| Description |
1 online resource (xiii, 251 pages) : illustrations |
| Series |
Lecture notes in mathematics ; volume 2310 |
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Lecture notes in mathematics (Springer-Verlag) ; 2310.
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| 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 |
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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 |
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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 |
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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 |
| Summary |
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 |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Description based upon print version of record |
| Subject |
Jacobians.
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Geometry, Algebraic.
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Curves, Algebraic.
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Geometría algebraica
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Curvas algebraicas
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Jacobianos
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Curves, Algebraic
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Geometry, Algebraic
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Jacobians
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| Form |
Electronic book
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| Author |
Rodríguez, Rubí E., 1953- author.
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| ISBN |
9783031101458 |
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3031101456 |
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