[7] Again, this normalization of Tm is for short-term convenience.

20


Proof: This is a direct computation, using the standard representatives and the cancellation lemma. Forf(z) =

∑n cn e

2πinz, using the standard representatives for Γ\Hm,

m1−2kTmf(z) =∑a,b,d

f∣∣∣2k

(a b0 d

)(z) (with ad = m, a, d > 0, b mod d)

=∑a,b,d

f(az + b

d

)d−2k =

∑n

cn∑a,b,d

e2πin az+bd d−2k =

∑n

cn∑a,d

d−2ke2πinaz/d∑b

e2πin bd

The inner sum over b is the sum of the character b→ e2πinb/d over Z/d. By the cancellation lemma, this is0 for non-trivial character, and is the order of the group, d, for trivial character. That character is trivial ifand only if d|n. Thus, replacing n by dn = mn/a,

Tmf(z) =∑a,d

d1−2k∑n

cnd e2πi(nd)az/d =

∑a,d

d1−2k∑n

cnd e2πinaz

Using d = m/a,

m1−2kTmf(z) = m1−2k∑

0<a|m

a2k−1∑n

cmnae2πinaz

Regrouping to identify the coefficients of the resulting Fourier expansion, replace na by N , and cancel them1−2k. ///

[7.4.2] Theorem: For m,m′ relatively prime, Tmm′ = Tm ◦ Tm′ , while

Tp` ◦ Tp = Tp`+1 + p1−2k · Tp`−1 = Tp ◦ Tp`

The Hecke operators commute, that is Tm ◦ Tm′ = Tm′ ◦ Tm. In particular, Tm ◦ Tm′ = Tm′m for coprimem,m′.

Proof: The case of coprime m,m′ can be seen from the effect on Fourier coefficients. Write c(n) for the nth

Fourier coefficient. The N th Fourier coefficient of Tm(Tm′f) is

∑a|m, a|N

a2k−1 · (mNa2th

Fourier coefficient of Tm′f) =∑

a|m, a|N

a2k−1 ·( ∑a′|m′, a′|mN

a2

a′2k−1 c(m′mNa2

a′2

))

=∑

a|m, a|N, a′|m′, a′|mNa2

a2k−1 · a′2k−1 · c(mm′Na2 · a′2

)For m,m′ coprime, a′|ma

Na is equivalent to a′|Na , or aa′|N . Also, for m,m′ coprime, the positive divisors d

of mm′ uniquely factor as the product gcd(d,m) · gcd(d,m′) of m and m′. Thus, letting d = aa′ in the sumin the right-hand side of the last equality,

(N th Fourier coefficient of TmT′mf) =

∑d|mm′, d|N

d2k−1 · c(mm′N

d2

)(for m,m′ coprime)

The right hand side is the N th Fourier coefficient of Tmm′f , as asserted.

The composition Tp ◦ Tp` with ` ≥ 1 is easily understood via the special representatives

Γ\Hp ≈ {(p 00 1

),

(1 b0 p

), with b mod p}

21


Γ\Hp` ≈ {(p`−t b

0 pt

), with b′ mod pt, 0 ≤ t ≤ `}

Multiplying these representatives, Tp` ◦ Tp is a sum over two types of products of special representatives:(p 00 1

)(p`−t b′

0 pt

)=

(p`+1−t pb′

0 pt

)and

(1 b0 p

)(p`−t b′

0 pt

)=

(p`−t b′ + ptb

0 pt+1

)The latter representatives are (

p(`+1)−(t+1) b′ + ptb0 pt+1

)and b′ + ptb runs through Z mod pt+1, demonstrating that these are among the special representatives forΓ\Hp`+1 . The only representative for Γ\Hp`+1 not obtained this way is(

p`+1 00 1

)which is obtained from among the first products of special representatives, with t = 0. The first type ofproducts with t ≥ 1 are all scalar multiples of special representatives for Γ\Hp`−1 :(

p`+1−t pb′

0 pt

)= p ·

(p`−t b′

0 pt−1

)but b′ mod pt, rather than mod pt−1, produces the representatives for Γ\Hp`−1 p times each. The extrascalar factor p does not alter the map given by the matrix, but it inserts an extra p−2k in the weight 2kaction on modular forms. Thus,

Tp` ◦ Tp = Tp`+1 + p1−2k · Tp`−1 (on weight 2k modular forms)

