### Arithmetic statistics of modular symbols

Heilbronn Number Theory Seminar

4th December 2019, 4:00 pm – 5:00 pm

Fry Building, 2.04

Mazur, Rubin, and Stein have formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols. We prove a refined version of this conjecture.

This is joint work with Yiannis Petridis.

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