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

2023/08/21 by Francesco Campagna, Campagna, Francesco, Gabriel A. Dill +1
Computer Science · Mathematics · #11G15 #11G18 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2308.10976

openalex publication_date 2023/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We characterise the integral affine plane curves over a finite field k with the property that all but finitely many of their 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_ℂ, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic.

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