An elliptic K3 surface associated to Heron triangles

A rational triangle is a triangle with rational sides and rational area. A Heron triangle is a triangle with integral sides and integral area. In this article we will show that there exist infinitely many rational parametrizations, in terms of s, of rational triangles with perimeter 2s(s + 1) and area s(s − 1). As a corollary, there exist arbitrarily many Heron triangles with all the same area and the same perimeter. The proof uses an elliptic K3 surface Y . Its Picard number is computed to be 18 after we prove that the Néron-Severi group of Y injects naturally into the NéronSeveri group of the reduction of Y at a prime of good reduction. We also give some constructions of elliptic surfaces and prove that under mild conditions a cubic surface in P can be given the structure of an elliptic surface by cutting it with the family of hyperplanes through a given line L. Some of these constructions were already known, but appear to have lacked proof in the literature until now.