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

DOI10.1515/CRELLE-2021-0013zbMath1532.11062arXiv1812.08378MaRDI QIDQ6311408

Asbjørn Christian Nordentoft

Publication date: 20 December 2018

Abstract: Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $kgeq4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain 'wide' families of automorphic $L$-functions, consisting of central values of the form $L(fotimeschi,1/2)$ with $chi$ a Dirichlet character.

Mathematics Subject Classification ID

Special values of automorphic (L)-series, periods of automorphic forms, cohomology, modular symbols (11F67) Fourier coefficients of automorphic forms (11F30) Langlands (L)-functions; one variable Dirichlet series and functional equations (11F66) Automorphic forms, one variable (11F12)

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