An analogue of Kida's formula for Mazur-Tate elements
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911289939329024 |
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| author | Pratap, Naman Ray, Anwesh |
| author_facet | Pratap, Naman Ray, Anwesh |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An analogue of Kida's formula for Mazur-Tate elements Pratap, Naman Ray, Anwesh Number Theory 11R23, 11G05 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. |
| title | An analogue of Kida's formula for Mazur-Tate elements |
| topic | Number Theory 11R23, 11G05 |
| url | https://arxiv.org/abs/2511.21979 |