An analogue of Kida's formula for Mazur-Tate elements

TL;DR

This paper establishes a Kida-type formula for Iwasawa invariants of Mazur-Tate elements attached to elliptic curves over number fields, extending previous results to more general reduction types and unifying known cases.

Contribution

It proves an analogue of Kida's formula for Mazur-Tate elements' Iwasawa invariants over abelian extensions, including cases with additive reduction.

Findings

Vanishing of the b5-invariant is preserved in extensions.

Explicit transition formula for b5 and b7 invariants.

Unification of Kida's formula for various p-adic L-functions.

Abstract

We prove an analogue of Kida's formula for the Iwasawa invariants of the Mazur-Tate elements attached to elliptic curves over $\mathbb{Q}$. Let $p$ be an odd prime and let $L/K$ be a Galois extension of abelian number fields with $p$-power Galois group. For an elliptic curve $E/\mathbb{Q}$, we study the Mazur-Tate elements over the finite layers of the cyclotomic $\mathbb{Z}_p$-extensions of $K$ and $L$. We show that the vanishing of the $\mu$-invariant is preserved in the extension: if the level-$n$ Mazur-Tate element over $K$ has $\mu = 0$, then the corresponding element over $L$ also has $\mu = 0$. Moreover, the associated $\lambda$-invariants satisfy an explicit transition formula. This parallels the work of Hachimori-Matsuno on Selmer groups and of Matsuno on $p$-adic $L$-functions. As an application, we obtain an analogue of Kida's formula for the analytic Iwasawa invariants associated to Pollack's signed $p$-adic $L$-functions. Since our results apply to elliptic curves with any reduction type at $p$ under mild hypotheses, including those with additive reduction, we also obtain a Kida-type formula for the $p$-adic $L$-functions constructed by Delbourgo for elliptic curves with unstable additive reduction. In particular, because Mazur-Tate elements approximate $p$-adic $L$-functions in the limit, our results unify all previously known cases of Kida's formula for analytic Iwasawa invariants.