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

Technical Summary: An Analogue of Kida's Formula for Mazur–Tate Elements

Problem Statement
The paper addresses the problem of establishing an analogue of Kida's formula for the Iwasawa invariants ($\mu$ and $\lambda$) associated with Mazur–Tate elements attached to elliptic curves over $\mathbb{Q}$. While Kida's classical formula relates the growth of class numbers in $p$-extensions of number fields to ramification data, and subsequent works by Hachimori–Matsuno and Matsuno extended this to Selmer groups and $p$-adic $L$-functions respectively, a unified formula for Mazur–Tate elements—finite-level analogues of $p$-adic $L$-functions—had not been fully established for elliptic curves with arbitrary reduction types at $p$. Specifically, the authors aim to determine how the vanishing of the $\mu$-invariant and the value of the $\lambda$-invariant behave when passing from an abelian number field $K$ to a $p$-extension $L$, without imposing restrictive hypotheses on the local behavior of the elliptic curve at $p$ (e.g., excluding additive reduction).

Methodology
The authors employ a combination of Iwasawa theory, modular symbols, and the arithmetic of Mazur–Tate elements. The core methodology involves:

1. Mazur–Tate Elements over Abelian Extensions: The authors define Mazur–Tate elements $\Theta_n(E/K)$ for an elliptic curve $E$ over an abelian extension $K/\mathbb{Q}$ at finite levels $n$ of the cyclotomic $\mathbb{Z}_p$-extension. These elements are constructed via modular symbols associated with the weight 2 newform $f$ corresponding to $E$ (via the Modularity Theorem).
2. Interpolation Properties: Utilizing Birch's Lemma and properties of modular symbols, the paper establishes that these elements interpolate special values of twisted $L$-functions $L(f/K, \chi, 1)$.
3. Local Analysis of Euler Factors: A critical technical step involves analyzing the compatibility of Mazur–Tate elements under projection maps between different levels of the cyclotomic tower. The authors define a local invariant $g_{\psi, n}(\ell)$ which measures the contribution of a prime $\ell$ to the change in the $\lambda$-invariant. This is derived by examining the Iwasawa-theoretic Euler factors $h_\ell(\psi, T)$ associated with the Hecke operators.
4. Character Summation: To transition from the base field $K$ to the extension $L$, the authors sum the local contributions $g_{\psi, n}(\ell)$ over all characters of the Galois group $\text{Gal}(K/\mathbb{Q})$. This summation relies on character-counting results for abelian extensions to relate the local behavior of the curve over $K$ to the ramification indices in $L$.
5. Reduction Types: The analysis distinguishes between split multiplicative reduction, good reduction with local $p$-torsion, and additive reduction. For additive reduction, the authors rely on the "potentially good ordinary" condition (Delbourgo's hypothesis) to relate Mazur–Tate elements to bounded $p$-adic $L$-functions.

Key Contributions and Results

• Main Theorem (Theorem 1.3): The paper proves a Kida-type formula for the Iwasawa invariants of Mazur–Tate elements. Let $L/K$ be a Galois extension of abelian number fields with a $p$-power Galois group. If the $\mu$-invariant of the Mazur–Tate element over $K$ vanishes ($\mu(\Theta_{n+n_L-n_K}(E/K)) = 0$), then the $\mu$-invariant over $L$ also vanishes. Furthermore, the $\lambda$-invariants satisfy:
  $$\lambda(\Theta_n(E/L)) = [L_\infty : K_\infty] \lambda(\Theta_{n+n_L-n_K}(E/K)) + \sum_{w \in P_1} f(w)(e(w)-1) + 2 \sum_{w \in P_2} f(w)(e(w)-1)$$
  where $P_1$ is the set of primes where $E$ has split multiplicative reduction, $P_2$ is the set of primes where $E$ has good reduction and non-trivial $p$-torsion, $e(w)$ is the ramification index, and $f(w)$ is the inertial degree.

• Generalization to Modular Forms: The result is generalized to Hecke eigencuspforms (Theorem 3.7), removing the need for the curve to be defined over $\mathbb{Q}$ in the strict sense, provided the reduction types are defined locally.

• Applications to $p$-adic $L$-functions:

  • Semistable Reduction: The authors show that their formula recovers Matsuno's result for $p$-ordinary elliptic curves and extends it to the case of good supersingular reduction (where $a_p(E)=0$), providing a Kida formula for the signed $p$-adic $L$-functions constructed by Pollack (Theorem 4.2).
  • Additive Reduction: Under the potentially good ordinary hypothesis of Delbourgo, the authors derive a Kida formula for the $p$-adic $L$-functions constructed by Delbourgo for elliptic curves with additive reduction (Theorem 4.5).

Significance and Claims
The paper claims that its results unify previously disparate cases of Kida's formula for analytic Iwasawa invariants. By working directly with Mazur–Tate elements, which approximate $p$-adic $L$-functions in the limit, the authors provide a framework that applies to elliptic curves with any reduction type at $p$ (including additive reduction) under mild hypotheses.

The authors state that this work parallels the algebraic results of Hachimori–Matsuno on Selmer groups but operates on the analytic side. A primary significance is the ability to handle cases where the existence of a bounded $p$-adic $L$-function was previously unknown or required specific conditions (such as the potentially good ordinary condition). The paper asserts that these formulae align with the philosophy of the refined main conjectures, suggesting that an algebraic analogue would imply refined conjectures for base fields that are abelian extensions of $\mathbb{Q}$. The authors explicitly note that their results do not invent new applications but rather provide the theoretical foundation to obtain growth formulae for Tate–Shafarevich groups in $p$-extensions of abelian number fields, contingent on the validity of the Main Conjecture.