John Cremona / Joan-Carles Lario / Jordi Quer / Kenneth Ribet (eds.)

Modular Curves and Abelian Varieties

Herausgegeben:Cremona, John; Lario, Joan-Carles; Quer, Jordi; Ribet, Kenneth

• Gebundenes Buch

Produktbeschreibung

This book presents lectures from a conference on "Modular Curves and Abelian Varieties" at the Centre de Recerca Matemàtica (Bellaterra, Barcelona). The articles in this volume present the latest achievements in this extremely active field and will be of interest both to specialists and to students and researchers. Many contributions focus on generalizations of the Shimura-Taniyama conjecture to varieties such as elliptic Q-curves and Abelian varieties of GL_2-type. The book also includes several key articles in the subject that do not correspond to conference lectures.

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It would be difficult to overestimate the influence and importance of modular forms, modular curves, and modular abelian varieties in the development of num ber theory and arithmetic geometry during the last fifty years. These subjects lie at the heart of many past achievements and future challenges. For example, the theory of complex multiplication, the classification of rational torsion on el liptic curves, the proof of Fermat's Last Theorem, and many results towards the Birch and Swinnerton-Dyer conjecture all make crucial use of modular forms and modular curves. A conference was held from July 15 to 18, 2002, at the Centre de Recerca Matematica (Bellaterra, Barcelona) under the title "Modular Curves and Abelian Varieties". Our conference presented some of the latest achievements in the theory to a diverse audience that included both specialists and young researchers. We emphasized especially the conjectural generalization of the Shimura-Taniyama conjecture to elliptic curves over number fields other than the field of rational numbers (elliptic Q-curves) and abelian varieties of dimension larger than one (abelian varieties of GL2-type).

Produktdetails

• Progress in Mathematics 224
• Verlag: Birkhäuser / Birkhäuser Basel / Springer, Basel
• Artikelnr. des Verlages: 978-3-7643-6586-8
• 2004
• Seitenzahl: 304
• Erscheinungstermin: 23. Februar 2004
• Englisch
• Abmessung: 241mm x 160mm x 24mm
• Gewicht: 650g
• ISBN-13: 9783764365868
• ISBN-10: 3764365862
• Artikelnr.: 12500516

Autorenporträt

John Cremona, Nottingham University, UK / Joan-Carles Lario, Universitat Politècnica de Catalunya, Barcelona, Spain / Jordi Quer, Barcelona, Spain / Kenneth Ribet, University of California, Berkeley, USA

Inhaltsangabe

Stable reduction of modular curves.- On p-adic families of automorphic forms.- ?-curves and abelian varieties of GL2-type from dihedral genus 2 curves.- The old subvariety of J0(NM).- Irreducibility of Galois actions on level 1 Siegel cusp forms.- On elliptic K-curves.- ?-curves and Galois representations.- On the local behaviour of ordinary modular Galois representations.- Arithmetic of ?-curves.- Serre's conjecture for mod 7 Galois representations.- Pairings in the arithmetic of elliptic curves.- Explicit parametrizations of ordinary and supersingular regions of X0(Pn).- Elliptic ?-curves with complex multiplication.- Abelian varieties over ? with large endomorphism algebras and their simple components over ?.- Abelian varieties over ? and modular forms.- Shimura curves embedded in Igusa's threefold.- Shafarevich-Tate groups of nonsquare order.