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

For distinct unitary cuspidal automorphic representations $\pi_1$ and $\pi_2$ for $\mathrm{GL}(2)$ over a number field $F$ and any $\alpha \in R$, 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 $k_i \geq 2$ and with trivial nebentypus, we obtain various upper bounds for the number of primes $p \leq x$ 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.

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In memory of Professor Richard Guy

The author was supported by a PIMS postdoctoral fellowship and the University of Lethbridge.

Received 17 May 2020

Accepted 24 December 2020

Published 29 September 2022