Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2602.08985

An omega result for the least negative Hecke eigenvalue

Youness Lamzouri
View original source ↗Read source document ↗Export metadata ↓

Abstract

We establish the existence of many holomorphic Hecke eigenforms $f$ of large weight $k$ for the full modular group, for which the least positive integer $n_f$ such that $λ_f(n_f)<0$ satisfies $n_f \ge (\log k)^{1-o(1)}.$ This is believed to be best possible up to the $o(1)$ term in the exponent, and improves on a result of Kowalski, Lau, Soundararajan and Wu, who showed that, when restricted to primes, the least prime $p$ such that $λ_f(p)<0$ can be as large as $(\log k)^{1/2+o(1)}$. We also discuss an extension of our result to primitive holomorphic cusp forms of weight $k$ and squarefree level $N\geq 1$.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Youness Lamzouri. 2026-06-01. An omega result for the least negative Hecke eigenvalue. https://arxiv.org/abs/2602.08985

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