The Arithmetic Geometry of Square-Sided Heron Triangles

Abstract

We study rational Heron triangles with two marked square sides using elliptic curves and K3 surfaces. An explicit quartic-to-elliptic correspondence parametrizes marked similarity classes by rational points satisfying a positivity condition, modulo \((x,y)\sim(x,-y)\). We determine the generic Mordell--Weil group, prove that every \(k\in\mathbf Q\setminus\{0,\pm1\}\) supports infinitely many scalene classes with exactly two square sides, and construct a primitive family with \(N(X)\gg X^{1/4}\). Requiring the third side to be square gives a genus-three Ciani quartic whose Jacobian is \(\mathbf Q\)-isogenous to a product of three elliptic curves. Geometrically, the two constructions give inequivalent elliptic fibrations on a single singular K3 surface, with geometric Mordell--Weil ranks \(2\) and \(0\). The minimal resolution of the all-square locus is a surface of general type with invariants \((K^2,p_g,q)=(2,3,0)\). Assuming weak Bombieri--Lang, parameters yielding a nondegenerate all-square triangle form a thin subset of \(\mathbf P^1(\mathbf Q)\).

42 pages