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Integral Representation of p-Class Groups In ℤp-Extensions and the Jacobian Variety
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- Journal:
- Canadian Journal of Mathematics / Volume 50 / Issue 6 / 01 December 1998
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1253-1272
- Print publication:
- 01 December 1998
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- Article
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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 
$L/k$.
