On the Distribution of 2-Selmer ranks of Quadratic Twists of Elliptic Curves over $\mathbb{Q}$

TL;DR

The work develops a generalized random alternating matrix model $M_{2k+r,oldsymbol{t}}^{ ext{Alt}}( frac{}{ ext{F}}_2)$ with up to two holes to describe the distribution of $2$-Selmer ranks in quadratic twists of elliptic curves with full rational $2$-torsion. It proves that, within each $oldsymbol{ extSigma}$-equivalence class, the distribution of $ ext{Sel}_2(E/Q)$ matches the corank distribution of matrices in the model, and that the moments converge accordingly, yielding an explicit essential-average size $3+ extstyleigl(igl|oldsymbol{t}igr|igr)$ for the essential $2$-Selmer group. The irreducible Markov-chain structure ensures positive density of attainable ranks and supports joint distributions with $oldsymbol{ extphi}$-Selmer groups; different classes can have distinct parameters, enriching the rank landscape. The results connect to problems such as $ heta$-congruent numbers, tiling numbers via $X_0(24)$, and modularity phenomena over imaginary quadratic fields, and they employ a blend of Rédei-matrix analysis, Smith’s box method, and Heath–Brown–Kane-type moment techniques to bridge algebraic and analytic perspectives.

Abstract

We characterize the distribution of 2-Selmer ranks of quadratic twists of elliptic curves over $\mathbb{Q}$ with full rational 2-torsion. We propose a new type of random alternating matrix model $M_{*,\mathbf t}^{\mathrm{Alt}}(\mathbb{F}_2)$ over $\mathbb{F}_2$ with 0, 1 or 2 ``holes'', with associated Markov chains, described by parameter $\mathbf t=(t_1,\cdots,t_s)\in\mathbb{Z}^s$ where $s$ is the number of ``holes''. We proved that for each equivalence classes of quadratic twists of elliptic curves: (1) The distribution of 2-Selmer ranks agrees with the distribution of coranks of matrices in $M_{*,\mathbf t}^{\mathrm{Alt}}(\mathbb{F}_2)$; (2) The moments of 2-Selmer groups agree with that of $M_{*,\mathbf t}^{\mathrm{Alt}}(\mathbb{F}_2)$, in particular, the average order of essential 2-Selmer groups is $3+\sum_i2^{t_i}$. Our work extends the works of Heath-Brown, Swinnerton-Dyer, Kane, and Klagsbrun-Mazur-Rubin where the matrix only has 0 ``holes'', the matrix model is the usual random alternating matrix model, and the average order of essential 2-Selmer groups is 3. A new phenomenon is that different equivalence classes in the same quadratic twist family could have different parameters, hence have different distribution of 2-Selmer ranks. The irreducible property of the Markov chain associated to $M_{*,\mathbf t}^{\mathrm{Alt}}(\mathbb{F}_2)$ gives the positive density results on the distribution of 2-Selmer ranks.

Theorems & Definitions (88)

• Theorem 1.1: Heath-Brown, Swinnerton-Dyer, Kane, and Klagsbrun-Mazur-Rubin
• Theorem 1.2
• Remark 1.3
• Proposition 1.4
• proof
• Conjecture 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Theorem 1.9