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.