Section: New Results

Class groups and other invariants of number fields

The article by H. Cohen and F. Thorne on Dirichlet series associated to quartic fields with given cubic resolvent has been published. This article gives an explicit formula for the Dirichlet series ${\sum }_K{\left|\Delta \left(K\right)\right|}^{-s}$, where the sum is over isomorphism classes of all quartic fields whose resolvent field is isomorphic to a fixed cubic field $k$.

The article [22] by H. Cohen and F. Thorne generalizes the work of A. Morra and the authors, on giving explicit formulas for the Dirichlet series generating function of $D_{\ell }$ extensions of odd prime degree $\ell$ with given quadratic resolvent. Over the course of the proof, the authors explain connections between their formulas and the Ankeny-Artin-Chowla conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other topics.

In her thesis, Iuliana Ciocanea-Teodorescu describes algorithms that answer questions arising in ring and module theory. The first main result of this thesis concerns the module isomorphism problem, how to compute a set of generators of minimal cardinality, and how to construct projective covers and injective hulls. The thesis also describe tests for module simplicity, projectivity, and injectivity, and constructive tests for existence of surjective module homomorphisms between two finite modules, one of which is projective. As a negative result, the problem of testing for existence of injective module homomorphisms between two finite modules, one of which is projective, is NP-complete. The last part of the thesis is concerned with finding a good working approximation of the Jacobson radical of a finite ring, that is, a two-sided nilpotent ideal such that the corresponding quotient ring is almost semisimple. The notion used to approximate semisimplicity is that of separability.

In her thesis [11], Pinar Kiliçer determines all CM curves of genus 2 defined over the reflex field. This extends the previous CM class number one problem for elliptic curves which asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring.