Equidistribution of $\alpha p^{\theta}$ with a Chebotarev condition and applications to extremal primes

We discuss a joint distribution result concerning the fractional part of $\alpha p^\theta$ for $\theta \in (0,1), \ \alpha>0$, where $p$ is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over $\mathbb{Q}$, under a zero-free region condition. As an application, for a fixed non-CM elliptic curve $E/\mathbb{Q}$, we derive an asymptotic formula for the number of primes at the extremes of the Sato-Tate measure modulo a large prime $\ell$. These are precisely the primes $p$ for which the Frobenius trace $a_p(E)$ satisfies the congruence $a_p(E)\equiv [2\sqrt{p}] \bmod \ell$. Recall that the case $a_p(E)=t$, with $t$ an integer, is the famous Lang-Trotter conjecture which is still wide open. (Joint work with Neha Prabhu)

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For password contact Pieter Moree (moree$@$mpim-bonn$.$mpg$.$de)