Congruences between derivatives of geometric \(L\)-functions. With an appendix by David Burns, King Fai Lai and Ki-Seng Tan
type: Item
instance of: scholarly article
title: Congruences between derivatives of geometric \(L\)-functions. With an appendix by David Burns, King Fai Lai and Ki-Seng Tan
zbMATH Open document ID: 1234.11080
published in: Inventiones Mathematicae
publication date: 2011-05-25T00:00:00Z
review text: The main theorem of this article establishes the truth of a deep conjecture concerning the leading terms of \(L\)-functions of Abelian extensions of global function fields. The conjecture, which we shall discuss in more detail below, is a refinement of a conjecture of Lichtenbaum relating the Weil-étale cohomology of curves over finite fields to the associated zeta function. It can also be seen as a function field analogue of a conjecture concerning \(L\)-functions of Galois extensions of number fields, namely the equivariant Tamagawa number conjecture of Burns-Flach in the setting of Artin \(L\)-functions at \(s= 0\). The theorem also implies the truth of a number of related conjectures, as is discussed in the article. Explicitly, all the following now hold in the function field setting: Chinburg's \(\Omega(3)\) conjecture for Abelian extensions, Chinburg's \(\Omega(1)\) conjecture for tamely ramified Abelian extensions, the congruence conjecture of [\textit{D. Burns}, Invent. Math. 169, No. 3, 451--499 (2007; Zbl 1133.11063)] (whence the title of the article), Rubin's integral refinement of Stark's conjecture, the refined class number formulas of Gross, Tate, and Aoki-Lee-Tan, and finally the refined \(p\)-adic abelian Stark conjecture of Gross. Let us now give a brief formulation of the theorem, omitting many details. We fix a finite Abelian extension \(F/k\) of global function fields with Galois group \(G\). The constructions take place in the Picard category \({\mathcal P}(R)\) of graded line bundles on \(\text{Spec}(R)\) with \(R= \mathbb Z[G]\) and \(R= \mathbb Q[G]\). The category \({\mathcal P}(R)\) has unit object \(\mathbf{1}_R\). Further, there is the determinant functor \({\mathbf d}_R\) of Grothendieck-Knudsen-Mumford which assigns to a perfect complex of \(R\)-modules an object of \({\mathcal P}(R)\). If \(C^*\) is a perfect \(\mathbb Q[G]\)-complex acyclic outside two consecutive degrees, then any isomorphism \(\lambda\) between the cohomology groups…
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R65
Mathematics Subject Classification ID: 19A31
Mathematics Subject Classification ID: 19B28
zbMATH DE Number: 5899561
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: leading terms
zbMATH Keywords: function fields
zbMATH Keywords: geometric Iwasawa theory
author: David Burns
reviewed by: Paul Buckingham
MaRDI profile type: Publication
full work available at URL: s00222-010-0286-3
OpenAlex ID: W2027593035
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DOI: 10.1007/S00222-010-0286-3
description: scientific article; zbMATH DE number 5899561