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$p$-class groups of certain extensions of degree $p$

Author: Christian Wittmann
Journal: Math. Comp. 74 (2005), 937-947
MSC (2000): Primary 11R29, 11R33, 11Y40
Published electronically: October 27, 2004
MathSciNet review: 2114656
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Abstract: Let $p$ be an odd prime number. In this article we study the distribution of $p$-class groups of cyclic number fields of degree $p$, and of cyclic extensions of degree $p$ of an imaginary quadratic field whose class number is coprime to $p$. We formulate a heuristic principle predicting the distribution of the $p$-class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to-$p$-part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.

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Additional Information

Christian Wittmann
Affiliation: Universität der Bundeswehr München, Fakultät für Informatik, Institut für Theoretische Informatik und Mathematik, 85577 Neubiberg, Germany

Keywords: Class groups, Galois modules, Cohen-Lenstra heuristics, numerical verifications
Received by editor(s): March 21, 2003
Received by editor(s) in revised form: March 27, 2004
Published electronically: October 27, 2004
Article copyright: © Copyright 2004 American Mathematical Society