Similarly, composing in the opposite direction, Tp ◦ Tp` gives products of special representatives(p`−t b′

0 pt

)(p 00 1

)=

(p`+1−t b′

0 pt

)and

(p`−t b′

0 pt

)(1 b0 p

)=

(p`−t pb′ + p`−tb

0 pt+1

)This time, the first type of product gives all the special representatives for Γ\Hp`+1 except(

1 b′

0 p`+1

)(with b′ mod p`+1)

The latter is produced by the second type of product with t = `. As before, for t < ` the second type ofproduct has a scalar factor p that can be pulled out:(

p`−t pb′ + p`−tb0 pt+1

)= p ·

(p(`−1)−t b′ + p(`−1)−tb

0 pt

)(with t < `)

The map b′ → b′ + p(`−1)−tb merely permutes b′ mod pt, so the sum over b mod p repeats the samerepresentative p times. The scalar multiple adds an extra factor p−2k to the automorphy factor, so again

Tp ◦ Tp` = Tp`+1 + p1−2k · Tp`−1 (on weight 2k modular forms)

This gives the formula for the composition, and shows commutativity. By induction, every Tp` is a polynomialin Tp, so these p-power Hecke operators commute with each other.

The general case of commutativity reduces to the coprime case and the prime power case. ///

22


[7.4.3] Corollary: A weight 2k modular form f with Fourier coefficients cn, which is an eigenfunction forTm with eigenvalue λm, has mth Fourier coefficient determined from λm and c1:

λm · c1 = cn

A simultaneous eigenfunction for all Hecke operators has c1 6= 0. Normalizing c1 = 1, the Fourier coefficientsof a simultaneous eigenfunction are weakly multiplicative:

cmn = cm · cn (for coprime m,n, when c1 = 1)

Proof: Taking N = 1 in the expression above for Fourier coefficients of Tmf restricts the sum over a|m anda|N to a = 1, and then ∑

0<a|m, a|1

a2k−1 cm·1a2

= cm

as claimed. If c1 = 0, then all Fourier coefficients are 0 and f itself is identically zero.

For simultaneous eigenfunction f , since c1 6= 0, we can take c1 = 1 without loss of generality. Then theproperty Tm ◦ Tn = Tmn of Hecke operators for coprime m,n gives λmλn = λmn, and

cm · cn = λm · λn = λmn = cmn

as claimed. ///

[7.4.4] Corollary: For a weight 2k cuspform f(z) =∑n≥1 cn e

2πinz which is a simultaneous eigenfunctionfor all Hecke operators, normalized so that c1 = 1, the associated L-function has an Euler product

L(s, f) =∑n

cnns

=∏p

1

1− cpp−s + p2k−1p−2s

Proof: Since c1 = 1, the nth Fourier coefficient is the eigenvalue λn of Tn. The weak multiplicativity gives

L(s, f) =∏p

(1 +

cpps

+cp2

p2s+cp3

p3s+)

Then the recursive relation Tp ◦ Tp` = Tp`+1 + p1−2kTp`−1 gives

cp`+1 = λp`+1 = λpλp` − p1−2kλp`−1 = cpcp` − p1−2kcp`−1

which sums the series for the pth Euler factor, just as for ∆. ///

[7.4.5] Corollary: For weights 2k = 12, 16, 18, 20, where the space of holomorphic cuspforms for SL2(Z)is one-dimensional, there is a unique cuspform f with leading Fourier coefficient 1, and the associated L-function L(s, f) =

∑n cnn has the expected Euler product expansion.

Proof: Since the space is one-dimensional, the unique (up to constant multiples) cuspform is unavoidablyan eigenfunction for all the Hecke operators. ///

[7.4.6] Corollary: The 0th Fourier coefficient of Tmf , with Fourier coefficients cn of f , is

0th Fourier coefficient of Tmf =( ∑a<a|m

a2k−1)· c0

23


Thus, the Hecke operators stabilize the space of cuspforms of weight 2k for SL2(Z). ///

[7.4.7] Corollary: The Eisenstein series E2k are simultaneous eigenfunctions for all Hecke operators.

Proof: The 0th Fourier coefficient of E2k is 1, and the previous corollary computes that the 0th Fouriercoefficient of TmE2k is σ2k−1(m). Thus, if E2k is to be an eigenfunction for Tm, the eigenvalue is σ2k−1(m),and the nth Fourier coefficient must be multiplied by σ2k−1(m).

