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

TL;DR

The paper develops an analogue of Kida's formula for Mazur–Tate elements attached to elliptic curves over abelian p-extensions. It proves that under mild hypotheses (Add) and (K-p), the μ-invariant of Mazur–Tate elements is preserved in p-extensions and provides an explicit λ-transition formula incorporating ramification data from primes where E has split multiplicative or nontrivial p-torsion behavior. The work unifies analytic and algebraic Iwasawa invariants across ordinary, supersingular, and additive reduction settings, and shows how prior Kida-type results for p-adic L-functions and Selmer groups follow from the Mazur–Tate framework. It also discusses applications to Delbourgo’s p-adic L-functions in additive reduction and outlines implications for refined main conjectures and Tate–Shafarevich group growth in p-extensions.

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 $μ$-invariant is preserved in the extension: if the level-$n$ Mazur-Tate element over $K$ has $μ= 0$, then the corresponding element over $L$ also has $μ= 0$. Moreover, the associated $λ$-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.

Figures (1)

• Figure 1: $g(T)$ decomposes as a product of Mazur-Tate elements $\Theta_{n+n_K}^{M_\psi}(f,\psi,T)$ for the extension $\mathbb{Q}_{(n+n_K)}/\mathbb{Q}$

Theorems & Definitions (37)

• Theorem 1.3: Kida's formula for elliptic curves
• Definition 2.1: Modular symbols
• Definition 2.2: Mazur--Tate elements
• Remark 2.3
• Lemma 2.4: Compatibility with $\pi^{M\ell,M}_n$
• proof
• Lemma 2.5: Birch's lemma
• proof
• Proposition 2.6
• Remark 2.7