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

TL;DR

This paper investigates the $d$-invariant in the Iwasawa theory of abelian varieties over function fields of positive characteristic, confirming conjectures and providing explicit formulas, especially for Jacobians and semistable varieties.

Contribution

It verifies Ulmer's conjecture relating the $d$-invariant to the Tate-Shafarevich group dimension and derives explicit formulas unconditionally for certain classes of abelian varieties.

Findings

Confirmed the $d$-invariant equals the Tate-Shafarevich group dimension.

Derived an explicit formula for the $d$-invariant involving Faltings height and Frobenius slopes.

Established the density of the $d=0$ locus in moduli spaces of elliptic surfaces.

Abstract

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 $\mathbb{Z}_p$-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.