Department of

# Mathematics

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

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2004          December 2004           January 2005
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6             1  2  3  4                      1
7  8  9 10 11 12 13    5  6  7  8  9 10 11    2  3  4  5  6  7  8
14 15 16 17 18 19 20   12 13 14 15 16 17 18    9 10 11 12 13 14 15
21 22 23 24 25 26 27   19 20 21 22 23 24 25   16 17 18 19 20 21 22
28 29 30               26 27 28 29 30 31      23 24 25 26 27 28 29
30 31


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