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Section: New Results

Class groups and other invariants of number fields

Participants : Karim Belabas, Jean-Paul Cerri, Henri Cohen, Pınar Kılıçer, Pierre Lezowski.

Ohno and Nakagawa have proved relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of 'extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. The paper [14] by Henri Cohen, Simon Rubinstein-Salzedo and Frank Thorne proves an identity relating certain degree fields with Galois groups $D$ and $F$, respectively, for any odd prime, giving another proof of the Ohno–Nakagawa relation between the counting functions of certain cubic fields.

Pınar Kılıçer and Marco Streng have solved a variant of the class number 1 problem for quartic CM fields with a geometric motivation [27] ; the question is whether a certain class group is trivial, which corresponds to a genus 2 curve with that complex multiplication being defined over a real-quadratic number field (instead of an extension). Using classical techniques provides a bound on the discriminant of such fields, which they refine taking ramification into account to obtain a practically useful bound. A carefully crafted enumeration algorithm finishes the proof.

In the article [28] , P. Lezowski studies the Euclidean properties of matrix algebras $M_n\left(R\right)$ over commutative rings $R$. In particular, he shows that for any integer $n>1$, $M_n\left(R\right)$ is a left and right Euclidean ring if and only if $R$ is principal. The proof is constructive and allows to better understand the Euclidean order types of matrix algebras. Similar ideas are also applied to prove $k$-stage Euclidean properties of $M_n\left(R\right)$, linking them with Bézout property for the ring $R$. The article [28] has been submitted to Journal of Algebra.

The article by Aurel Page on the computation of arithmetic Kleinian groups has appeared [21] .