Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves

TL;DR

This paper improves the understanding of the distribution of primes for which the reduction of a non-CM elliptic curve has trivial Tate-Shafarevich group, refining previous asymptotic formulas under GRH.

Contribution

It introduces a new argument that enhances the error term in the asymptotic formula for primes with trivial Tate-Shafarevich groups of elliptic curve reductions.

Findings

Improved error term in asymptotic formula under GRH.

Refined distribution estimates for primes related to Tate-Shafarevich groups.

Enhanced analytical techniques for elliptic curve reductions.

Abstract

Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes $p\le x$ such that the reduction of $\E$ modulo p has a trivial Tate-Shafarevich group. Recent results of A. C. Cojocaru and C. David lead to a better error term. We introduce a new argument in the scheme of the proof which gives further improvement.