# Tate Classes and Endoscopy for GSp4

# Tate Classes and Endoscopy for GSp4

**Meeting ID: 920 2195 5230**

**Passcode: The three-digit integer that is the cube of the sum of its digits.**

Weissauer proved using the theory of endoscopy that the Galois representations associated to classical modular forms of weight two appear in the middle cohomology of both a modular curve and a Siegel modular threefold. Correspondingly, there are large families of Tate classes on the product of these two Shimura varieties, and it is natural to ask whether one can construct algebraic cycles giving rise to these Tate classes. It turns out that a natural algebraic cycle generates some, but not all, of the Tate classes: to be precise, it generates exactly the Tate classes which are associated to generic members of the endoscopic L-packets on GSp4. In the non-generic case, one can at least show that all the Tate classes arise from Hodge cycles. I'll explain these results and their proofs, which rely on the theta correspondence.