The Arithmetic Geometry of Square-Sided Heron Triangles

2026/05/31 by Yangcheng Li
#math.NT

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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 \((x,y)∼(x,-y)\). We determine the generic Mordell--Weil group, prove that every \(k∈\mathbf Q∖\0,±1\\) supports infinitely many scalene classes with exactly two square sides, and construct a primitive family with \(N(X)≫ X^1/4\). 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 \((K²,p_g,q)=(2,3,0)\). Assuming weak Bombieri--Lang, parameters yielding a nondegenerate all-square triangle form a thin subset of \(\mathbf P¹(\mathbf Q)\).

Citations

• Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties
• Two lectures on the arithmetic of K3 surfaces
• Elliptic Surfaces
• An elliptic K3 surface associated to Heron triangles
• Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces

Related

• An elliptic K3 surface associated to Heron triangles
• Heron Quadrilaterals via elliptic curves
• Heron triangles with two rational medians and Somos-5 sequences
• Explicit coverings of families of elliptic surfaces by squares of curves
• Euclidean Geometry and Elliptic Curves