A Diophantine Criterion for the Shafarevich-Tate Groups of Elliptic Curves from Heron Triangles

Abstract

The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar Diophantine equation p u^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), which arise from Heron triangles, for primes p = 1 (mod 8) where q = (p^2+1)/2 is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either rank(E_p(Q)) = 2 with trivial Shafarevich-Tate group or rank = 0 with III(E_p/Q)[2] = (Z/2Z)^2.