This monograph studies decompositions of the Jacobian of a smooth projective curve, induced by the action of a finite group, into a product of abelian subvarieties. The authors give a general theorem on how to decompose the Jacobian which works in many cases and apply it for several groups, as for groups of small order and some series of groups. In many cases, these components are given by Prym varieties of pairs of subcovers. As a consequence, new proofs are obtained for the classical bigonal and trigonal constructions which have the advantage to generalize to more general situations. Several isogenies between Prym varieties also result.
Author(s): Herbert Lange, Rubí E. Rodríguez
Series: Lecture Notes in Mathematics, 2310
Publisher: Springer
Year: 2022
Language: English
Pages: 260
City: Cham
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
4.2.3 The Irreducible Rational Representations of S3
4.2.4 Curves with an S3-action: Decomposition of J"0365J
4.2.5 The Degree of the Isogeny ψ
5 Covers of Degree 4
5.1 Cyclic Covers of Degree 4
5.2 The Klein Group of Order 4
5.2.1 Decompositions of J"0365J
5.2.2 Degrees of Some Isogenies
5.2.3 Proof of Proposition 5.2.5
5.3 The Dihedral Group of Order 8
5.3.1 Ramification and Genera
5.3.2 Decompositions of J"0365J
5.3.3 An Isogeny Coming from an Action of a Quotient Group
5.3.4 A Generalization of the Bigonal Construction
5.4 The Bigonal Construction
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
5.8 Another Generalization of the Trigonal Construction
5.8.1 Statement and Preparations
5.8.2 The Trigonal Construction
5.8.3 The Degree of γ in the General Principally Polarized Case
5.8.4 The Non-Principally Polarized Case
6 Some Series of Group Actions
6.1 Cyclic Covers
6.1.1 Notation and First Results
6.1.2 Cyclic Covers of Degree p
6.1.3 Cyclic Covers of Degree p2
6.1.4 Cyclic Covers of Degree pq
6.2 Covers with Dihedral Group Action
6.2.1 The Irreducible Rational Representations of Dn
6.2.2 Curves with Dp-Action: Ramifications and Genera
6.2.3 Curves with Dp-Action: Decompositions of J"0365J
6.2.4 The Degree of the Isogeny φ
6.2.5 Curves with D2α-Action, α≥2
6.2.6 Curves with a D2p-Action: Ramification and Genera
6.2.7 Curves with D2p-Action: Decomposition of J"0365J
6.2.8 Some Isogenies Between Prym Subvarieties
6.3 Semidirect Products of a Group of Order 3 by Powers of K4
6.3.1 The Group Gn
6.3.2 The Diagram of Subcovers of a Curve with Gn-Action
6.3.3 A Jacobian Isogenous to the Product a Big Number of Jacobians
7 Some Special Groups and Complete Decomposability
7.1 The Alternating Group A5
7.1.1 The Irreducible Rational Representations of A5
7.1.2 Ramification and Genera
7.1.3 Decompositions of J"0365J
7.1.4 Some Isogenies Between Prym Varieties of Subcovers
7.2 Groups with Schur Index Larger Than One
7.2.1 The Group of Hamiltonian Quaternions
7.2.2 A Group of Order 24
7.3 Completely Decomposable Jacobians
7.3.1 The Theorem of Ekedahl-Serre
7.3.2 Examples
7.3.3 Intermediate Covers
Bibliography
Index
