An arithmetic site for the rings of integers of algebraic number fields

Schmidt, Alexander

doi:10.1007/bf01232391

Cited by 2 publications

(1 citation statement)

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“…The reason for this generality is that for arithmetic and geometric purposes, it is advantageous to allow for more general topologies than the Zariski topology; for instanceétale topology [6] or the positive topology [8]. (Of course, we could use any topological space as base here, i.e., not necessarily restricting our discussion to schemes.…”

Section: Global Hom-lie Structuresmentioning

confidence: 99%

Arithmetic Witt-hom-Lie algebras

Larsson¹

2009

J Generalized Lie Theory Appl

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This paper is concerned with explaining and further developing the rather technical definition of a hom-Lie algebra given in a previous paper which was an adaption of the ordinary definition to the language of number theory and arithmetic geometry. To do this we here introduce the notion of Witt-hom-Lie algebras and give interesting arithmetic applications, both in the Lie algebra case and in the hom-Lie algebra case. The paper ends with a discussion of a few possible applications of the developed hom-Lie language.

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Section: Global Hom-lie Structuresmentioning

confidence: 99%

Arithmetic Witt-hom-Lie algebras

Larsson¹

2009

J Generalized Lie Theory Appl

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On $p$-Adic Zeta-Functions Associated to the Positive Topology of Algebraic Number Fields

Schmidt

1997

Publ. Res. Inst. Math. Sci.

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In [S] we introduced a new Grothendieck topology (called positive topology) over the rings of integers of algebraic number fields. If K is a finite extension of Q then Spec(0^) furnished with the positive topology shows a behavior very similar to the etale site over a complete curve over a finite field.In this paper we are going to investigate /7-adic zeta functions associated to the positive topology. Following the analogy it is natural to expect a functional equation. The aim of this paper is to show that such a functional equation holds true as soon as a primitive/?*' 1 root of unity is contained in K.If K/k is an abelian extension then the positive /?-adic zeta-function of K splits into a product of positive p-adic L-functions attached to the characters of K/k in the usual way. If the field k is totally real and "totally /7-real" (e.g. k = Q, see below for the definition) we calculate (at least up to the Greenberg conjecture) the positive j?-adic L-functions L p p os (s,\l/) in terms of the analytic /7-adic L-functions defined by Kubota/Leopoldt and Deligne/Ribet. §1. Review of the Positive TopologyIn this section we want to summarize the basic definitions and results about the positive topology over the rings of integers of algebraic number fields. Details and proofs can be found in [S]. Let p be a prime number. We

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An arithmetic site for the rings of integers of algebraic number fields

Cited by 2 publications

References 12 publications

Arithmetic Witt-hom-Lie algebras

Arithmetic Witt-hom-Lie algebras

On $p$-Adic Zeta-Functions Associated to the Positive Topology of Algebraic Number Fields

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