Almost all elliptic curves are Serre curves

2006/11/03 by Nathan Jones, Jones, Nathan · 4 citations
Computer Science · Mathematics · #11G05 #11R45 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11R45

paper · pdf · doi:10.48550/arxiv.math/0611096

26 pages

arxiv created 2006/11/03 · openalex publication_date 2006/11/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using a multidimensional large sieve inequality, we obtain a bound for the mean square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove a theorem to the effect that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.

Cited by

• Distribution questions for isogeny graphs over finite fields
• The possible adelic indices for elliptic curves admitting a rational cyclic isogeny
• Counting elliptic curves with prescribed entanglements
• Elliptic curves with surjective adelic Galois representations

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