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

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.