The geometric distribution of Selmer groups of elliptic curves over function fields

2020/03/17 by Tony Feng, Feng, Tony, Aaron Landesman +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2003.07517

openalex publication_date 2020/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fix a positive integer n and a finite field \mathbb F_q. We study the joint distribution of the rank of E, the n-Selmer group of E, and the n-torsion in the Tate-Shafarevich group of E as E varies over elliptic curves of fixed height d ≥ 2 over \mathbb F_q(t). We compute this joint distribution in the large q limit. We also show that the "large q, then large height" limit of this distribution agrees with the one predicted by Bhargava-Kane-Lenstra-Poonen-Rains.

Citations

• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• Revêtements étales et groupe fondamental (SGA 1)

Cited by

• The monodromy of cyclic Pryms
• Quadratic spaces and Selmer groups of abelian varieties with multiplication

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