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

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 André-Oort conjecture for $Y(1{)}_{\mathbb{C}}^2$, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic.

12 pages, simplified the proof of the main result

References in corpus (3)

• Two variants of the support problem for products of abelian varieties and tori
• Around the support problem for Hilbert class polynomials
• Counting torsion points on subvarieties of the algebraic torus