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Integral Representation of p -Class Groups In ℤ p -Extensions and the Jacobian Variety

Published online by Cambridge University Press:  20 November 2018

Pedro Ricardo López-Bautista
Affiliation:
Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Av. San Pablo No. 180, Col. Reynosa Tamaulipas, Azcapotzalco D.F., C.P. 02200 México email: rlopez@hp9000a1.uam.mx
Gabriel Daniel Villa-Salvador
Affiliation:
Departamento de Matemáticas Centro de Investigación y de Estudios Avanzados, del I.P.N., Apartado Postal 14-740, 07000 México, D.F. email: villa@math.cinvestav.mx
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Abstract

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For an arbitrary finite Galois $p$ -extension $L/K$ of ${{\mathbb{Z}}_{p}}$ -cyclotomic number fields of $\text{CM}$ -type with Galois group $G=\text{Gal}(L/K)$ such that the Iwasawa invariants $\mu _{K}^{-},\,\mu _{L}^{-}$ are zero, we obtain unconditionally and explicitly the Galois module structure of $C_{L}^{-}\,(p)$ , 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 $G=\text{Gal}(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$ -torsion part of the Jacobian variety ${{J}_{L}}(p)$ associated to $L/k$ .

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1998

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