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arXiv · 1909.00511

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

King-Fai Lai·Ignazio Longhi·Takashi Suzuki·Ki-Seng Tan·Fabien Trihan
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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.

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King-Fai Lai, Ignazio Longhi, Takashi Suzuki, Ki-Seng Tan, Fabien Trihan. 2019-09-02. On the $\mu$-invariants of abelian varieties over function fields of positive characteristic. https://doi.org/10.2140/ant.2021.15.863

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