$\ell$-Galois special subvarieties and the Mumford-Tate conjecture

TL;DR

This paper introduces $ ext{ell}$-Galois special subvarieties as an $ ext{ell}$-adic analog of special subvarieties, proves their equivalence with Hodge-theoretic notions under certain conditions, and applies these results to the Mumford-Tate conjecture in various contexts.

Contribution

It defines $ ext{ell}$-Galois special subvarieties, establishes their equivalence with Hodge-theoretic special subvarieties under a monodromy condition, and applies this to progress on the Mumford-Tate conjecture.

Findings

The $ ext{ell}$-Galois exceptional locus is a countable union of algebraic subvarieties.

For large $n$ and $d$, the Mumford-Tate conjecture holds on a dense open subset of the moduli space.

The Mumford-Tate conjecture for abelian varieties is equivalent to a local structural conjecture about $ ext{ell}$-Galois special subvarieties.

Abstract

We introduce $\ell$-Galois special subvarieties as an $\ell$-adic analog of the Hodge-theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that both notions are equivalent. We study some properties of these subvarieties and prove this equivalence for subvarieties satisfying a simple monodromy condition. As applications, we show that the $\ell$-Galois exceptional locus is a countable union of algebraic subvarieties and, if the derived group of the generic Mumford-Tate group of a family is simple, its part of positive period dimension coincides with the Hodge locus of positive period dimension. We use this to prove that for $n$ and $d$ sufficiently large, the absolute Mumford-Tate conjecture in degree $n$ holds on a dense open subset of the moduli space of smooth projective hypersurfaces of degree $d$ in $\mathbb{P}^{n+1}$, with the exception of hypersurfaces defined over number fields. Finally, we show that the Mumford-Tate conjecture for abelian varieties is equivalent to a conjecture about the local structure of $\ell$-Galois special subvarieties in $\mathcal{A}_g$.