#### Vol. 15, No. 4, 2021

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

### King-Fai Lai, Ignazio Longhi, Takashi Suzuki, Ki-Seng Tan and Fabien Trihan

Vol. 15 (2021), No. 4, 863–907
##### 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 ${ℤ}_{p}$-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.

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Iwasawa theory, abelian variety, Selmer group, $\mu$-invariant, elliptic surface