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

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 $M_{\ast ,\mathbf{t}}^{\mathrm{Alt}}({\mathbb{F}}_2)$ 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 $M_{\ast ,\mathbf{t}}^{\mathrm{Alt}}({\mathbb{F}}_2)$; (2) The moments of 2-Selmer groups agree with that of $M_{\ast ,\mathbf{t}}^{\mathrm{Alt}}({\mathbb{F}}_2)$, in particular, the average order of essential 2-Selmer groups is $3+{\sum }_i{2}^{t_i}$. 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 $M_{\ast ,\mathbf{t}}^{\mathrm{Alt}}({\mathbb{F}}_2)$ gives the positive density results on the distribution of 2-Selmer ranks.

51 pages