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.

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).