Quadratic twists of tiling number elliptic curves

Abstract: A positive integer $n$ is called a tiling number if the equilateral triangle can be dissected into $nk^2$ congruent triangles for some integer $k$. An integer $n>3$ is tiling number if and only if at least one of the elliptic curves $E^{(\pm n)}:\pm ny^2=x(x-1)(x+3)$ has positive Mordell-Weil rank. Let $A$ denote one of the two curves. In this paper, using Waldspurger formula and an induction method, for $n\equiv 3,7\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=#2\mathrm{Cl}(\mathbb{Q}(\sqrt{-m}))$ as $m$ runs over factors of $n$. Together with $2$-descent method which express $\mathrm{dim}_{\mathbb{F}_2}\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ in terms of the corank of a matrix of $\mathbb{F}_2$-coefficients, we show that for $n\equiv 3,7\mod 24$ positive square-free, the analytic Sha of $A$ being odd is equivalent to that $\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ being trivial, as predicted by the BSD conjecture. We also show that, among the residue classes $3$, resp. $7\mod 24$, the subset of $n$ such that both of $E^{(n)}$ and $E^{(-n)}$ have analytic Sha odd is of limit density $0.288\cdots$ and $0.144\cdots$, respectively, in particular, they are non-tiling numbers. This exhibits two new phenomena on tiling number elliptic curves: firstly, the limit density is different from the general phenomenon on elliptic curves predicted by Bhargava-Kane-Lenstra-Poonen-Rains; secondly, the joint distribution has different behavior among different residue classes.