For n > 0, the nth Fourier coefficient of E2k is a uniform multiple C2k ·σ2k−1(n) of σ2k−1, where the constantis C2k = (2πi)2k/(2k − 1)! ζ(2k). For N ≥ 1, the theorem gives

N th Fourier coefficient of TmE2k = C2k

∑0<a|m, a|N

a2k−1∑

0<d|mNa2

d2k−1

We must show that the latter is σ2k−1(m) · C2kσ2k−1(N), that is, that∑0<a|m, a|N

a2k−1∑

0<d|mNa2

d2k−1 = σ2k−1(m) · σ2k−1(N)

It suffices to treat m and N being powers of a fixed prime p, since σ2k−1 has the weak multiplicativity [8]

σ2k−1(r · s) = σ2k−1(r) · σ2k−1(s) (for coprime r, s)

Of course,

σ2k−1(p`) =∑

0≤i≤`

(pi)2k−1

Thus, with m = pu and N = pv, we must show that∑0≤i≤u, i≤v

(pi)2k−1∑

0≤j≤u+v−2i

(pj)2k−1 =∑

0≤i≤u

(pi)2k−1 ·∑

0≤j≤v

(pj)2k−1

The clutter can be reduced by letting X = p2k−1, and we need to prove∑0≤i≤u, i≤v

Xi∑

0≤j≤u+v−2i

Xj =∑

0≤i≤u

Xi ·∑

0≤j≤v

Xj

Summing the finite geometric series, and taking u ≤ v without loss of generality, the left-hand side is

∑0≤i≤u

Xi 1−Xu+v−2i+1

1−X=

1

1−X

(1−Xu+1

1−X−Xu+v+1 1−X−(u+1)

1−X−1

)

=1

1−X

(1−Xu+1

1−X+Xu+v+2 1−X−(u+1)

1−X

)=

1−Xu+1 +Xu+v+2 −Xv+1

(1−X)2=

1−Xu+1

1−X· 1−Xv+1

1−Xwhich is the right-hand side. Thus, the Eisenstein series is a simultaneous eigenfunction for all the Heckeoperators. ///

[7.4.8] Remark: The Hecke operators do not tell anything new about the Eisenstein series, since we hadto use details about the sum-of-powers-of-divisors function σ2k−1(n) to prove the eigenfunction property.In contrast, for weights 12, 16, 18, 20 where the space of cuspforms is one-dimensional, the above discussion

[8] The weak multiplicativity of σ`(n) is not hard to verify: for m,n relatively prime, a positive divisor d of mn

uniquely factors as a product of positive divisors gcd(m, d) and gcd(n, d) of m and n.

24


imparts significant information about the Fourier coefficients. For weights higher than 20, there is furthernon-trivial content in proof that spaces of cuspforms have bases of simultaneous eigenfunctions, as will beaccomplished in the following section.

[7.4.9] Remark: A motivation to further renormalize Hecke operators may be found in the renormalizationof L(s, f) to have functional equation s→ 1− s rather than s→ 2k − s. That is, if we decide to write

L(s, f) =∑n

cn/n2k−1

2

ns

so that the functional equation is s→ 1− s, then we might want

λm · c1 = cn/n2k−1

2

rather than λm · c1 = cn. That is, we would want to normalize the Hecke operators as

Tmf =1

m2k−1

2

·m2k−1∑

δ∈Γ\Hm

f |2kδ = m2k−1

2

∑δ∈Γ\Hm

f |2kδ

[7.4.10] Remark: The glaring problem remains, to prove that spaces of holomorphic cuspforms of weights2k for SL2(Z) have bases of simultaneous eigenfunctions for all Hecke operators Tm. This will be resolvedby showing that the Hecke operators are self-adjoint with respect to a natural inner product on spaces ofcuspforms, demonstrated below.

8. Petersson inner product, Poincare series

We showed that the space of holomorphic cuspforms of weight 2k for SL2(Z) is finite-dimensional. We nowsee that it has a natural inner product.

