Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants

TL;DR

This paper characterizes certain algebraic curves over finite fields related to elliptic curves with isomorphic endomorphism rings, proving a finite field analogue of the André-Oort conjecture and applying it to the modular support problem.

Contribution

It provides a complete characterization of algebraic curves over finite fields that relate to elliptic curves with isomorphic endomorphism rings, extending the André-Oort conjecture to finite fields.

Findings

Characterization of algebraic curves over finite fields with special elliptic curve properties

Proof of a finite field analogue of the André-Oort conjecture for $Y(1)^2$

Application to solving the modular support problem in positive characteristic

Abstract

We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the Andr\'e-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of Andr\'e. We use our result to solve the modular support problem for function fields of positive characteristic.