Density of algebraic points on products of curves

Abstract: A general philosophy in arithmetic geometry is that ``geometry governs arithmetic." In the setting of algebraic curves over a number field k, a geometric invariant, the genus, controls the arithmetic property of an infinitude of $k$-rational points. Instead of considering only $k$-rational points, however, one can study closed points (Galois orbits of geometric points) which contain all the essential information about the curve over $k$. The density degree set of a nice variety $X$ over $k$ is the set of all positive integers $d$ such that the set of degree $d$ points on $X$ is Zariski dense. For nice curves over $k$, the structure of the density degree set is known to reflect the geometry of the curve. Far less is known in the case of surfaces. In this talk, I'll discuss recent work on the study of Zariski density of algebraic points on surfaces in a case that builds upon our understanding for curves. Our primary focus is the product of two nice curves over $k$. This is joint work with Fu, Gazaki, Porzio, Rawson, and Vogt.

number theory

Audience: researchers in the topic

Five College Number Theory Seminar