The $\mu$-invariant change for abelian varieties over finite $p$-extensions of global fields

TL;DR

This paper investigates how the $mbda$-invariant of an ordinary abelian variety changes over finite Galois $p$-extensions of global fields, providing explicit bounds, asymptotic growth descriptions, and numerical evidence.

Contribution

It extends previous work by explicitly bounding local Galois cohomology sizes and describing the asymptotic behavior of $mbda$-invariants in $p$-extensions of global fields.

Findings

Explicit bounds for local Galois cohomology sizes at supersingular places.

Asymptotic growth descriptions of $mbda$-invariants in $bz_p^d$-extensions.

Numerical evidence supporting the theoretical results.

Abstract

We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $\mu$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size $\delta_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of $\delta_v$ along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for the change of $\mu$-invariants of $A$ from the tower $L/K$ to the tower $LK'/K'$. Finally, we present numerical evidence supporting these results.