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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Notes
Eichler integrals of integral weight cusp forms were some of the earliest examples of quantum modular forms considered by Zagier [13, Sect. 11] (see also Lee [9]). In these works, the discrepancy (4.1) is required to extend to a polynomial instead of just a continuous function. Now, one gets a non-trivial result since the Eichler integrals are certainly not restrictions of polynomials (not even restriction of smooth functions).
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Acknowledgements
I would like to thank Dorian Goldfeld and Columbia University for their hospitality and the reviewer for useful comments.
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Nordentoft, A.C. A note on additive twists, reciprocity laws and quantum modular forms. Ramanujan J 56, 151–162 (2021). https://doi.org/10.1007/s11139-020-00270-1
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DOI: https://doi.org/10.1007/s11139-020-00270-1