[8.1] SL2(R)-invariant integral on H An SL2(R)-invariant integral on compactly-supported,

continuous, complex-valued functions [9] f on H should be given by

f −→∫H

f(z) ϕ(x, y) dx dy

for positive, real-valued ϕ giving the invariance∫H

f(gz) ϕ(x, y) dx dy =

∫H

f(z) ϕ(x, y) dx dy (for all g ∈ SL2(R), for all f ∈ Coc (H))

That is, ϕ(x, y) should compensate for change-of-measure that occurs in the change of variables replacing zby g−1z in the integral. Invariance under translation and dilations(

1 t0 1

)(x+ iy) −→ (x+ t) + iy and

(t 00 t−1

)(x+ iy) −→ t2x+ it2y

already completely determine ϕ(x, y), as follows. The translation invariance implies that ϕ(x, y) isindependent of x. The dilation invariance gives

ϕ(y) dx dy = ϕ(t2y) d(t2x) d(t2y) = ϕ(t2y) t4 dx dy

[9] The collection of continuous, compactly-supported, complex-valued functions on a topological space X is usually

denoted Coc (X).

25


Letting y = 1 gives ϕ(t2) = t−4. Thus, already this degree of invariance gives an essentially unique candidatedx dyy2 for an SL2(R)-invariant measure, and SL2(R)-invariant integral

f −→∫H

f(z)dx dy

y2(for f ∈ Coc (H))

From a naive viewpoint, we could hope for the best, and check: [10]

[8.1.1] Claim: The measure dx dy/y2 on H is SL2(R)-invariant.

[8.1.2] Remark: Again, this argument is only a place-holder, since far better ways to understand thisinvariance will be exhibited later. However, this argument is easy and possibly reassuring.

Proof: We already have invariance under P =

(∗ ∗0 ∗

)⊂ G = SL2(R). It is economical to note that the

P and w =

(0 −11 0

)generate G. Indeed, we have the Bruhat decomposition [11]

G = P t PwP

Indeed, the big cell PwP is exactly the collection of

(a bc d

)with c 6= 0. To prove G-invariance of the given

measure, it suffices to prove invariance under

w : x+ iy −→ −1

x+ iy=

−xx2 + y2

+ iy

x2 + y2

The jacobian determinant is

det

∂

∂x

−xx2 + y2

∂

∂x

y

x2 + y2

∂

∂y

−xx2 + y2

∂

∂y

y

x2 + y2

= det

−1

x2 + y2+

2x2

(x2 + y2)2

−2xy

(x2 + y2)2

2xy

(x2 + y2)2

1

x2 + y2− 2y2

(x2 + y2)2

= (x2 + y2)−2 · det

(−(x2 + y2) + 2x2 −2xy

2xy (x2 + y2)− 2y2

)= (x2 + y2)−2 ·

(− (x2 + y2)2 + 2(x2 + y2)2 − 4x2y2 + 4x2y2

)= 1

giving invariance. ///

[8.1.3] Remark: Such a computation should not be misunderstood as suggesting anything miraculous aboutthis outcome. As noted earlier, in fact the full G-invariance of a P -invariant measure on G/K is inevitable,from a somewhat more sophisticated viewpoint.

[10] In reality, from a slightly more sophisticated viewpoint, we could almost-immediately obtain the invariant measure

on H ≈ SL2(R)/SO(2,R) from Haar measure on P and K in an Iwasawa decomposition G = PK. Here, P would be

the group generated by translations and dilations, and K = SO(2,R). Nevertheless, we give an explicit computation

to see that the translation-and-dilation-invariant measure is fully invariant.

[11] For G = SL2(R), this decomposition was known long before F. Bruhat’s work, but it was only in the 1950s

and 1960s that Bruhat and others saw the relevance of such decompositions even in more general semi-simple and

reductive Lie groups and p-adic groups, for representation theory.

26


[8.2] Petersson inner products Because Im(γz) = Im(z)/|cz + d|2, for two elliptic modular forms f, gof weight 2k, the function f(z)g(z)|y|2k is Γ = SL2(Z)-invariant, thus descends to the quotient Γ\H.

Let F be the standard fundamental domain for SL2(Z). The Petersson inner product on holomorphicmodular forms of weight 2k for SL2(Z) can be written

〈f, g〉 =

∫Γ\H

f(z) g(z)dx dy

y2=

∫F

f(z) g(z)dx dy

y2

We have seen that cuspforms are exponentially decreasing as y → +∞, so the integral over F is absolutelyconvergent.

