Kida’s formula for split prime $\mathbb{Z}_p$-extensions over imaginary quadratic fields
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 27-55

Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ splits into $\mathfrak{p}$ and $\mathfrak{p}^{\ast }$. Then there exists a uniquely defined $\mathbb{Z}_p$-extension $N_{\infty }/k$ such that the prime ideal $\mathfrak{p}^{\ast }$ does not ramify. For a finite extension $K/k$, we call $K_{\infty }=KN_{\infty }$ the split prime $\mathbb{Z}_{p}$-extension corresponding to $\mathfrak{p}$. We prove an analogue of Kida’s formula for the split prime $\mathbb{Z}_p$-extensions. As an application, we apply this formula to $\mathfrak{p}$-ramified Iwasawa modules and determine the isomorphism classes of unramified Iwasawa modules associated to $\mathbb{Z}_p$-extensions over $k$.

Soit $p$ un nombre premier impair et $k$ un corps quadratique imaginaire dans lequel $p$ est décomposé en $\mathfrak{p}$ et $\mathfrak{p}^{\ast }$. Alors il existe une unique $\mathbb{Z}_p$-extension $N_{\infty }/k$ telle que l’idéal premier $\mathfrak{p}^{\ast }$ soit non ramifié. Pour une extension finie $K/k$, nous appelons $K_{\infty }=KN_{\infty }$ la $\mathbb{Z}_{p}$-extension de type split prime correspondant à $\mathfrak{p}$. Nous démontrons un analogue de la formule de Kida pour les $\mathbb{Z}_p$-extensions de type split prime. Comme application, nous appliquons cette formule aux modules d’Iwasawa $\mathfrak{p}$-ramifiés et déterminons les classes d’isomorphisme des modules d’Iwasawa non ramifiés associés aux $\mathbb{Z}_p$-extensions sur $k$.

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DOI : 10.5802/jtnb.1353
Classification : 11R23
Keywords: Iwasawa modules, Iwasawa $\lambda $-invariants, Kida’s formula
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Kazuaki Murakami. Kida’s formula for split prime $\mathbb{Z}_p$-extensions over imaginary quadratic fields. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 27-55. doi: 10.5802/jtnb.1353
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