English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$,leads to many interesting predictions about the distribution of $p$-classgroups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic toa non-abelian setting by considering, for each imaginary quadratic field $K$,the Galois group of the $p$-class tower of $K$, i.e.$G_K:=\mathrm{Gal}(K_\infty/K)$ where $K_\infty$ is the maximal unramified$p$-extension of $K$. By class field theory, the maximal abelian quotient of$G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\geq 1$, wegive a heuristic of Cohen-Lentra type for the maximal $p$-class $c$ quotient of$\G_K$ and thereby give a conjectural formula for how frequently a given$p$-group of $p$-class $c$ occurs in this manner. In particular, we predictthat every finite Schur $\sigma$-group occurs as $G_K$ for infinitely manyfields $K$. We present numerical data in support of these conjectures.

Heuristics for $p$-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst