Ordinary and almost ordinary Prym varieties

We study the p-rank stratification of the moduli space of Prym varieties in characteristic p > 0. For arbitrary primes p and l with l not equal p and integers g >= 3 and 0 <= f <= g, the first theorem generalizes a result of Nakajima by proving that the Prym varieties of all the unramified Z/l-covers of a generic curve X of genus g and p-rank f are ordinary. Furthermore, when p >= 5 and l = 2, the second theorem implies that there exists a curve of genus g and p-rank f having an unramified double cover whose Prym has p-rank f' for each g/2 - 1 <= f' <= g - 2; (these Pryms are not ordinary). Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the l-torsion group scheme with the theta divisor of the Jacobian of a generic curve X of genus g and p-rank f.