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).
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).
Abelian varieties and their moduli are a topic of increasing importance in today`s mathematics, applications ranging from algebraic geometry and number theory to mathematical physics. This collection...
A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the...
Abelian varieties can be classified via their moduli. In positive characteristic the structure of the p-torsion-structure is an additional, useful tool. For that structure supersingular abelian...
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