2:00 pm Thursday, October 21, 2010
Algebra, Number Theory, and Combinatorics: L-functions and random matrices: Two recent results by Eduardo Dueñez (UT San Antonio) in RLM 9.166
In this talk we present two recent results about the underlying random matrix-theoretical symmetry in zeros of L-functions. The first result pertains to the Riemann zeta function (joint with Farmer, Froehlich, Hughes, Mezzadri, Phan). Motivated philosophically by Levinson's method, which employs zeros of the derivative of the zeta function to obtain critical zero-density estimates, we study the distribution of zeros of the derivative of characteristic polynomials of unitary random matrices and apply it to model those of the zeta function. The second result pertains to the phenomenon of "central" repulsion of critical zeros in families of elliptic curves (joint with Huynh, Keating, Miller and Snaith). This repulsion manifests itself as a deviation of numerical data from the expected limiting Katz-Sarnak statistics of low-lying critical zeros in families of quadratic twists of a given elliptic curve. We collect data for twists of the curves of conductors 11 and 19, and present some evidence in support of a random-matrix discretized model for the L-zero statistics that is a natural analogue of the Waldspurger/Kohnen-Zagier discretization of central L-values. Submitted by
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