Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

Ramacher, Pablo; Wakatsuki, Satoshi

doi:10.1016/j.aim.2022.108372

Advances in Mathematics

2022

DOI: 10.1016/j.aim.2022.108372

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Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

Pablo Ramacher

Satoshi Wakatsuki

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Cited by 1 publication

References 27 publications

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Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

Sugiyama¹

2021

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For a totally real number field F and its adèle ring A F , let π vary in the set of irreducible cuspidal automorphic representations of PGL 2 (A F ) corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power L-functions L(s, Sym r (π)) weighted by special values of symmetric square L-functions L( z+1 2 , Sym 2 (π)) at z ∈ [0, 1] in the level aspect. If 0 < z 1, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power L-functions for F = Q with harmonic weight. Hence our result is regarded as a zinterpolation of Ricotta and Royer's result. If z = 0, density of low-lying zeros weighted by central values is a different type only when r = 2, and it does not appear in random matrix theory as Katz and Sarnak predicted. Moreover, we propose a conjecture on weighted density of low-lying zeros of L-functions by special L-values.In the latter part, Appendices A, B and C are dedicated to the comparison among several generalizations of Zagier's parameterized trace formula. We prove that the explicit Jacquet-Zagier type trace formula (the ST trace formula) by Tsuzuki and the author recovers all of Zagier's, Takase's and Mizumoto's formulas by specializing several data. Such comparison is not so straightforward and includes non-trivial analytic evaluations.

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Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

Sugiyama¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

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Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

Cited by 1 publication

References 27 publications

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

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