On Hypothesis H of Rudnick and Sarnak

2025/07/28 by Jiang, Yujiao · 4 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.20653

Abstract

We prove Hypothesis H in full generality for \rm GL_n over any number field. This result is a consequence of our stronger effective bound on Euler products involving Rankin--Selberg coefficients at prime ideal powers. The proof rests on a new analytic method, which employs a power sieve over number fields and an iterative argument to bypass the functoriality barrier that had restricted prior results to n≤ 4. As applications, we unconditionally establish the GUE statistics for automorphic L-function zeros, provide the first effective polynomial bound for the strong multiplicity one problem for coefficients, and resolve the Selberg orthogonality conjecture with stronger error terms.

Citations

• Simultaneous extreme values of zeta and L-functions
• Fourier coefficients of GL(N) automorphic forms in arithmetic progressions
• Counting rational points on smooth cyclic covers
• Upper bounds on L-functions at the edge of the critical strip
• A Theorem on Analytic Strong Multiplicity One
• Second order average estimates on local data of cusp forms
• On the absolute convergence of the spectral side of the Arthur trace\n formula for GL(n)
• Zeros of principal L-functions and random matrix theory

Cited by

• Zeros of Polynomials in Derivatives of Automorphic L-functions
• Decorrelation of SO(5)× SO(2) Bessel periods for symmetric cubes of GL(2)
• Large sieves for GL_n and applications
• Selberg orthogonality for half-integral weight modular forms

Related

• Large sieves for GL_n and applications
• A Mild Tchebotarev Theorem for GL(n)
• The Analytic Strong Multiplicity One Theorem for GL_m(A_K)
• A Prime Number Theorem for Rankin-Selberg L-functions over Number fields
• Effective Multiplicity One for GL(n)