Tate classes and endoscopy for $\operatorname{GSp}_4$ over totally real fields

Abstract

The theory of endoscopy predicts the existence of large families of Tate classes on certain products of Shimura varieties, and it is natural to ask in what cases one can construct algebraic cycles giving rise to these Tate classes. This paper takes up the case of Tate classes arising from the Yoshida lift: these are Tate cycles in middle degree on the Shimura variety for the group $\operatorname{Res}_{F/\mathbb Q} (\operatorname{GL}_2 \times \operatorname{GSp}_4)$, where $F$ is a totally real field. A special case is the family of Tate classes which reflect the appearance of two-dimensional Galois representations in the middle cohomology of both a modular curve and a Siegel modular threefold. We show that a natural algebraic cycle generates exactly the Tate classes which are associated to \emph{generic} members of the endoscopic $L$-packets on $\operatorname{GSp}_{4,F}$. In the non-generic case, we give an alternate construction, which shows that the predicted Tate classes arise from Hodge cycles.