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

Francesco Campagna, Gabriel Andreas Dill

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.

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 with the property that all but finitely many of their -points have coordinates that are -invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the André-Oort conjecture for , which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic.
Paper Structure (5 sections, 11 theorems, 4 equations)

This paper contains 5 sections, 11 theorems, 4 equations.

Table of Contents

  1. Introdvctio
  2. Variatio -- a finite field variant of Lang's question
  3. Repetitio -- finite fields and complex multiplication
  4. Probatio -- proof of Theorem \ref{['thm:main_thm']}
  5. Peroratio

Key Result

Theorem 1.1

Let $p \in \mathbb{N}$ be a prime and let $F$ denote an algebraic closure of the finite field $\mathbb{F}_p$. Let $\mathcal{C} \subseteq \mathbb{A}^2_F$ be an integral closed curve satisfying the following property: for all but finitely many points $(x_1,x_2) \in \mathcal{C}(F)$ we have that $x_1$ a

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Scanlon-Voloch
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 11 more