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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
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.