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

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.

Accepted for publication in Algebra & Number Theory. No changes in the text from v3. 47 pages

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Cited by in corpus (4)

• Finiteness and cofiniteness of fine Selmer groups over function fields
• On $\mu$-invariants and isogenies for abelian varieties over function fields
• Non-commutative Iwasawa theory of abelian varieties over global function fields
• Iwasawa theory of fine Selmer groups over global fields