Central values of additive twists of Maa{\ss} forms $L$-functions - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2022

Central values of additive twists of Maa{\ss} forms $L$-functions

Résumé

In the present paper we study the central values of additive twists of Maa{\ss} forms $L$-series. In the case of the modular group, we show that the additive twists (when averaged over denominators) are asymptotically normally distributed. This supplements the recent work of Petridis--Risager which settled an averaged version of a conjecture of Mazur--Rubin concerning modular symbols. The methods of the present paper combine dynamical input due to Bettin and the first named author with the new fact that the additive twists define quantum modular forms in the sense of Zagier. This latter property is shown for a general discrete, co-finite group with cusps. Our results also has a number of arithmetic applications; in the case of Hecke congruence groups the quantum modularity implies certain reciprocity relations for twisted moments of twisted ${\rm GL}_2$-automorphic $L$-functions, extending results of Conrey and the second named author. In the case of cuspidal Maa{\ss} forms for the modular group, we also obtain a calculation of certain wide moments of twists of the $L$-function of the Maa{\ss} form.
Fichier principal
Vignette du fichier
maass-modsymb.pdf (703.53 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04052946 , version 1 (30-03-2023)

Identifiants

Citer

Sary Drappeau, Asbjørn Christian Nordentoft. Central values of additive twists of Maa{\ss} forms $L$-functions. 2022. ⟨hal-04052946⟩
29 Consultations
28 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More