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Texas A&M Number Theory Seminar

Department of Mathematics
Milner 317
Wednesdays, 12:30-1:30 PM

Stephan Baier

Jacobs University Bremen

Wednesday, November 7, 2007

Milner 317, 12:30PM


Abstract: The philosophy of Random Matrix Theory is that statistics associated to zeros of a natural family of L-functions can be modeled by the statistics of eigenvalues of large random matrices in a suitable linear group. Here we consider the statistics of zeros of Hasse-Weil L-functions associated to elliptic curves over the rationals near the central point s = 1. In particular, we draw conclusions about the zeros at the central point itself, which by the conjecture of Birch-Swinnerton-Dyer contain important arithmetical information about the relevant elliptic curve. Building on previous work of Matthew Young on this subject, we establish that if the Riemann hypothesis holds for the said Hasse-Weil L-functions, then a positive proportion of elliptic curves over the rationals have analytic rank equal to algebraic rank and finite Tate-Shafarevic group, in accordance with the Birch-Swinnerton-Dyer conjecture. Matthew Young had obtained this result, with better estimates, under the additional condition that GRH holds for Dirichlet and symmetric square L-functions.

This is joint work with Liangyi Zhao.