Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants
Volume 218 / 2025
Acta Arithmetica 218 (2025), 215-229
MSC: Primary 11G15; Secondary 11G18
DOI: 10.4064/aa240123-12-11
Published online: 13 March 2025
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 $Y(1)^2_\mathbb {C}$, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic.