[8.2.1] Remark: There is an issue of what is being integrated-over. Since f(z)g(z)|y|2k is Γ-invariant,integrating it over any fundamental domain for Γ will give the same result. In fact, it is better to describean intrinsic way to integrate functions on the quotient Γ\H, without choosing or knowing any fundamentaldomain. Nevertheless, for the moment, it suffices to think of the integral on Γ\H as being an integral over afundamental domain such as F .

[8.3] Holomorphic Poincare series The holomorphic Eisenstein series of weight 2k are explicit, andcomplement the cuspforms inside the space of holomorphic modular forms of weight 2k for Γ = SL2(Z).Holomorphic Poincare series are more complicated constructions producing cuspforms.

In terms of the operator (f |2kγ)(z) = (cz + d)−2kf(γz), the holomorphic Eisenstein series can be expressed

E2k =∑

γ∈Γ∞\Γ

1|2kγ = 12

∑coprime c,d

1

(cz + d)2k(with Γ∞ =

(∗ ∗0 ∗

)⊂ Γ = SL2(Z))

Using a more rapidly decreasing Γ∞-invariant function e2πinz in place of 1, but still invariant under Γ∞, thenth Poincare series Pn(z) with n ≥ 1 is

Pn(z) =∑

γ∈Γ∞\Γ

e2πinz|2kγ = 12

∑coprime c,d

e2πin az+bcz+d

(cz + d)2k(with a, b such that ad− bc = 1)

Since ∣∣∣ e2πinγ(z)

(cz + d)2k

∣∣∣ =e−2πn·Im(γz)

|cz + d|2k≤ 1

|cz + d|2k(for y > 0)

absolute convergence of Poincare series, uniformly on compacts, follows from that of the Eisenstein series.

[8.3.1] Claim: The holomorphic Poincare series Pn with n ≥ 1 are cuspforms.

Proof: Holomorphic cuspforms of weight 2k are rapidly decreasing as y → +∞, while from the Fourierexpansion and easy estimates, E2k(x + iy) goes to 1 as y → +∞. Thus, it suffices to show that Poincareseries go to 0 as y →∞.

The subsum over terms with c 6= 0 is readily estimated in absolute value:

∑coprime c,d; c 6=0

∣∣∣ e2πin az+bcz+d

(cz + d)2k

∣∣∣ ≤ ∑c 6=0, d∈Z

1

|cz + d|2k=

∑c 6=0, d∈Z

1

((cx+ d)2 + (cy)2)k

A sum of this form is dominated by the obvious integral:∑d∈Z

1

((cx+ d)2 + (cy)2)k�∫R

dt

(t2 + (cy)2)k=

1

|cy|2k−1

∫R

dt

(t2 + 1)k� 1

|cy|2k−1

∫R

dt

(t2 + 1)k

27


The sum over c 6= 0 is bounded by a constant multiple of 1/y2k−1. This goes to 0 as y → +∞ for 2k > 2.

The subsum with c = 0 is just e2πinz, which goes to 0 as y → +∞, for n > 0. Thus, Pn(z) is a holomorphiccuspform. ///

Every linear functional λ on a finite-dimensional inner-product space V is of the form λ(v) = 〈v, vλ〉 for

unique vλ ∈ V . [12] The linear functionals∑n cne

2πinz −→ cn are essentially given by Poincare series:

[8.3.2] Claim: For cuspform f(z) =∑n≥1 cn e

2πinz, and mth Poincare series Pm(z),

〈f, Pm〉 = m1−2k · cm ·Γ(2k − 1)

(4π)2k−1

Proof: This is a direct computation:

〈f, Pm〉 = =

∫Γ\H

f(z)( ∑γ∈Γ∞\Γ

e2πimz|2kγ)y2k dx dy

y2=

∫Γ∞\H

f(z) e2πimz y2k dx dy

y2

by unwinding: for any irredundant representatives {γ} for Γ∞\Γ, and for any fundamental domain F for Γ,⋃γ γF is a fundamental domain for Γ∞, since⋃

δ∈Γ∞

δ( ⋃γ∈Γ∞\Γ

γF)

=⋃δ∈Γ∞

⋃γ∈Γ∞\Γ

δγF =⋃γ∈Γ

γF = H

With a simple choice of fundamental domain for Γ∞, namely, {z : 0 ≤ x ≤ 1, 0 < y < +∞}, the integral is∫ ∞y=0

