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Unbounded average Selmer ranks of elliptic curves in torsion families

Tristan Phillips

TL;DR

The work addresses unbounded growth phenomena for average p-Selmer sizes in torsion-limited families of elliptic curves over number fields, leveraging genus-zero modular curves X_1(M,MN) to parameterize curves with prescribed torsion data.A central methodological contribution is an Erdős–Kac-type analysis of Selmer ratio distributions, combined with a modular-curve counting framework that imposes prescribed local conditions via weighted projective stacks, enabling precise asymptotics for sums of Selmer ratios across height-bounded families.The authors establish that the average size of Sel_p(E) can grow like a positive power of log(B) under explicit hypotheses, and they extend the framework to composite-degree isogenies and higher d-Selmer groups, providing concrete θ-exponents pulled from local-density data.These results unify and extend earlier unbounded-Selmer phenomena in special-parameter families, highlight the impact of local Tamagawa-number behavior, and supply explicit constants via modular-curve counts and local-density computations.

Abstract

Let $M$ and $N$ be positive integers for which the modular curve $X_1(M,MN)$ 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 $\mathbb{Z}/M\mathbb{Z} \oplus \mathbb{Z}/MN\mathbb{Z}$. In many cases, it is shown that this average is unbounded.

Unbounded average Selmer ranks of elliptic curves in torsion families

TL;DR

The work addresses unbounded growth phenomena for average p-Selmer sizes in torsion-limited families of elliptic curves over number fields, leveraging genus-zero modular curves X_1(M,MN) to parameterize curves with prescribed torsion data.A central methodological contribution is an Erdős–Kac-type analysis of Selmer ratio distributions, combined with a modular-curve counting framework that imposes prescribed local conditions via weighted projective stacks, enabling precise asymptotics for sums of Selmer ratios across height-bounded families.The authors establish that the average size of Sel_p(E) can grow like a positive power of log(B) under explicit hypotheses, and they extend the framework to composite-degree isogenies and higher d-Selmer groups, providing concrete θ-exponents pulled from local-density data.These results unify and extend earlier unbounded-Selmer phenomena in special-parameter families, highlight the impact of local Tamagawa-number behavior, and supply explicit constants via modular-curve counts and local-density computations.

Abstract

Let and be positive integers for which the modular curve has genus , and let be a prime divisor of . This article gives asymptotic lower bounds for the average size of the -Selmer group of elliptic curves over a number field, with torsion subgroup . In many cases, it is shown that this average is unbounded.

Paper Structure

This paper contains 18 sections, 18 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Number fields
  4. Distribution of norms of prime ideals in congruence classes
  5. Modular curves
  6. Weighted projective stacks
  7. Heights
  8. Isogenies
  9. Selmer groups
  10. Selmer ratios and Tamagawa numbers
  11. Counting elliptic curves with prescribed torsion and local conditions
  12. Distribution of Selmer ratios
  13. Asymptotics for sums of Selmer ratios
  14. Isogenies of prime degree
  15. Isogenies of composite degree
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $X_1(M,MN)$ has genus zero, and suppose that $p$ is a prime divisor of $MN$. Then there exists an explicit constant $\theta$ (see Table tab:mean_variance_II), depending only on $M$, $N$, $K$, and $p$, such that If additionally then $\theta$ is positive, implying that the average size of the $p$-Selmer group is unbounded.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Remark 2.3
  • Proposition 2.4: BN22
  • Example 2.5
  • Proposition 2.6: Sil09
  • Example 2.7
  • ...and 25 more