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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).
Serge Lang was an iconic figure in mathematics, both for his own
important work and for the indelible impact he left on the field of
mathematics, on his students, and on his colleagues. Over the
course of his career, Lang traversed a tremendous amount of
mathematical ground. As he moved from subject to subject, he found
analogies that led to important questions in such areas as number
theory, arithmetic geometry, and the theory of negatively curved
spaces. Lang's conjectures will keep many mathematicians occupied
far into the future. In the spirit of Lang's vast contribution to
mathematics, this memorial volume contains articles by prominent
mathematicians in a variety of areas of the field, namely Number
Theory, Analysis, and Geometry, representing Lang's own breadth of
interest and impact. A special introduction by John Tate includes a
brief and fascinating account of the Serge Lang's life. This
volume's group of 6 editors are also highly prominent
mathematicians and were close to Serge Lang, both academically and
personally. The volume is suitable to research mathematicians in
the areas of Number Theory, Analysis, and Geometry.
Serge Lang was an iconic figure in mathematics, both for his own
important work and for the indelible impact he left on the field of
mathematics, on his students, and on his colleagues. Over the
course of his career, Lang traversed a tremendous amount of
mathematical ground. As he moved from subject to subject, he found
analogies that led to important questions in such areas as number
theory, arithmetic geometry, and the theory of negatively curved
spaces. Lang's conjectures will keep many mathematicians occupied
far into the future. In the spirit of Lang's vast contribution to
mathematics, this memorial volume contains articles by prominent
mathematicians in a variety of areas of the field, namely Number
Theory, Analysis, and Geometry, representing Lang's own breadth of
interest and impact. A special introduction by John Tate includes a
brief and fascinating account of the Serge Lang's life. This
volume's group of 6 editors are also highly prominent
mathematicians and were close to Serge Lang, both academically and
personally. The volume is suitable to research mathematicians in
the areas of Number Theory, Analysis, and Geometry.
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).
This volume is the offspring of a week-long workshop on "Galois
groups over Q and related topics," which was held at the
Mathematical Sciences Research Institute during the week March
23-27, 1987. The organizing committee consisted of Kenneth Ribet
(chairman), Yasutaka Ihara, and Jean-Pierre Serre. The conference
focused on three principal themes: 1. Extensions of Q with finite
simple Galois groups. 2. Galois actions on fundamental groups,
nilpotent extensions of Q arising from Fermat curves, and the
interplay between Gauss sums and cyclotomic units. 3.
Representations of Gal(Q/Q) with values in GL(2)j deformations and
connections with modular forms. Here is a summary of the conference
program: * G. Anderson: "Gauss sums, circular units and the
simplex" * G. Anderson and Y. Ihara: "Galois actions on 11"1 ( ***
) and higher circular units" * D. Blasius: "Maass forms and Galois
representations" * P. Deligne: "Galois action on 1I"1(P-{0, 1, oo})
and Hodge analogue" * W. Feit: "Some Galois groups over number
fields" * Y. Ihara: "Arithmetic aspect of Galois actions on 1I"1(P
- {O, 1, oo})" - survey talk * U. Jannsen: "Galois cohomology of
i-adic representations" * B. Matzat: - "Rationality criteria for
Galois extensions" - "How to construct polynomials with Galois
group Mll over Q" * B. Mazur: "Deforming GL(2) Galois
representations" * K. Ribet: "Lowering the level of modular
representations of Gal( Q/ Q)" * J-P. Serre: - Introductory Lecture
- "Degree 2 modular representations of Gal(Q/Q)" * J.
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