Algebraic cycles and Tate classes on Hilbert modular varieties

TL;DR

This paper investigates the structure of Tate classes on motives associated with Hilbert modular varieties, providing dimension computations and conditions under which these classes are algebraic, thus linking automorphic forms to algebraic cycles.

Contribution

It computes the dimension of Tate classes in certain motives from Hilbert modular varieties and establishes conditions for these classes to be algebraic cycles.

Findings

Dimension of Tate classes computed for specific motives.

Conditions identified under which Tate classes are algebraic cycles.

Results connect automorphic representations with algebraic geometry.

Abstract

Let $E/\mathbb{Q}$ be a totally real number field that is Galois over $\mathbb{Q}$, and let $\pi$ be a cuspidal, nondihedral automorphic representation of $\mathrm{GL}_2(\mathbb{A}_E)$ that is in the lowest weight discrete series at every real place of $E$. The representation $\pi$ cuts out a "motive" $M_\mathrm{et}(\pi^{\infty})$ from the $\ell$-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If $\ell$ is sufficiently large in a sense that depends on $\pi$ we compute the dimension of the space of Tate classes in $M_\mathrm{et}(\pi^{\infty})$. Moreover if the space of Tate classes on this motive over all finite abelian extensions $k/E$ is at most of rank one as a Hecke module, we prove that the space of Tate classes in $M_\mathrm{et}(\pi^{\infty})$ is spanned by algebraic cycles.