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arXiv · 2308.10976

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

Francesco Campagna·Gabriel Andreas Dill
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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.

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Francesco Campagna, Gabriel Andreas Dill. 2023-08-21. Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants. https://arxiv.org/abs/2308.10976

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