Central values of additive twists of modular $L$-functions

Asbjorn Nordentoft

Published 2018-12-20Version 1

Additive twists of a modular $L$-function are important invariants associated to a cusp form, since the additive twists encode the Eichler-Shimura isomorphism. In this paper we prove that additive twists of $L$-functions associated to cusp forms $f$ of even weight are asymptotically normally distributed. This generalizes a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we present applications to the moments of $L(f\otimes \chi,1/2)$ supplementing recent work of Blomer-Fouvry-Kowalski-Michel-Mili{\'c}evi{\'c}-Sawin.