On derivatives of Artin \(L\)-series
type: Item
instance of: scholarly article
title: On derivatives of Artin \(L\)-series
zbMATH Open document ID: 1239.11128
published in: Inventiones Mathematicae
publication date: 2011-11-22T00:00:00Z
review text: Stark conjectures on units in global fields predict the rationality of certain quotients ``leading term of an \(L\)-function divided by a regulator''. So-called refined Stark conjectures make integrality statements about these quotients. Whilst the original conjectures were formulated for any Galois extension \(K/k\), the refined versions mostly assumed the extensions to be abelian (for example in the Rubin-Stark conjectures). The article under review is a breakthrough in several respects. It formulates a ``refined non-abelian Stark conjecture'' (Conj. 2.4.1), relates it to the earlier versions, and it provides rather strong evidence for the truth of this conjecture. To wit, the author proves that the Leading Term Conjecture for L-functions (also called ETNC for an appropriate motive) implies this Conjecture 2.4.1. Moreover, the article gives plenty of unconditional results on the conjecture and its consequences, and provides very detailed information concerning relations to other work. In this review we have to be selective, because of the length and encompassing character of this article. Let us begin by trying to sketch the statement of the pivotal Conjecture 2.4.1, also called the Central Conjecture (CC). Some notation is necessary. Always, \(S\) is a finite set of places of \(k\) containing the infinite and the ramified places. Let \(U_{K,S}\) denote the \(S\)-units of \(K\) and \(X_{K,S}\) the kernel of augmentation on the free \(\mathbb Z\)-module with basis \(S(K)\). There is also a finite auxiliary set \(T\), disjoint from \(S\), and an equivariant \(S\)-truncated \(T\)-modified \(L\)-function, whose definition we will not recall here. These lead to equivariant leading terms \(\theta^{(r)}(0)\) depending on all the data. Then for any \(G\)-linear map \(\phi: U_{K,S} \to X_{K,S}\), one has the product \(h:=\theta^{(r)}(0) R(\phi)\), where \(R(\phi)\) is the ``determinant'' (more precisely: the reduced norm) of \(\phi\) followed by \textit{the inverse of} the…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 5976095
zbMATH Keywords: Artin L-series
zbMATH Keywords: refined Stark conjectures
zbMATH Keywords: annihilators
author: David Burns
MaRDI profile type: Publication
full work available at URL: s00222-011-0320-0
OpenAlex ID: W2050101534
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DOI: 10.1007/S00222-011-0320-0
description: scientific article; zbMATH DE number 5976095