Decomposition of Jacobians by Prym varieties / / Herbert Lange, Rubí E. Rodríguez

• Record Nr.: UNINA9910632490603321
• Autore: Lange H (Herbert), <1943->
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 3-031-10145-6
• Descrizione fisica: 1 online resource (261 pages)
• Collana: Lecture Notes in Mathematics ; ; v.2310
• Disciplina: 905
• Soggetti: Curves, Algebraic; Corbes algebraiques; Geometria algebraica; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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 -- 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.