Number Theory Seminar: Tate Classes and Endoscopy for GSp₄

SEMINARS, NUMBER THEORY

Speaker:

Naomi Sweeting - Harvard University

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. For this talk, I'll focus on the behavior of the algebraic cycle class. NB: This talk is independent of the one in last week's number theorists' seminar.