On a question of H. A. Schwarz

Abstract

This paper is a contribution to the study of tetrahedra with rational side lengths and rational volume. The problem can be formulated as a study of a hyperelliptic variety attached to the Cayley-Menger determinant. The problem posed by Schwarz was as to whether this variety is rational over the rationals. In a thesis that was never published Otto Schulz proved in 1912 that the variety was rational over $\mathbb{Q}(\sqrt{-1})$. This does not seem to have appeared eleswhere nor, apparently, has the question been studied since then. In this paper we take up an idea of Kurt Heegner to reprove the theorem using the theory of Weddle surfaces. Although this does not lead to a final answer to Schwarz question it seems likely that it is negative.

arXiv commentThe mathematics described here was presented and discussed at the conference "Thermodynamics Formalism- Applications to Geometry, Number Theory and Stochastics" (8th-12th July, 2019) at the Mittag-Leffler Institute, Djursholm This is the revised version following the referee's report