Chebotarev Density Theorem and Extremal Primes for non-CM elliptic curves

Amita Malik, Neha Prabhu

Published 2020-12-23Version 1

For a fixed non-CM elliptic curve $E$ over $\mathbb{Q}$ and a prime $\ell$, we prove an asymptotic formula on the number of primes $p \leq x$ for which the Frobenius trace $a_p(E)$ satisfies the congruence $a_p(E)\equiv [2\sqrt{p}] \pmod \ell$. In order to achieve this, we establish a joint distribution concerning the fractional part of $\alpha p^\theta$ for $\theta \in [0,1], \alpha>0$, and primes $p$ satisfying the Chebotarev condition. As a corollary, we also obtain upper bounds for the number of extremal primes. The results rely on GRH for Dedekind zeta functions for Galois extensions of number fields