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arXiv · 2601.00660

Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions

Chengliang Guo
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Abstract

In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we establish asymptotic formulas for these moments. Our results give an interesting non-equidistribution phenomenon over the full fundamental domain. In fact, this independent equidistribution should be true in a compact set. We further investigate this behaviour by examining a truncated version involving truncated Eisenstein series. Additionally, we propose a conjecture on the joint value distribution of Eisenstein series. The proofs are based on the bounds of the shifted mixed moments of $L$-functions.

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Chengliang Guo. 2026-01-02. Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions. https://arxiv.org/abs/2601.00660

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