An omega result for the least negative Hecke eigenvalue

TL;DR

The paper addresses signs of Hecke eigenvalues for holomorphic cusp forms and proves omega-type lower bounds for the least index $n_f$ with $\lambda_f(n_f)<0$. The authors combine a sharply localized trigonometric polynomial with a Petersson-trace framework to force many forms to have small negative-angle parameters $\theta_f(p)$ for primes up to a chosen bound $z$, and then deduce that $\lambda_f(p^m)>0$ for all $p^m\le z$, yielding $n_f\gg \frac{\log k}{(\log\log k)^2}$. They show that a substantial portion of forms (as large as $|\mathcal{H}_k| \exp(-5 \frac{\log k \log_3 k}{(\log_2 k)^3})$) satisfy this, and also obtain a similar bound when restricting to primes, along with an extension to primitive cusp forms of squarefree level $N$. These results improve upon prior bounds (e.g., $n_f\ll k^{3/4}$) and align with the conjectured $(\log k)^{1+o(1)}$-scale behavior for the maximal $n_f$, while strengthening the analogy with the least quadratic non-residue problem and extending to wider families of forms.

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 $N\geq 1$.

Theorems & Definitions (9)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['Thm:Main']}
• proof : Proof of Theorem \ref{['Thm:Main2']}