An application of the effective Sato-Tate conjecture

TL;DR

This paper extends Murty's effective approach to the Sato-Tate conjecture from elliptic curves to general motives, providing explicit bounds on prime distributions related to Frobenius traces.

Contribution

It generalizes the effective Sato-Tate conjecture to arbitrary motives using Murty's analysis, under certain hypotheses, and applies it to bound primes for elliptic curves.

Findings

Conditional upper bound of $O((\log N)^2 (\log \log 2N)^2)$ for the smallest prime with opposite Frobenius traces.

Extension of Murty's effective Sato-Tate results to general motives.

Application to bounding primes related to elliptic curve Frobenius traces.

Abstract

Based on the Lagarias-Odlyzko effectivization of the Chebotarev density theorem, Kumar Murty gave an effective version of the Sato-Tate conjecture for an elliptic curve conditional on analytic continuation and Riemann hypothesis for the symmetric power $L$-functions. We use Murty's analysis to give a similar conditional effectivization of the generalized Sato-Tate conjecture for an arbitrary motive. As an application, we give a conditional upper bound of the form $O((\log N)^2 (\log \log 2N)^2)$ for the smallest prime at which two given rational elliptic curves with conductor at most $N$ have Frobenius traces of opposite sign.