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

Abstract

For distinct unitary cuspidal automorphic representations ${\pi }_1$ and ${\pi }_2$ for $\mathrm{GL}(2)$ over a number field $F$ and any $\alpha \in \mathbb{R}$, let ${\mathcal{S}}_{\alpha }$ be the set of primes $v$ of $F$ for which ${\lambda }_{{\pi }_1}(v)\ne e^{i\alpha }{\lambda }_{{\pi }_2}(v)$, 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.

Accepted by Math. Res. Lett. in Dec. 2020