An application of the effective Sato-Tate conjecture

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.