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On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture

Barinder S. Banwait, Xiaoyu Huang

TL;DR

This work identifies non-CM elliptic curves with conductor up to $N<500{,}000$ that admit infinitely many quadratic twists satisfying the full Birch–Swinnerton-Dyer conjecture by encoding the $p$-part and $2$-part BSD conditions into explicit algorithms and applying them to the LMFDB. The authors provide a comprehensive dataset of BSD-satisfying twists, extending prior results up to conductor 150 and enabling unconditional data on $| ext{Sha}(E_d)|$ for these families. They also test the Radziwiłł–Soundararajan conjecture for the distribution of $ ext{log}(| ext{Sha}(E_d)|/ ext{sqrt}|d|)$, finding that generic twists align with a Gaussian limit while the BSD-satisfying subfamily exhibits a systematic bias due to arithmetic restrictions. The results highlight how local arithmetic constraints shape statistical distributions in twist families and demonstrate the value of algorithmic BSD verification for producing large, analyzable datasets in arithmetic statistics.

Abstract

Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the $L$-functions and Modular Forms Database (LMFDB), identifying all elliptic curves $E$ of conductor at most $500{,}000$ that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwiłł and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily.

On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture

TL;DR

This work identifies non-CM elliptic curves with conductor up to that admit infinitely many quadratic twists satisfying the full Birch–Swinnerton-Dyer conjecture by encoding the -part and -part BSD conditions into explicit algorithms and applying them to the LMFDB. The authors provide a comprehensive dataset of BSD-satisfying twists, extending prior results up to conductor 150 and enabling unconditional data on for these families. They also test the Radziwiłł–Soundararajan conjecture for the distribution of , finding that generic twists align with a Gaussian limit while the BSD-satisfying subfamily exhibits a systematic bias due to arithmetic restrictions. The results highlight how local arithmetic constraints shape statistical distributions in twist families and demonstrate the value of algorithmic BSD verification for producing large, analyzable datasets in arithmetic statistics.

Abstract

Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the -functions and Modular Forms Database (LMFDB), identifying all elliptic curves of conductor at most that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwiłł and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily.
Paper Structure (11 sections, 3 theorems, 5 equations, 3 figures, 2 algorithms)

This paper contains 11 sections, 3 theorems, 5 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Quadratic Twist Families That Satisfy the Full BSD Conjecture
  3. $p$-part of BSD for Odd Primes
  4. $2$-part of BSD for $E_d$
  5. Algorithms
  6. The curves up to conductor 150
  7. Verifying the conjecture of Radziwiłł and Soundararajan
  8. Background on the Conjecture
  9. Verifying the conjecture
  10. Statistical Diagnostics for Normality
  11. Observed Deviation in the BSD Family

Key Result

Theorem 1.1

Let $\mathcal{C}$ be the set of elliptic curves listed in banwait2024quadtwist. For every $E \in \mathcal{C}$, there exists an explicit infinite family of quadratic twists $\{E_d\}$ such that each $E_d$ satisfies the full Birch and Swinnerton-Dyer conjecture.

Figures (3)

  • Figure 3.1: Distribution of $17{,}531$$Z_d^{\textup{an}}$ values for generic twists of 46a1 with $|d| \le 100{,}000$. The histogram (light blue) shows the raw counts, while the Kernel Density Estimate (solid blue line) provides a smooth approximation that closely tracks the theoretical standard Gaussian prediction (black dashed). A time-lapse visualization of the convergence to $\mathcal{N}(0,1)$ is available at https://html-preview.github.io/?url=https://github.com/cocoxhuang/ants_xvii/blob/main/ants_xvii/rs_conjecture/output/46a1_maxd100000_nf100_20260116_185105_pdf.html.
  • Figure 3.2: Goodness-of-fit diagnostics for curve 46a1 as functions of the truncation bound on $|d|$ where the maximum bound $X$ increases to $100,000$. The KS and Wasserstein distances decrease with increasing bound indicating improved distributional agreement to normality.
  • Figure 3.3: Distribution of normalized $Z_d$ for $1008$ of the the unconditionally BSD-satisfying twists of 46a1. The observed distribution (blue) is clearly shifted to the right of the generic Radziwiłł--Soundararajan prediction (black dashed). This shift reflects the bias introduced by restricting to real quadratic fields with split primes at the conductor. A time-lapse visualization of the convergence to $\mathcal{N}(0,1)$ is available at https://html-preview.github.io/?url=https://github.com/cocoxhuang/ants_xvii/blob/main/ants_xvii/rs_conjecture/output/46a1_restricted_nf50_20260115_014127_pdf.html.

Theorems & Definitions (8)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5