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

Technical Summary: Identification of Elliptic Curves Admitting Infinitely Many BSD-Satisfying Twists

Problem Statement
The Birch and Swinnerton-Dyer (BSD) conjecture remains one of the central open problems in number theory. While significant probabilistic progress has been made regarding the average rank of elliptic curves (establishing that at least 66% satisfy the rank part of the conjecture), these results do not identify specific curves where the full conjecture holds. Historically, explicit infinite families of curves satisfying the full BSD conjecture were known primarily for curves with Complex Multiplication (CM). For non-CM curves, the landscape was expanded by the work of Burungale–Skinner–Tian–Wan, who proved the existence of infinite families of quadratic twists of non-CM elliptic curves satisfying the full BSD conjecture. However, their work provided only a limited explicit list (conductor up to 150) and did not detail the explicit algorithmic procedure or implementation required to verify the necessary local conditions for identifying such curves.

Methodology
The authors bridge the gap between theoretical existence and explicit computation by translating the theoretical conditions established by Burungale–Skinner–Tian–Wan into executable algorithms. The methodology proceeds in two main stages:

1. Algorithmic Filtering (Algorithm 1): The authors encode the hypotheses required for the $p$-part of the BSD conjecture to hold for rank 0 quadratic twists.

   • Odd Primes ($p > 2$): They utilize results from Skinner–Urban, Zhang, and Burungale–Skinner–Tian–Wan to verify the $p$-part. This involves checking conditions on the Galois representation $\rho_{E,p}$, the existence of specific ramified primes, and the behavior of the curve at supersingular primes.
   • The Prime 2: They synthesize results from [5] and [22] to establish conditions for the 2-part of the BSD conjecture. This requires verifying the rank is 0, checking the Manin constant, and analyzing the 2-torsion structure $E(\mathbb{Q})[2]$ and the discriminant sign. Specific conditions are checked for cases where $E(\mathbb{Q})[2] \cong \mathbb{Z}/2\mathbb{Z}$ versus $E(\mathbb{Q})[2] = 0$.
   • Implementation: These checks are implemented in SageMath and applied to the entire LMFDB database of elliptic curves with conductor $N < 500,000$.

2. Twist Verification (Algorithm 2): For curves passing the initial filter, the authors implement a second algorithm to identify specific quadratic twists $E_d$ (within a range $[-1000, 1000]$) that satisfy the full BSD conjecture. This involves verifying local conditions on the twist parameter $d$, such as square-freeness, congruence conditions modulo 4 or 8, and splitting behavior of primes in specific number fields.

Key Contributions

• Explicit Algorithmic Realization: The primary contribution is the translation of the non-constructive existence proofs of Burungale–Skinner–Tian–Wan into explicit, executable algorithms (Algorithm 1 and Algorithm 2).
• Extensive Database Identification: Applying these algorithms to the LMFDB, the authors identify 36,687 elliptic curves with conductor $N < 500,000$ that unconditionally admit infinitely many quadratic twists satisfying the strong BSD conjecture. This represents approximately 0.1% of all curves in that range.
• Refinement of Existing Lists: The authors clarify and refine the list of curves up to conductor 150 presented in [4, Example 1]. They demonstrate that four curves previously listed (62a1, 66b1, 105a1, 141c1) do not currently satisfy the unconditional criteria due to the absence of applicable theorems for their specific 2-torsion structures and the inapplicability of results from [5] (due to the order of vanishing of the L-value).
• Statistical Analysis of the Shafarevich–Tate Group: Using the newly identified dataset where $|\text{Ш}(E_d)|$ is known unconditionally, the authors test the conjecture of Radziwiłł and Soundararajan regarding the Gaussian distribution of $\log(|\text{Ш}(E_d)|/\sqrt{|d|})$.

Results

• Dataset Generation: The authors produced a dataset of elliptic curves and their specific twists where the full BSD conjecture holds unconditionally. This dataset is unique because the order of the Shafarevich–Tate group is derived without assuming the BSD conjecture (as the conjecture is proven for these specific families).
• Verification of Radziwiłł–Soundararajan (Generic Case): For generic twists of a representative curve (46a1) with root number +1, the distribution of normalized values converges to a standard Gaussian distribution $N(0,1)$, consistent with the conjecture, though finite-sample skewness is observed.
• Observation of Systematic Bias (BSD-Satisfying Subfamily): When restricting the analysis to the specific subfamily of twists that satisfy the full BSD conjecture (denoted $D_{BSD}$), the distribution deviates significantly from the generic prediction. The authors observe a systematic positive bias and a shift toward a bimodal shape.
  • Cause of Bias: This deviation is attributed to the rigid arithmetic constraints defining $D_{BSD}$ (e.g., $d$ must be positive, $d \equiv 1 \pmod 8$, and prime factors must split in specific ways). These constraints force the twists to lie in real quadratic fields with specific splitting behaviors, altering the density of primes and the underlying probabilistic model compared to the generic case.

Significance and Claims
The paper claims to provide the first large-scale, unconditional identification of non-CM elliptic curves admitting infinite families of BSD-satisfying twists. By moving from probabilistic existence to explicit enumeration, the authors create a unique dataset for studying the Shafarevich–Tate group.

The significance of the statistical findings lies in the observation that while the Radziwiłł–Soundararajan conjecture holds for generic families, it requires modification (specifically regarding mean and variance) when applied to families defined by strict local arithmetic conditions. The authors note that this does not invalidate the conjecture but highlights its sensitivity to subfamily restrictions. They suggest that with further data, the distribution for the restricted family might still approach a Gaussian form, albeit with different parameters, warranting further investigation.

The work is supported by a public code repository built upon the LMFDB infrastructure, allowing for the reproduction of the identification of these curves and the verification of the statistical claims.