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

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.

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 ...

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.