Refinements of strong multiplicity one for $\mathrm{GL}(2)$

TL;DR

This paper refines strong multiplicity one results for $ ext{GL}(2)$ automorphic representations, establishing new lower bounds on the density of primes where their Fourier coefficients differ and providing bounds on primes with specific coefficient relations.

Contribution

It introduces improved density bounds for primes distinguishing non-twist-equivalent $ ext{GL}(2)$ automorphic representations and refines existing results on Fourier coefficient comparisons.

Findings

Lower Dirichlet density of $rac{1}{16}$ for $ ext{GL}(2)$ representations

Lower densities of $rac{2}{13}$ and $rac{1}{11}$ for non-twist-equivalent cases

Upper bounds on primes with equal squared Fourier coefficients

Abstract

For distinct unitary cuspidal automorphic representations $\pi_1$ and $\pi_2$ for $\mathrm{GL}(2)$ over a number field $F$ and any $\alpha\in\Bbb{R}$, let $\mathcal{S}_{\alpha}$ be the set of primes $v$ of $F$ for which $\lambda_{\pi_1}(v)\neq e^{i\alpha} \lambda_{\pi_2}(v)$, where $\lambda_{\pi_i}(v)$ is the Fourier coefficient of $\pi_i$ at $v$. In this article, we show that the lower Dirichlet density of $\mathcal{S}_\alpha$ is at least $\frac{1}{16}$. Moreover, if $\pi_1$ and $\pi_2$ are not twist-equivalent, we show that the lower Dirichlet densities of $\mathcal{S}_\alpha$ and $ \cap_\alpha\mathcal{S}_\alpha$ are at least $\frac{2}{13}$ and $\frac{1}{11}$, respectively. Furthermore, for non-twist-equivalent $\pi_1$ and $\pi_2$, if each $\pi_i$ corresponds to a non-CM newform of weight $k_i\ge 2$ and with trivial nebentypus, we obtain various upper bounds for the number of primes $p\le x$ such that $\lambda_{\pi_1}(p)^2 = \lambda_{\pi_2}(p)^2$. These present refinements of the works of Murty-Pujahari, Murty-Rajan, Ramakrishnan, and Walji.