On $\mu$-invariants and isogenies for abelian varieties over function fields

TL;DR

This paper explores how Iwasawa d-invariants of abelian varieties over function fields change under isogeny, providing formulas and invariance results for the Birch--Swinnerton-Dyer conjecture without relying on previous Kato--Trihan results.

Contribution

It derives formulas for d-invariants under isogeny and proves the invariance of the BSD conjecture over function fields without Kato--Trihan's results.

Findings

Formulas for d-invariants under isogeny

Invariance of BSD conjecture over function fields

No reliance on Kato--Trihan's results

Abstract

We give several formulas for how Iwasawa $\mu$-invariants of abelian varieties over unramified $\mathbb{Z}_{p}$-extensions of function fields change under isogeny. These are analogues of Schneider's formula in the number field setting. We also prove that the validity of the Birch--Swinnerton-Dyer conjecture (including the leading coefficient formula) over function fields is invariant under isogeny, without using the result of Kato--Trihan.