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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

**Title:**

**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.