Integral representations of p-class groups in \({\mathbb{Z}}_ p\)-extensions, semisimple differentials and Jacobians
For a ramified p-extension of cyclotomic \({\mathbb{Z}}_ p\)-number fields with Galois group G, the \({\mathbb{Z}}_ p[G]\)-module structure of the minus part of the p-primary ideal class group is obtained when the p-th roots of unity are not present. When the p-th roots of unitary are present, the integral representation is determined if all the decomposition groups of the ramified primes are contained in one of them and this one is normal. Analogous results are obtained for a wildly ramified p-extension of algebraic function fields of one variable over an algebraically closed field k of characteristic \(p>0\). Assuming the existence of a unique maximal decomposition group, the k[G]-module structure of the holomorphic semisimple differentials is obtained. For this family is also obtained the \({\mathbb{F}}_ p[G]\)-module structure of the elements of order dividing p in the null class group, \({\mathbb{F}}_ p\) denoting the field with p elements. Finally, the structure of the p-art of the Jacobian variety associated with the overfield, is determined as \({\mathbb{Z}}_ p[G]\)-module for the same class.
cyclotomic \({\mathbb Z}_ p\)-extensions
cyclotomic \({\mathbb Z}_ p\)-number fields
scientific article; zbMATH DE number 4128941
scientific article