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

Abstract

We give several formulas for how Iwasawa $μ$-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.

Theorems & Definitions (51)

• Theorem 1.1: Proposition \ref{['0049']}, Theorem \ref{['0004']}
• Theorem 1.2: Theorem \ref{['0030']}
• Theorem 1.3: Theorem \ref{['0031']}
• Theorem 1.4: Theorem \ref{['0047']}
• Theorem 1.5: Theorem \ref{['0051']}
• Proposition 3.1
• proof
• Definition 4.1
• Proposition 4.2
• proof