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On a question of Hayes concerning integrality of Brumer elements

Content Provider	Springer Nature Link
Author	Greither, Cornelius
Copyright Year	2015
Abstract	To every abelian Galois extension K / k of number fields with group G, one can associate a so-called Brumer element $$\theta _{K/k,S}$$ of the rational group ring $$\mathbb Q[G]$$ (depending on another technical parameter S). In a certain sense this element can be thought of as an equivariant version of the class number $$h_K$$ . There is a slight problem since this element is only “almost” integral (that is, it may have non-integral coefficients). The potential denominators always divide $$w_K$$ , the number of roots of unity in K. Hayes (Contemp Math 358:193–205, 2004) raised the question whether the Brumer element will be p-integral under certain suitable hypotheses, one of which implies that p divides $$h_K$$ ; of course only situations with $$p\|w_K$$ are of interest. This paper answers this question in the negative. We start with an “almost counterexample” over $$\mathbb Q$$ (for $$k=\mathbb {Q}$$ there are no true counterexamples) and manufacture true counterexamples in which k is a suitable real quadratic field. Using deep recent results on the distribution of class numbers (Bhargava et al.) one can show that the method in fact yields infinitely many counterexamples.Pour toute extension abélienne K / k de corps de nombres ayant G pour groupe de Galois, on sait construire un élément $$\theta _{K/k,S}$$ (appelé de Brumer) dans l’anneau de groupe $$\mathbb Q[G]$$ à coefficients rationnels. (Cet élément aussi dépend d’un paramètre auxiliaire S.) Dans un certain sens cet élément est une version équivariante du nombre $$h_K$$ de classes. Il y a un petit problème puisque $$\theta _{K/k,S}$$ n’est pas tout à fait entier (c’est-à-dire, à coefficients dans $$\mathbb Z$$ ). On peut toutefois l’écrire avec un dénominateur commun qui divise $$w_K$$ , le nombre de racines de l’unité contenues dans K. En 2004 D. Hayes posa la question: Est-ce que la p-intégralité de l’élément de Brumer est entraînée par quelques hypothèses bien adaptées (l’une parmi elles impliquant que le nombre $$h_K$$ est divisible par p)? Bien sûr, seuls les cas où p divise $$w_K$$ sont intéressants. Cet article établit une réponse négative à cette question. Pour commencer on regarde un “presque-contre-exemple” sur $$\mathbb Q$$ (il n’y a pas de vrai contre-exemple sur le corps des rationnels), et on le modifie pour trouver de vrais contre-exemples pour lesquels k est un corps réel quadratique. En utilisant de profonds résultats récents sur la distribution des nombres de classes (Bhargava et al.) on peut même montrer qu’il y a une infinité de contre-exemples.
Starting Page	97
Ending Page	104
Page Count	8
File Format	PDF
ISSN	21954755
Journal	Annales mathématiques du Québec
Volume Number	41
Issue Number	1
e-ISSN	21954763
Language	English
Publisher	Springer International Publishing
Publisher Date	2015-11-23
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Integrality Class numbers, class groups, discriminants Algebra L-functions Analysis Brumer elements Mathematics Class groups Number Theory Zeta functions and $L$-functions of number fields Cyclotomic extensions
Content Type	Text
Resource Type	Article
Subject	Mathematics

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