Arithmetic upon an algebraic surface

Abstract

The title of my lecture is, I am afraid, probably misleading and certainly too ambitious. For, on the one hand, the connection between arithmetic and geometry suggested by it is not the modern development in divisors theory, but an application of algebraic geometry for arithmetical purposes. On the other hand, I shall confine the subject of this lecture to cubic surfaces in ordinary space, considered in the rational domain, so that a proper title would be for instance The geometry of ternary cubic Diophantine equations. I prefer, however, the more ambitious and inaccurate one, as suggesting the possibility of similar investigations for other surfaces, possibly considered in more general arithmetical fields. The short time at my disposal does not allow me to dwell on such extensions. I mention only that I have already completed an extensive arithmetical research on quartic surfaces; and that the whole subject—arithmetic upon an algebraic surface—seems to me to be so wide in scope, that I can envisage the possibility of further important developments. Let us consider an ordinary space, where coordinates (x, y, z) are introduced and points at infinity are defined in the usual way. I call rational an algebraic surface, or curve, or point set of this space when it can be represented by one or more algebraic equations with rational coefficients,