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Index
abelian variety 24, 169, 186 Abel-Jacobi map 75, 149, 161,
162, 166 Adams operations 82, 94 adequate equivalence 179 admissible
epimorphism 88 metric 144 modular unit 105 monomorphism 88
Arakelov divisor 144 Arithmetic Progressions 9
cap product 73, 95 character 19
algebraic Hecke 209 Chern 74, 80, 99 Dirichlet 9 Hecke 35, 36, 209
Chern class 80, 96 class number 7, 28 Class Number Formula 11, 12,
107 Class Number Problem
GauB 8,34 classifying space 90 cohomology
absolute Hodge 155, 159 continuous etale 164
Deligne-Beilinson 62, 83, 128
motivic 55, 56, 83, 101, 104, 109, 128
parabolic motivic 219 complex
absolute Hodge 103 Bloch-Suslin 102, 109, 110,
112 cone 63 Deligne 62 Deligne-Beilinson 64 Goncharov 115 Hodge 157 homological 69 polarizable Hodge 157
complex multiplication 33, 130 conductor 9, 15, 20, 32, 35, 59,
210 cone construction 158 comveau
filtration by - 152 Conjecture( s)
Beilinson 21, 129, 138, 148, 153, 160
Beilinson-Bloch 123, 164 Beilinson-Jannsen 161 Birch & Swinnerton-
Dyer 40, 56, 128, 148,
234
203 Deligne 127 Fermat 5 Grothendieck 153, 175, 176,
177,178 Hard Lefschetz 176 Hasse-Weil 33 Hodge 83, 153, 161 Hodge 1)- 154 Morde1l48 Shimura-Tanyama-Weil 46,
204 Shafarevich 49 Standard 33, 59 Tate 49,' 137, 153, 166, 188,
199 Weil29 Zagier 117
critical point 125 cup product 66 current 68, 144, 154 curve
elliptic 23, 123, 127, 129 modular 8, 39, 140
cusp 27, 204 cycle
absolute Hodge 169, 174 Hirzebruch-Zagier 139 Hodge 169
cycle map 75, 102, 165, 196
de Rham conjugation 122 dilogarithm 110, 116, 118 discriminant 11, 24, 27
Eisenstein series 131 Eisenstein-Kronecker-Lerch 132
endomorphism ring 26 eta-function
Weierstra6 30
Index
Euler-Poincare characteristic 79 Euler product 9, 59 exact category 88 exact functor 88
fibre functor 190 filtration
gamma 95, 149 Hodge 53, 54, 189 weight 53, 103,
155, 186, 189, 195 Frobenius 17, 57 functional equation 9, 11, 20,
32, 58 fundamental class 73, 84,
162, 196, 200
good proper cover 197 Green's function 42 Grossencharacter 209 group
arithmetic Chow 144 Bloch 113, 115, 117 Chow 75, 136 decomposition 17 Galois 14 generalized Chow 149 homotopy 90 ideal class 7, 28 inertia 18, 58 Mordell-Weil 40 Tate-Shafarevich 43, 50 Weil-Chatelet 44
Hasse principle 43
Index
Heegner point 8, 205 height 48 Hilbert modular surface 138 Hodge structure 143
mixed 155 Q-rational 186
homology absolute Hodge 159 continuous f-adic 165 Deligne 69, 70 motivic 56
homotopy property 92
intermediate Jacobian 65, 162 intersection
arithmetic 143 intersection number 41, 145, 147 isogeny 25
dual 26 trivial 26
Karoubian envelope 88, 171, 180 Klein function 42 Kronecker dimension 198
L-function 58 Artin 17 Dirichlet 9 Hasse-Weil 30, 50 Hecke 34, 37, 38, 209
lambda-ring 82 Langlands Program 19 linear variety 223
metrized line bundle 144 model
Neron 33 regular 124
235
regular arithmetic 143 motive 21, 53, 54, 63, 125, 172,
174 I-motive 53 Artin 128, 186 Deligne 174 Dirichlet 215 effective 172, 186 Lefschetz 125, 172 mixed 53, 104, 157, 191 Tate 54, 168, 173
nerve 89 node 27 number field
cyclotomic 6 imaginary quadratic 6
origin 23
pairing 67 intersection 145
period 41, 211 Beilinson 125 Deligne 127
plus-construction 85 Poincare duality 125 Poincare duality theory 71, 82,
128, 153, 189, 193 polylogarithm 110, 111, 116 prime
irregular 7 regular 7
principal triviality 73, 196 projection formula 92, 179
Q-structure 106
ramification index 26
236
rank 105, 173 realization 125, 168, 186
