Central values of additive twists of Maa{\ss} forms $L$-functions

TL;DR

This paper investigates the distribution of additive twists of Maa{ ext} forms' L-functions, showing they are asymptotically normal and revealing their quantum modular properties, with implications for automorphic L-functions and reciprocity laws.

Contribution

It demonstrates that additive twists of Maa{ ext} forms are quantum modular forms and establishes their asymptotic normality, extending previous conjectures and linking to arithmetic reciprocity relations.

Findings

Additive twists are asymptotically normally distributed.

Additive twists define quantum modular forms for general groups.

Quantum modularity leads to reciprocity relations for twisted L-functions.

Abstract

In the present paper we study the central values of additive twists of Maa{\ss} forms $L$-series. In the case of the modular group, we show that the additive twists (when averaged over denominators) are asymptotically normally distributed. This supplements the recent work of Petridis--Risager which settled an averaged version of a conjecture of Mazur--Rubin concerning modular symbols. The methods of the present paper combine dynamical input due to Bettin and the first named author with the new fact that the additive twists define quantum modular forms in the sense of Zagier. This latter property is shown for a general discrete, co-finite group with cusps. Our results also has a number of arithmetic applications; in the case of Hecke congruence groups the quantum modularity implies certain reciprocity relations for twisted moments of twisted ${\rm GL}_2$-automorphic $L$-functions, extending results of Conrey and the second named author. In the case of cuspidal Maa{\ss} forms for the modular group, we also obtain a calculation of certain wide moments of twists of the $L$-function of the Maa{\ss} form.