Session S21 - Galois representations and automorphic forms

Wednesday, July 21, 17:40 ~ 18:20 UTC-3

Generalized theta series and the central values of $L$-functions of Hilbert modular forms

Nicolás Sirolli

CONICET / Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The central values of twisted $L$-functions of a modular form, after the work of Waldspurger, are known to be related to Fourier coefficients of half-integral weight modular forms.

A solution to the problem of computing effectively these half-integral weight modular forms was proposed by Gross, using ternary theta series attached to quaternion algebras. This construction imposed restrictions on the levels of the modular forms involved, as well as quadratic restrictions on the twisting discriminants.

In this work we show how to remove these restrictions by constructing families of generalized theta series giving the central values. Furthermore, our results hold over arbitrary totally real fields.

Joint work with Gonzalo Tornaría (Universidad de la República, Uruguay).

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