## 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, i.e., a_{p}(E) = ±[2^{√}p^{]}. Assuming that all the symmetric power L-functions associated to E have analytic continuation for all s ∈ C and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal 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, and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where a_{p}(E) is fixed because of the Sato-Tate distribution.

Original language | English (US) |
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Pages (from-to) | 929-943 |

Number of pages | 15 |

Journal | Proceedings of the American Mathematical Society |

Volume | 148 |

Issue number | 3 |

DOIs | |

State | Published - 2020 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics