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arXiv · 1909.09665

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

Asbjorn Christian Nordentoft
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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.

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Asbjorn Christian Nordentoft. 2020-10-23. A note on additive twists, reciprocity laws and quantum modular forms. https://doi.org/10.1007/s11139-020-00270-1

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