∫ 1

x=0

∑n

cne2πinz ·e−2πimx e−2πmy y2k dx dy

y2=

∫ ∞y=0

∫ 1

0

∑n

cn e−2πny e2πinx ·e−2πimx e−2πmy y2k dx dy

y2

=

∫ ∞y=0

cm e−2πmy e−2πmy y2k dy

y2= cm

∫ ∞y=0

y2k−1 e−4πmy dy

y= cm ·

Γ(2k − 1)

(4πm)2k−1

by mutual orthogonality of distinct exponentials in x. ///

[8.4] Self-adjointness of Hecke operators

The Hecke operators Tm on weight 2k holomorphic cuspforms are provably self-adjoint with respect to thePetersson inner product. They commute with each other, so there is an orthonormal basis of simultaneousHecke eigenvectors, whose L-functions have Euler products.

The seemingly elementary but unexplanatory arguments for the self-adjointness will not be given here.Rather, a more modern [13] proof is given in a supplement. The idea can be introduced as follows.

[12] This linear algebra fact is readily verified: given a non-zero linear functional λ on a finite-dimensional inner-

product space V , there is vo orthogonal to kerλ and such that λ(vo) = 1. For v ∈ V , v − λ(v) · vo is in the kernel of

λ, so is orthogonal to vo. Then

0 = 〈v − λ(v) · vo, vo〉 = 〈v, vo〉 − λ(v)〈vo, vo〉

shows that λ(v) = 〈v, v1〉 for v1 = vo/〈vo, vo〉.

[13] Here more modern merely means post-1960.

28


Since Tmn = Tm ◦ Tn for coprime m,n, it suffices to show that the p-power Hecke operators Tp` are self-adjoint for prime powers p`. As noted earlier, a complication in the definition of Hecke operators is that thesummands in Tp`f are not modular forms of level one, that is not for Γ(1) = SL2(Z), but mostly of higherlevel p`, that is, only for the congruence subgroup

Γ(p`) = {(a bc d

)=

(1 00 1

)mod p`}

For ` unbounded, there is no bound on the level of the summands. This suggests considering the largerspace inner-product space V of weight 2k holomorphic cuspforms for all prime-power congruence subgroupsΓ(p`). In the supplement, we show that the group GL2(Qp), with Qp the p-adic rational numbers, actsreasonably on V , and the p-power Hecke operators can be rewritten as integral operators whose adjoints areeasily determined by a change-of-variables.

Such a discussion also introduces the role of representation theory of the groups GL2(Qp) in the theory ofmodular forms. In that context, an invariant measure on GL2(Qp) is needed, and is easily provided. In thatcontext, we also clarify the invariant measure on H as arising from an invariant measure on GL2(R). See thesupplement for this discussion.

Bibliography

[Deligne 1974] P. Deligne, La conjecture de Weil, I, Inst. Hautes Etudes Sci. Publ. Math. 43 (1974),273-307.

[Fricke-Klein 1897] R. Fricke, F. Klein, Vorlesungen uber die Theorie der automorphen Funktionen. ErsterBand: Die gruppentheoretischen Grundlagen, Teubner, Leipzig, 1897.

[Fricke-Klein 1912] R. Fricke, F. Klein, Vorlesungen uber die Theorie der automorphen Functionen. ZweiterBand: Die funktionentheoretischen Ausfuhrungen und die Anwendungen. 1. Lieferung: Engere Theorie derautomorphen Funktionen, Teubner, Leipzig, 1897.

[Gunning 1959/62] R.C. Gunning, Lectures on Modular Forms, notes by A. Brumer, Princeton Univ. Press,1962.

[Hecke 1937] E. Hecke, Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwick-lungen, I, II, Math. Ann. 114 (1937) no. 1, 1-28, 316-351.

[Mordell 1917] L. J. Mordell, On Mr. Ramanujan’s empirical expansions of modular functions, Proc. Camb.Phil. Soc. 19 (1917), 117-124.

[Petersson 1930] H. Petersson, Theorie der automorphen Formen beliebiger reeler Dimension und ihreDarstellung durch eine neue Art Poincarescher Reihen, Math. Ann. 103 (1930, 369-436.

[Poincare 1882] H. Poincare, Theorie des groupes fuchsiens, Acta Math. 1 (1882), 1-62.

[Poincare 1883] H. Poincare, Memoire sur les groupes kleineens, Acta Math. 3 (1883), 49-92.

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