On Totally Split Primes in High-Degree Torsion Fields of Elliptic Curves

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