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arXiv · 2404.12982

Non-vanishing of geodesic periods of automorphic forms

Petru Constantinescu·Asbjørn Christian Nordentoft
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Abstract

We prove that one hundred percent of the closed geodesic periods of a Hecke--Maa{\ss} cusp form for the modular group are non-vanishing when ordered by length. We present applications to the non-vanishing of central values of Rankin--Selberg $L$-functions. Similar results for holomorphic forms for general Fuchsian groups of finite covolume with a cusp are also obtained, as well as results towards normal distribution. Our new key ingredient is to relate the distributions of closed geodesic periods and vertical line integrals via graph theory.

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Petru Constantinescu, Asbjørn Christian Nordentoft. 2024-04-19. Non-vanishing of geodesic periods of automorphic forms. https://arxiv.org/abs/2404.12982

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