Betti 168 crystalline 192 de Rham 168 geometric 90 integral mixed 189 i-adic 168 pure 190
reduction 27 additive 27 bad 27 good 27 multiplicative 27 semi-stable 27 unstable 27
regulator 13, 107, 214 Beilinson 109 Borel 107 elliptic 41
regulator map 67, 104, 106, 107, 108, 135, 160, 200
Beilinson 108, 129 Borel 106
Riemann Hypothesis 10 root number 10, 21
sigma-function WeierstraB 42
spectral sequence Hochschild-Serre 165 localization exact 120 Quillen 92
support 152 symbol
Jacobi 15 tame 120
Tamagawa number 51 Tate module 187, 188
graded 188
Tate twist 125, 168 Theorem
Index
Artin's Reciprocity 19 Atiyah-Singer Index 80 Borel 107 Borel-Beilinson 109 Deligne 169 Deninger 132 Deuring 37, 38, 133 Dirichlet's Unit 12, 106, 135 Fermat's Last 5, 47 Gillet-R-R 100, 200 Goncharov 116 Gross-Zagier 3, 4, 203, 206 Grothendieck-R-R 63 Hirzebruch-R-R 79 Jannsen 163 Kronecker-Weber 15 Kummer 7 Mordell-Weil 39, 148 Purity 93 Ramakrishnan 141 Zagier 113
torsor 44, 84 trilogarithm 113, 115
volume 106, 129
WeierstraB equation 24 global minimal 27 minimal 27
weight 126, 194, 209
Z -function 28 Weil28
zeta-function 28 Dedekind 10 Riemann 10, 127 WeierstraB 42
Edited by Klas Diederich
Band D 1: H. Kraft: Geometrische Methoden in der Invariantentheorie
Band D 2: J. Bingener: Lokale Modulraume in der analytischen Geometrie 1
Band D 3: J. Bingener: Lokale Modulraume in der analytischen Geometrie 2
Band D4: G. Barthel/F. Hirzebruch/T. Hofer: Geradenkonfigurationen und Aigebraische Flachen*
Band D5: H. Stieber: Existenz semiuniverseller Deformationen in der komplexen Analysis
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"A publication of the Max-Planck-Institut fOr Mathematik, Bonn
From Gauss to Painleve A modern theory of special functions
Edited by Katsunori Iwasaki, Hironobu Kimura, Shin Shimomura, and Masaaki Yoshida
1991. XII, 347 pp. (Aspects of Mathematics, Vol. E 16; ed. by Klas Diederich) Hardcover. ISBN 3-528-06355-6 ISSN 0179-2156
FromGa Pain ve
to
This book - dedicated to Tosihusa Kimura on the occasion of his sixtieth birthday - gives an introduction to the modern theory of special functions. It focuses on the nonlinear Painleve differential equation and its solutions, the so-called Painleve functions. It contains modern treatments of the Gauss hypergeometric differential equation, monodromy of second order Fuchsian equations and nonlinear differential equations near singular points. The book starts from an elementary level requiring only basic notions of
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Complex Analysis Dedicated to H. Grauert
Proceedings of the International Workshop 1990 Edited by Klas Diederich (Ed.)
1991. X, 341 pp. (Aspects of Mathematics, Vol. E 17; ed. by KJas Diederich) Hardcover. ISBN 3-528-06413-7
Klas Diederich I EdJ
Compl x Analysis
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This volume contains the Proceedings of the International Workshop "Complex Analysis", which was held from February 12-16, 1990, in Wuppertal (Germany) in honour of H. Grauert, one of the most creative mathematicians in Complex-Analysis of this century. In complete accordance with the width of the work of Grauert the book contains research notes and longer articles of many important mathematicians from all areas of Complex Analysis (Altogether there are 49 articles in the volume). Some of the main subjects are: Cauchy-Riemann Equations with estimates,
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