On the mu-invariants of abelian varieties over function fields of positive characteristic

Let A be an abelian variety over a global function field K of characteristic p. We study the mu-invariant appearing in the Iwasawa theory of A over the unramified Z(p)-extension of K. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the mu-invariant) in terms of other quantities including the Faltings height of A and Frobenius slopes of the numerator of the Hasse-Weil L-function of A/K assuming the conjectural Birch-Swinnerton-Dyer formula. Our next result is to prove this mu-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "mu = 0" locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.