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Modular Curves and Abelian Varieties : Progress in Mathematics - John Cremona

Modular Curves and Abelian Varieties

Progress in Mathematics

By: John Cremona (Editor), Joan-Carles Lario (Editor), Jordi Quer (Editor), Kenneth Ribet (Editor)

Hardcover Published: 23rd February 2004
ISBN: 9783764365868
Number Of Pages: 289

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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).

Stable reduction of modular curvesp. 1
On p-adic families of automorphic formsp. 23
Q-curves and abelian varieties of GL[subscript 2]-type from dihedral genus 2 curvesp. 45
The old subvariety of J[subscript 0](NM)p. 53
Irreducibility of Galois actions on level 1 Siegel cusp formsp. 75
On elliptic k-curvesp. 81
Q-curves and Galois representationsp. 93
On the local behaviour of ordinary modular Galois representationsp. 105
Arithmetic of Q-curvesp. 125
Serre's conjecture for mod 7 Galois representationsp. 141
Pairings in the arithmetic of elliptic curvesp. 151
Explicit parametrizations of ordinary and supersingular regions of X[subscript 0](p[superscript n])p. 165
Elliptic Q-curves with complex multiplicationp. 181
Abelian varieties over Q with large endomorphism algebras and their simple components over Qp. 189
Abelian varieties over Q and modular formsp. 241
Shimura curves embedded in Igusa's threefoldp. 263
Shafarevich-Tate groups of nonsquare orderp. 277
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783764365868
ISBN-10: 3764365862
Series: Progress in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 289
Published: 23rd February 2004
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 1.33

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