Department of

Mathematics


Seminar Calendar
for events the day of Monday, December 6, 2004.

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Monday, December 6, 2004

12:00 pm in 106 Lincoln Hall,Monday, December 6, 2004

Visibility of Shafarevich-Tate Groups of Modular Abelian Varieties at Higher Level

William Stein (Harvard University)

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.

2:00 pm in 241 Altgeld Hall,Monday, December 6, 2004

Matrix Integrals and KdV Tau Functions

Richard Corrado (UIUC Physics)

Abstract: We continue the discussion of intersection theory on the moduli space of Riemann surfaces, 2d topological gravity and integrable hierarchies of KdV type. In particular, we will discuss Kontsevich's matrix integral representation of the string partition function, which he proved was a KdV tau function.