On a remark of Serre

2025/08/14 by Katie Ahrens, Ahrens, Katie, Jon Grantham +1
Computer Science · Mathematics · #11G20 #11Y55 #14H52 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #and 11A41

paper · doi:10.48550/arxiv.2508.11048

openalex publication_date 2025/08/14 · openalex created_date 2026/04/30 · openalex updated_date 2026/07/28

Abstract

Inspired by a remark of Serre, we extend the search for primes p such that the maximum Hasse bound for the number of points on an elliptic curve over \mathbbF_p⁵ is not achieved. We then give a list of all q<10⁷⁰ such that the Hasse bound is not achieved over \mathbbF_q. We explore the heuristics for how many such numbers should exist in each case. Finally, look at similar criteria for genus 2 and 3 curves.

Citations

• Primes of the Form m²+1 and Goldbach's `Other Other' Conjecture

Related

• Extremal primes of elliptic curves without complex multiplication
• Elliptic curves maximal over extensions of finite base fields
• Maximal curves of genus 5 over finite fields
• A uniform bound on the smallest surjective prime of an elliptic curve
• New bounds on the maximum number of points on genus-4 curves over small finite fields