ASBJØRN CHRISTIAN NORDENTOFT A. Additive twists of modular L-functions are important invariants associated to the space of cusp forms, since the additive twists encode the Eichler-Shimura isomorphism. In this paper we prove that additive twists of an L-function associated to a cusp form f of even weight are asymptotically normally distributed. This generalizes a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we present applications to the moments of L(f ⊗ χ, 1/2) supplementing recent work of Blomer-Fouvry-Kowalski-Michel-Milićević-Sawin.

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 ≥ 4 {k\geq 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 ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

In this paper, we study hybrid subconvexity bounds for class group 𝐿-functions associated to quadratic extensions K/\mathbb{Q} (real or imaginary). Our proof relies on relating the class group 𝐿-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E(z,1/2+it)\ll_{\varepsilon}y^{1/2}(\lvert t\rvert+1)^{1/3+\varepsilon}, y\gg 1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.

Shifted convolution sums play a prominent rôle in analytic number theory. Here these sums are considered in the context of holomorphic Hecke eigenforms. We investigate pointwise bounds, mean-square bounds consistent with the optimal conjectural bound, and find asymptotics on average for their variance.

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