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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

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Band D4: G. Barthel/F. Hirzebruch/T. Hofer: Geradenkonfigurationen und Aigebraische Flachen*

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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

differential equations, function theory and group theory. Graduate students should be able to work with the text.

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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,

q-convexity, CR structures, deformation theory, envelopes of holo­morpy, function algebras, complex group . actions, Hodge theory, instantons, Kahler geometry, Lefschetz theorems, holomorphic map­pings, Nevanlinna theory, complex singularities, twistor theory, unifor­mization.

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