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Table of Contents
Intro
Preface
Acknowledgements
Contents
Notations
1 Introduction
2 Preliminaries and Basic Results
2.1 Line Bundles on Abelian Varieties
2.2 Polarized Abelian Varieties
2.3 Endomorphisms of Abelian Varieties
2.4 The Weil Form on K(L)
2.5 Symmetric Idempotents
2.6 Abelian Subvarieties of a Polarized Abelian Variety
2.6.1 The Principally Polarized Case
2.6.2 The Case of an Arbitrary Polarization
2.7 Poincaré's Reducibility Theorem
2.8 Complex and Rational Representations of Finite Groups
2.9 The Isotypical and Group Algebra Decompositions
2.9.1 Generalities
2.9.2 Induced Action on the Tangent Space
2.10 Action of a Hecke Algebra on an Abelian Variety
3 Prym Varieties
3.1 Finite Covers of Curves
3.1.1 Definitions and Elementary Results
3.1.2 The Signature of a Galois Cover
3.1.3 The Geometric Signature of a Galois Cover
3.2 Prym Varieties of Covers of Curves
3.2.1 Definition of Prym Varieties
3.2.2 Polarizations of Prym Varieties
3.2.3 The Degrees of the Decomposition Isogeny
3.2.4 Degrees of Isogenies Arising from a Decomposition of f: C""0365C →C
3.3 Two-Division Points of Prym Varieties of Double Covers
3.4 Prym Varieties of Pairs of Covers
3.5 Galois Covers of Curves
3.5.1 Jacobians and Pryms of Intermediate Covers
3.5.2 Isotypical and Group Algebra Decompositions of Intermediate Covers
3.5.3 Decomposition of the Tangent Space of the Prym Variety Associated to a Pair of Subgroups
3.5.4 The Dimension of an Isotypical Component
4 Covers of Degree 2 and 3
4.1 Covers of Degree 2
4.2 Covers of Degree 3
4.2.1 Cyclic Covers of Degree 3
4.2.2 Non-cyclic Covers of Degree 3: The Galois Closure
5.4.1 Definition and First Properties
5.4.2 Determination of the Bigonal Construction in the Non-Galois Case
5.4.3 The Bigonal Construction over C =P1
5.4.4 Pantazis' Theorem
5.5 The Alternating Group of Degree 4
5.5.1 Ramification and Genera
5.5.2 Decompositions of J
5.5.3 A Generalization of the Trigonal Construction
5.6 The Trigonal Construction for Covers with Group A4
5.7 The Symmetric Group S4
5.7.1 Ramification and Genera
5.7.2 Decomposition of J""0365J
5.7.3 Isogenies Arising from Actions of Subgroups of S4
5.7.4 An Isogeny Arising from the Action of a Quotient of S4.
Preface
Acknowledgements
Contents
Notations
1 Introduction
2 Preliminaries and Basic Results
2.1 Line Bundles on Abelian Varieties
2.2 Polarized Abelian Varieties
2.3 Endomorphisms of Abelian Varieties
2.4 The Weil Form on K(L)
2.5 Symmetric Idempotents
2.6 Abelian Subvarieties of a Polarized Abelian Variety
2.6.1 The Principally Polarized Case
2.6.2 The Case of an Arbitrary Polarization
2.7 Poincaré's Reducibility Theorem
2.8 Complex and Rational Representations of Finite Groups
2.9 The Isotypical and Group Algebra Decompositions
2.9.1 Generalities
2.9.2 Induced Action on the Tangent Space
2.10 Action of a Hecke Algebra on an Abelian Variety
3 Prym Varieties
3.1 Finite Covers of Curves
3.1.1 Definitions and Elementary Results
3.1.2 The Signature of a Galois Cover
3.1.3 The Geometric Signature of a Galois Cover
3.2 Prym Varieties of Covers of Curves
3.2.1 Definition of Prym Varieties
3.2.2 Polarizations of Prym Varieties
3.2.3 The Degrees of the Decomposition Isogeny
3.2.4 Degrees of Isogenies Arising from a Decomposition of f: C""0365C →C
3.3 Two-Division Points of Prym Varieties of Double Covers
3.4 Prym Varieties of Pairs of Covers
3.5 Galois Covers of Curves
3.5.1 Jacobians and Pryms of Intermediate Covers
3.5.2 Isotypical and Group Algebra Decompositions of Intermediate Covers
3.5.3 Decomposition of the Tangent Space of the Prym Variety Associated to a Pair of Subgroups
3.5.4 The Dimension of an Isotypical Component
4 Covers of Degree 2 and 3
4.1 Covers of Degree 2
4.2 Covers of Degree 3
4.2.1 Cyclic Covers of Degree 3
4.2.2 Non-cyclic Covers of Degree 3: The Galois Closure
5.4.1 Definition and First Properties
5.4.2 Determination of the Bigonal Construction in the Non-Galois Case
5.4.3 The Bigonal Construction over C =P1
5.4.4 Pantazis' Theorem
5.5 The Alternating Group of Degree 4
5.5.1 Ramification and Genera
5.5.2 Decompositions of J
5.5.3 A Generalization of the Trigonal Construction
5.6 The Trigonal Construction for Covers with Group A4
5.7 The Symmetric Group S4
5.7.1 Ramification and Genera
5.7.2 Decomposition of J""0365J
5.7.3 Isogenies Arising from Actions of Subgroups of S4
5.7.4 An Isogeny Arising from the Action of a Quotient of S4.