Joint IAS/Princeton University Number Theory Seminar

We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some interesting relations between shtuka spaces and function field arithmetic. For instance, there should be a notion of modularity for the r-fold product of an elliptic curve. We verify these predictions in a few cases. This is partly joint work with Noam Elkies.