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Chapter 6Complex multiplication in highergeneraWe trusted in the God who created the integersand created what we call arithmetic invariants.Arithmetic Variety of Moduli for Genus TwoJun-Ichi Igusa [139]Contents6.1 Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.1.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . 1616.1.2 Theta functions and Riemann forms . . . . . . . . . . . . . . . . . . . 1616.1.3 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.1.4 Picard variety and polarizations . . . . . . . . . . . . . . . . . . . . . 1646.1.5 Homomorphisms and the Rosati involution . . . . . . . . . . . . . . . 1676.2 Class groups and units . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.2.2 Computation of the Picard and unit groups . . . . . . . . . . . . . . . 1706.2.3 Multiplication of fractional i<strong>de</strong>als by finite idèles . . . . . . . . . . . . 1716.3 Complex multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.3.1 CM field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.3.2 Reflex field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.3.3 CM abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.3.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.5 Dual abelian variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.6 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.3.7 The main theorems of complex multiplication . . . . . . . . . . . . . . 1796.4 Class polynomials for genus 2 . . . . . . . . . . . . . . . . . . . . . . 1806.4.1 Jacobian variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.4.2 Igusa invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.4.3 Igusa class polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.4.4 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
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2012-ENST-003EDITE de ParisDoctorat
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Jean-Pierre Flori: Boolean function
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AbstractThe core of this thesis is
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AcknowledgmentsVladimir. — Quand
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ContentsList of symbols and notatio
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Contentsxv4 Efficient characterizat
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List of figures1.1 The filter model
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List of symbols and notationGeneral
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List of symbols and notationxxil(t)
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List of symbols and notationxxiiidi
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List of symbols and notationxxvu
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2 Introduction1 Mathematics and cry
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4 Introductiongiving more general b
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Chapter 1Boolean functions incrypto
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1.1. Cryptographic criteria for Boo
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1.2. Families of Boolean functions
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20 Chapter 2. On a conjecture about
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Part IIBent functions and pointcoun
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96 Chapter 3. Bent functions and al
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98 Chapter 3. Bent functions and al
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100 Chapter 3. Bent functions and a
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Page 217 and 218: Bibliography[1] Milton Abramowitz a
Page 219 and 220: Bibliography 193[28] Reinier Martij
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Page 223 and 224: Bibliography 197[87] Andreas Enge a
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Page 227 and 228: Bibliography 201[147] Kiran Sridhar
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Bibliography 209[270] Gerard van de
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IndexAAbelian variety . . . . . . .
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Index 213Division polynomial . . .
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Index 215SSemi-bent function . . .
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218 Résumé longEstragon. — Alor
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220 Résumé long⊕ ⊕ ⊕s i+L
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222 Résumé long1.2 D’une conjec
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224 Résumé longAfin d’utiliser
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226 Résumé longSon graphe est rep
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228 Résumé longLa caractérisatio
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230 Résumé longThéorème 2.4 (Th
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232 Résumé longprécédent. Dans
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234 Résumé longProposition 2.13.
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236 Résumé longFigure 8 - Un tore
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238 Résumé longAlgorithme 1 - Cal
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240 Résumé longThéorème 3.8 ([2
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