## EVENT DETAILS AND ABSTRACT

**Colloquium**
**Title:** Visibility of Shafarevich-Tate Groups of Modular

**Speaker:** Professor William Stein

**Speaker Info:** Harvard University

**Brief Description:**

**Special Note**:

**Abstract:**

I will begin by introducing the Birch and Swinnerton-Dyer
conjecture in the context of abelian varieties attached to modular
forms, and discuss some of the main results about it. I will then
introduce Mazur's notion of visibility of Shafarevich-Tate groups and
explain some of the basic facts and theorems. Cremona, Mazur, Agashe,
and myself carried out large computations about visibility for modular
abelian varieties of level N in J_0(N). These computations addressed
the following question: If A is a modular abelian variety of level N,
how much of the Shafarevich-Tate group of A is modular of level N,
i.e., visible in J_0(N). The results of these computations suggest
that often much of the Shafarevich-Tate group is NOT modular of level
N. This suggests asking if every element is modular of level N*m, for
some auxiliary integer m, and if so, what can one say about the set of
such m? I will finish the talk with some new data and thoughts about
this last question, which is still very much open.

**Date:** Monday, January 17, 2005

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Prof. Elton P. Hsu

**Contact email:** elton@math.northwestern.edu

**Contact Phone:** 847-491-8541

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