Alina Bucur of the University of California, San Diego

Title: Effective Sato-Tate under GRH

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 the analytic continuation and the Riemann hypothesis for all the symmetric power L-functions. Using similar techniques, Kedlaya and I obtained a similar conditional effectivization of the generalized Sato-Tate conjecture for an arbitrary motive. As an application, we obtained a conditional upper bound of the form O((logN)^2(loglogN)^2) for the smallest prime at which two given rational elliptic curves with conductor at most N have Frobenius traces of opposite sign. In this talk, I will discuss how to improve this bound to the best possible in terms of N and under slightly weaker assumptions. Our new approach extends to abelian varieties. This is joint work with Kiran Kedlaya and Francesc Fite.