On totally split primes in high-degree torsion fields of elliptic curves

Abstract

Analogously to primes in arithmetic progressions to large moduli, we can study primes that are totally split in extensions of $\mathbb{Q}$ of high degree. Motivated by a question of Kowalski we focus on the extensions $\mathbb{Q}(E[d])$ obtained by adjoining the coordinates of $d$-torsion points of a non-CM elliptic curve $E/\mathbb{Q}$. A prime $p$ is said to be an outside prime of $E$ if it is totally split in $\mathbb{Q}(E[d])$ for some $d$ with $p<|\text{Gal}(\mathbb{Q}(E[d])/\mathbb{Q})| = d^{4-o(1)}$ (so that $p$ is not accounted for by the expected main term in the Chebotarev Density Theorem). We show that for almost all integers $d$ there exists a non-CM elliptic curve $E/\mathbb{Q}$ and a prime $p<|\text{Gal}(\mathbb{Q}(E[d])/\mathbb{Q})|$ which is totally split in $\mathbb{Q}(E[d])$. Furthermore, we prove that for almost all $d$ that factorize suitably there exists a non-CM elliptic curve $E/\mathbb{Q}$ and a prime $p$ with $p^{0.2694} < d$ which is totally split in $\mathbb{Q}(E[d])$. To show this we use work of Kowalski to relate the question to the distribution of primes in certain residue classes modulo $d^2$. Hence, the barrier $p < d^4$ is related to the limit in the classical Bombieri-Vinogradov Theorem. To break past this we make use of the assumption that $d$ factorizes conveniently, similarly as in the works on primes in arithmetic progression to large moduli by Bombieri, Friedlander, Fouvry, and Iwaniec, and in the more recent works of Zhang, Polymath, and the author. In contrast to these works we do not require any of the deep exponential sum bounds (ie. sums of Kloosterman sums or Weil/Deligne bound). Instead, we only require the classical large sieve for multiplicative characters. We use Harman's sieve method to obtain a combinatorial decomposition for primes.