On special elements in higher algebraic -theory and the Lichtenbaum–Gross Conjecture (original) (raw)

On special elements in higher algebraic K-theory and the Lichtenbaum-Gross Conjecture

2011

We conjecture the existence of special elements in odd degree higher algebraic K-groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of Artin L-functions of finite dimensional complex representations. We prove this conjecture in certain important cases and also provide other evidence (both theoretical and numerical) in its support.

1 on Special Elements in Higher Algebraic K-Theory and the Lichtenbaum-Gross Conjecture

2016

On The Determination Of Number Fields By Artin L-Functions

arXiv: Number Theory, 2015

Let kkk be a number field, K/kK/kK/k a finite Galois extension with Galois group GGG, chi\chichi a faithful character of GGG. We prove that the Artin L-function L(s,chi,K/k)L(s,\chi,K/k)L(s,chi,K/k) determines the Galois closure of KKK over Q\QQ. In the special case k=Qk=\Qk=Q it also determines the character chi\chichi.

Are number fields determined by Artin L-functions?

Journal of Number Theory, 2016

Let k be a number field, K/k a finite Galois extension with Galois group G, χ a faithful character of G. We prove that the Artin L-function L(s, χ, K/k) determines the Galois closure of K over Q. In the special case k = Q it also determines the character χ.

Conjectures on the logarithmic derivatives of Artin L-functions II

arXiv: Algebraic Geometry, 2018

We formulate a general conjecture relating Chern classes of subbundles of Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin L-functions of number fields. This conjecture may be viewed as a far-reaching generalisation of the (Lerch-)Chowla-Selberg formula computing logarithms of periods of elliptic curves in terms of special values of the Gamma\\GammaGamma-function. We prove several special cases of this conjecture in the situation where the involved Artin characters are Dirichlet characters. This article contains the computations promised in the article {\\it Conjectures sur les d\\'eriv\\'ees logarithmiques des fonctions L d'Artin aux entiers n\\'egatifs}, where our conjecture was announced. We also give a quick introduction to the Grothendieck-Riemann-Roch theorem and to the geometric fixed point formula, which form the geometric backbone of our conjecture.

A Note on the Artin Conjecture

2006

Let K/Q be a Galois extension of Q and ρ : Gal(K/Q) −→ GL(n,C) a nontrivial irre-ducible representation of its Galois group. E. Artin [1] associated to this data an L-function L(s,ρ), defined for Res > 1, which he conjectured to continue analytically to an entire func-tion on the whole ...

IwasawaL-functions of varieties over algebraic number fields

Inventiones Mathematicae, 1983

A fascinating task in algebraic number theory is the study of the values of various complex L-functions (Dedekind zeta function, Artin L-function, Hasse-Weil L-function ...) at integer points. It has often turned out that these values are essentially rational numbers (see ). Therefore it is of course a fundamental problem to give arithmetic interpretations of these numbers. One possibility of attack on this problem seems to be the following: First interpolate these rational numbers by a p-adic L-function and then relate this function to the characteristic power series of an "Iwasawa module" which is naturally associated with the underlying arithmetic problem. But this program is extremely difficult and has been fully established only in special cases (recent work of Mazur/Wiles concerning the "main conjecture" in cyclotomic Iwasawa theory). For this reason, we pursue in this paper the much simpler problem of calculating, up to a p-adic unit, the values of the above mentioned characteristic power series at integer points. A considerable body of work has already been done in this direction. Of course, the results we obtain contain much of this earlier work, and also are compatible with known conjectures about the corresponding values of the complex L-functions. Let X be a proper smooth scheme over an algebraic number field k; let [r be an algebraic closure of k, );: =X x k, and Gk: =Gal(k/k) the absolute Galois group of k. Obviously H~ 2g) is a Gk-mOdule finitely generated and free over Z. It defines by duality an algebraic torus T(X) over k (for example, T(Spec(k)) is the multiplicative group G m over k). In Part II of this paper we shall define and study the Iwasawa L-functions of an arbitrary algebraic torus T over k. In the case T= T(X) they should be viewed as the 0-dimensional Iwasawa Lfunctions of X, because they depend only on the 0-cohomology of X. Their complex analogue is the 0-dimensional L-function of X in the sense of Serre , which is nothing else but the Artin L-function (in the sense of [6] which differs slightly from the original one) associated with the representation of G k on H ~ ()~, @).

On the existence of special elements in odd KKK-theory groups

arXiv: Number Theory, 2020

Let kkk be an imaginary quadratic number field, and F/kF/kF/k a finite abelian extension of Galois group GGG. We investigate the relationship between the conjectural special elements introduced in \cite{Burns-DeJeu-Gangl} and ETNC in the semi-simple case. This provides a partial proof of the conjecture for F/kF/kF/k under certain conditions.