Section: New Results

Number and function field enumeration

Participants : Henri Cohen, Anna Morra, Pieter Rozenhart.

In joint work with R. Scheidler and M. Jacobson, P. Rozenhart has generalized Belabas's algorithm for tabulating cubic number fields to cubic function fields [30] . This generalization required function field analogues of the Davenport-Heilbronn Theorem and of the reduction theory of binary cubic and quadratic forms. As an additional application, they have modified the tabulation algorithm to compute 3-ranks of quadratic function fields by way of a generalisation of a theorem due to Hasse. The algorithm, whose complexity is quasi-linear in the number of reduced binary cubic forms up to some upper bound $X$, works very well in practice. A follow-up article [29] describes how to use these results to compute 3-ranks of quadratic function fields, in particular yielding examples of unusually high 3-rank.

H. Cohen and A. Morra [12] have obtained an explicit expression for the Dirichlet generating function associated to cubic extensions of an arbitrary number field with a fixed quadratic resolvent. As a corollary, they have proved refinements of Malle's conjecture in this context.