Extremal primes of elliptic curves without complex multiplication

TL;DR

This paper establishes the first non-trivial upper bounds on the number of extremal primes for elliptic curves over Q without complex multiplication, assuming automorphy and GRH, using equidistribution and measure refinement techniques.

Contribution

It provides new upper bounds for extremal primes of non-CM elliptic curves by leveraging automorphy, GRH, and refined equidistribution estimates.

Findings

First non-trivial upper bounds for extremal primes

Utilizes explicit equidistribution for Sato-Tate measure

Refines estimates based on small measure of extremal primes

Abstract

Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. Assuming that all the symmetric power L-functions associated to E are automorphic and satisfy the Generalized Riemann Hypothesis, we give the first non-trivial upper bounds for the number of such primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner (arXiv:1305.5283) and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.