Unbounded average Selmer ranks of elliptic curves in torsion families

2025/12/18 by Tristan Phillips, Phillips, Tristan
Computer Science · Mathematics · #11G07 #11G18 #11G50 #14G35 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Primary 11G05 #Secondary 11K65

paper · pdf · doi:10.48550/arxiv.2512.16120

openalex publication_date 2025/12/18 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28

Abstract

Let M and N be positive integers for which the modular curve X₁(M,MN) has genus 0, and let p be a prime divisor of MN. This article gives asymptotic lower bounds for the average size of the p-Selmer group of elliptic curves over a number field, with torsion subgroup ℤ/Mℤ ⊕ ℤ/MNℤ. In many cases, it is shown that this average is unbounded.

Citations

• Selmer groups of families of elliptic curves with an ℓ-isogeny
• Genus formulas for families of modular curves
• The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias
• The distribution of the Tamagawa ratio in the family of elliptic curves\n with a two-torsion point
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• The average number of elements in the 4-Selmer groups of elliptic curves is 7
• Local invariants of isogenous elliptic curves
• Ternary cubic forms having bounded invariants, and the existence of a\n positive proportion of elliptic curves having rank 0
• Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

Related

• On average sizes of Selmer groups and ranks in families of elliptic curves having marked points
• The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• Average size of Selmer group in large q limit
• The average number of elements in the 4-Selmer groups of elliptic curves is 7