$A$-analyticity of separatricies of foliations

TL;DR

This paper proves that under certain algebraic conditions, invariant formal curves of foliations on surfaces are actually algebraic, providing a criterion for their algebraicity based on p-th power closure properties.

Contribution

It establishes that invariant formal curves are A-analytic if the foliation is closed under p-th powers for almost all primes, extending previous algebraicity criteria.

Findings

Invariant formal curves are A-analytic under the given conditions.

Provides an algebraicity criterion for invariant curves.

Builds on Bost's prior work to extend algebraicity results.

Abstract

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let $L$ be a foliation on $X$. We prove that if $L$ is closed under $p$-th powers for almost all primes $p$, then any $L$-invariant smooth formal curve is $A$-analytic. Building on prior work of Bost we obtain an algebraicity criterion for those curves.