A note on additive twists, reciprocity laws and quantum modular forms

TL;DR

This paper demonstrates that additive twists of cuspidal L-functions form quantum modular forms, establishing a reciprocity law for their moments and connecting quantum modularity to functional equations of weight 2 cusp forms.

Contribution

It generalizes recent results by proving quantum modularity of additive twists and deriving a new reciprocity law for twisted moments of cuspidal L-functions.

Findings

Additive twists' central values form quantum modular forms.

A reciprocity law for the first moment of multiplicative twists is established.

Quantum modularity at infinity relates to functional equations of weight 2 cusp forms.

Abstract

We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.