Rational points on symmetric squares of constant algebraic curves over function fields

TL;DR

This paper investigates the distribution and bounds of rational points on symmetric squares of constant algebraic curves over function fields, providing explicit bounds and analyzing Frobenius descent effects.

Contribution

It introduces new methods to describe rational points on symmetric squares over function fields and establishes explicit bounds, with insights into Frobenius descent and limitations of generalizations.

Findings

Explicit bounds on the number of rational points on symmetric squares.

Description of adelic points surviving Frobenius descent.

Examples showing limitations of extending results to all subvarieties.

Abstract

We consider smooth projective curves C/$\mathbb{F}$ over a finite field and their symmetric squares $C^{(2)}$. For a global function field $K/\mathbb{F}$, we study the $K$-rational points of $C^{(2)}$. We describe the adelic points of $C^{(2)}$ surviving Frobenius descent and how the $K$-rational points fit there. Our methods also lead to an explicit bound on the number of $K$-rational points of $C^{(2)}$ satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.