An elliptic K3 surface associated to Heron triangles

Abstract

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 2 − 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 Néron-Severi 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 3 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.

Finiteness results for K3 surfaces over arbitrary fields

“…Proof. Let s p be the specialization homomorphism Pic YQ → Pic YF p , which is injective by [21,Proposition 6.2]. Then by [8,Theorem 1.4] the cokernel of s p is torsionfree.…”

Explicit moduli spaces for congruences of elliptic curves

“…The main tool we wish to use (see [26,Proposition 6.2]) is that there is an injective map NS(X) → NS(X p ) that preserves the intersection pairing.…”

Nontrivial elements of Sha explained through K3 surfaces

“…The reason for stating that result separately is that one can compute the rank of the Néron-Severi group in explicit cases using methods from [21] and [22].…”