Diophantine Criterion for Non-trivial Shafarevich-Tate Groups
Vinodkumar Ghale, Gayatri Panicker, Debopam Chakraborty
TL;DR
The paper analyzes the elliptic curves $E_p: y^2 = x(x-1)(x+p^2)$ associated with a quartic Diophantine framework, establishing a precise dichotomy between rank $2$ with trivial $\Sha[2]$ and rank $0$ with $\Sha[2] \cong (\mathbb{Z}/2\mathbb{Z})^2$ for primes $p \equiv 1 \pmod{8}$ with $q=(p^2+1)/2$ prime. This dichotomy is governed by the solvability of the Diophantine equation $p(x^2-py^2)^2 = z^2+4x^2y^2$ with $z$ odd, connecting a generalized Fermat-type problem to the arithmetic of $E_p$ via a 2-descent analysis that yields a fixed $2$-Selmer rank $s_2=2$. The work also provides a geometric interpretation in terms of Heron triangles: $E_p$ corresponds to the Heronian family with $n=p$ and $ au=p^{-1}$, so solvability implies infinitely many nondegenerate triangles of area $p$, while failure implies only degenerate cases. Overall, the results give an explicit mechanism to construct elliptic curves with nontrivial $2$-torsion in the Shafarevich–Tate group and illuminate the deep link between Diophantine solvability, Selmer groups, and arithmetic geometry of rational triangles.
Abstract
The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with \textit{generalized Fermat's equation}. In this work, we are interested in the integer solutions of a similar quartic equation $pu^{2} = v^{2}+w^{2}$. For a particular form of $u,v$, and $w$, we prove that the elliptic curves $E_p: y^2 = x(x-1)(x+p^2)$, for primes $p \equiv 1 \pmod{8}$ where $q = (p^2+1)/2$ is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either $\mathrm{rank}(E_p(\mathbb{Q})) = 2$ with trivial Shafarevich-Tate group or $\mathrm{rank} = 0$ with $\Sha(E_p/\mathbb{Q})[2] \cong (\mathbb{Z}/2\mathbb{Z})^2$.