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

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 $k\ge 4$) 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 $L(f\otimes \chi ,1/2)$ with $\chi$ a Dirichlet character.

arXiv comment38 pages, small changes according to the referee's comments (accepted for publication in Crelle)

References in corpus (5)

• High moments of the Estermann function
• The second moment theory of families of L-functions
• Dynamics of continued fractions and distribution of modular symbols
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Cited by in corpus (3)

• Distribution of modular symbols in ${\mathbb{H}}^3$
• On the distribution of periods of holomorphic cusp forms and zeroes of period polynomials
• Non-Random Behaviour in Sums of Modular Symbols