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Annales mathématiques du Québec

, Volume 37, Issue 1, pp 79–108 | Cite as

The analogue of the Gauss class number problem in motivic cohomology

  • Caroline Junkins
  • Manfred Kolster
Original Paper
  • 131 Downloads

Abstract

The classical Gauss problem of determining all imaginary quadratic number fields of class number one has an analogue involving finite motivic cohomology groups attached to the ring of integers \(o_F\) in a totally real number field \(F\). In the classical situation, the value of the zeta function of \(F\) at \(0\) can be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the class number. For a totally real field \(F\) and an even integer \(n \ge 2\), the value of the zeta-function of \(F\) at \(1-n\) can also be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the order \(h_n(F)\) of the motivic cohomology group \(H_\mathcal M ^2(o_F,\mathbb Z (n))\). This order is always divisible by \(2^d\), where \(d\) is the degree of \(F\). We determine all totally real fields \(F (\ne \mathbb{Q })\) and all even integers \(n \ge 2\), for which the quotient \(\frac{h_n(F)}{2^d}\) is equal to 1. There are no fields for \(n \ge 6\), there is only the field \(\mathbb{Q }(\sqrt{5})\) for \(n = 4\), and there are 11 fields for \(n=2\). The motivic cohomology groups \(H_\mathcal M ^2(o_F,\mathbb Z (n))\) contain the canonical subgroups \(WK^\mathcal M _{2n-2}(F)\), called motivic wild kernels, which are analogous to Tate–Shafarevic groups. For even integers \(n \ge 4\), there is again only the case \(n = 4\) and the field \(\mathbb{Q }(\sqrt{5})\), for which the motivic wild kernel \(WK^\mathcal M _{2n-2}(F)\) vanishes. However, for \(n=2\), among the totally real fields \(F\) of degrees between \(2\) and \(9\), we found 21 of them, for which \(WK^\mathcal M _{2}(F)\) vanishes, the largest degree being \(5\). A basic assumption in our approach is the validity of the \(2\)-adic Main Conjecture in Iwasawa theory for the trivial character, which so far has only been proven for abelian number fields (by A. Wiles).

Keywords

Special values of zeta-functions Gauss problem Motivic cohomology Algebraic \(K\)-theory Iwasawa theory 

Résumé

Un problème classique de Gauss a été de déterminer tous les corps de nombres quadratiques imaginaires ayant un nombre de classes égal à 1. Ce problème a un analogue pour les groupes de cohomologie motivique finis rattachés à l’anneau \(o_F\) des entiers algébriques d’un corps de nombres \(F\) totalement réel. Dans le cas classique, les valeurs de la fonction zêta de Dedekind évaluée en 0 peuvent s’écrire comme quotients de deux entiers,  non nécessairement copremiers entre eux, et avec le numérateur égal au nombre de classes. Dans la situation d’un corps \(F\) totalement réel, avec \(n\) un entier pair \(\ge 2\), les valeurs de la fonction zêta de \(F\) en \(1-n\) peuvent s’écrire comme quotients de deux entiers, non nécessairement copremiers entre eux, mais avec le numérateur égal à l’ordre \(h_n(F)\) des groupes de cohomologie motivique \(H_\mathcal M ^2(o_F,{Z}(n))\). Ces ordres sont toujours divisibles par \(2^d\), où \(d\) est le degré de \(F\). Nous déterminons tous les corps de nombres \(F\) (\(\ne \mathbb Q \)) totalement réels, et tous les entiers pairs \(n\ge 2\), pour lesquels \(\frac{h_n(F)}{2^d}=1\). Il n’y a aucun corps pour \(n\ge 6\), il n’y a que le corps \(\mathbb Q (\sqrt{5})\) pour \(n=4\), et il y a 11 corps totalement réels pour \(n=2\). Les groupes de cohomologie motivique \(H_\mathcal M ^2(o_F,\mathbb Z (n))\) contiennent des sous-groupes canoniques \(WK^\mathcal M _{2n-2}(F)\), appelés noyaux sauvages motiviques, qui sont les analogues des groupes de Tate-Shafarevich. Pour les entiers pairs \(n\ge 4\), une fois de plus le noyau sauvage motivique \(WK^\mathcal M _{2n-2}(F)\) s’annule seulement pour le corps \(\mathbb Q (\sqrt{5})\) et pour \(n=4\). Cependant, pour \(n=2\), parmi les corps de nombres totalement réels \(F\) de degrés entre \(2\) et \(9\), nous en avons trouvé 21 pour lesquels \(WK^\mathcal M _{2}(F)\) s’annnule, le plus grand degré de ces corps étant \(5\). Une hypothèse de base de notre méthode est la validité de la conjecture principale 2-adique de la théorie d’Iwasawa, qui jusqu’à présent n’a été prouvée que pour les corps de nombres abéliens (par A. Wiles).

Mathematics Subject Classification

11R23 11R42 11R70 11Y40 

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Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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