Non-euclidean Pythagorean Triples, a Problem of Euler, and Rational Points on K3 Surfaces

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler’s work on the second problem with a modern point of view. 1. Problem I: Pythagorean triples An ordinary Pythagorean triple is a triple (a, b, c) of positive integers satisfying a + b = c. Finding these is equivalent, by the Pythagorean theorem, to finding right triangles with integral sides. Since the equation is homogeneous, the problem for rational numbers is the same, up to a scale factor. Some Pythagorean triples, such as (3, 4, 5), have been known since antiquity. Euclid [6, X.28, Lemma 1] gives a method for finding such triples, which leads to a complete solution of the problem. The primitive Pythagorean triples are exactly the triples of integers (m − n, 2mn,m + n) for various choices of m,n (up to change of order). Diophantus in his Arithmetic [5, Book II, Problem 8] mentions the problem of writing any (rational) number as the sum of squares. This inspired Fermat to write his famous “last theorem” in the margin. Expressed in the language of algebraic geometry, the equation x + y = z describes a curve in the projective plane. This curve is parametrized by a projective line according to the assignment (in homogeneous coordinates) (1) (m,n) 7→ (m − n, 2mn,m + n). The rational points on the curve correspond to primitive Pythagorean triples, which explains why the same parametrization appeared above. Now let us consider the analogous question in a non-Euclidean geometry. In the hyperbolic plane, if one uses the multiplicative distance function instead of its logarithm (which is more common), it makes sense to ask for hyperbolic right triangles whose sides all have rational numbers a, b, c as lengths. It then follows from the formulae of hyperbolic trigonometry, see [4, 42.2, 42.3], that the sines and cosines of the angles of these triangles are also rational. The hyperbolic analogue of the Pythagorean theorem [4, 42.3(f)] tells us sin a · sin b = sin c, If A,B are two points in the Poincaré model of a hyperbolic plane, and if P,Q are the ends (on the defining circle) of the hyperbolic line containing A and B, then μ(AB) = (AB, PQ) is the multiplicative distance function for the segment AB. Here (AB, PQ) denotes the cross-ratio of the four points in the ambient Euclidean plane [4, 39.10].