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# Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety

Published:1998-12-01
Printed: Dec 1998
• Pedro Ricardo López-Bautista
• Gabriel Daniel Villa-Salvador
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## Abstract

For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $\mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$.
 Keywords: ${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure
 MSC Classifications: 11R33 - Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] 11R23 - Iwasawa theory 11R58 - Arithmetic theory of algebraic function fields [See also 14-XX] 14H40 - Jacobians, Prym varieties [See also 32G20]

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