Algebraicity of special values of \(L\)-functions
type: Item
instance of: scholarly article
title: Algebraicity of special values of \(L\)-functions
zbMATH Open document ID: 0666.12008
DOI: 10.1007/BF01394148
author: Pierre Colmez
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143649
review text: Let \(F\) be a finite extension of degree \(n\) of an imaginary quadratic field \(K\). Deligne has conjectured the algebraicity of special values of \(L\)-functions attached to certain Hecke characters of type \(A_ 0\) of \(F\). These conjectures were proved by Harder. The purpose of this paper is to provide a more elementary, though still very technical, proof of these conjectures under some special, but still fairly general, assumptions. The main theorem: Let \(\psi\) be a Hecke character of \(K\) whose infinity type is \(\bar z,\) and \(\vartheta\) a Dirichlet character of \(F\). Set \(\phi_ k=\vartheta (\psi^ k\circ N_{F/K})\). Let \(t\) be an integer. Then the value of the \(L\)-function of \(\phi_ k\) at \(t\) divided by \(\omega^{nk}\pi^{n(t-k)}\) is an algebraic number, where \(\omega\) is the real period of an elliptic curve with complex multiplication by \(k\) defined over \(\overline {\mathbb Q}\), when (i) \(n=2\), \(1\leq t\leq k\), \(F\) arbitrary, or (ii) \(n\geq 3\), \(t=k\), and the image of the action of \(\mathrm{Gal}(\overline{K}/K)\) on the embeddings of \(F\) in \(\mathbb C\) (lying over a fixed embedding of \(K\) in \(\mathbb C\) contains the alternating group of order \(n\). An explicit formula for the special values is given, in terms of derivatives of generalized Kronecker-Eisenstein series. The author's tools include a refined version of Shintani's method to find the sum of certain series indexed by lattices of units, and a distribution-theoretic study of the resulting generalized Kronecker-Eisenstein series.
reviewed by: Avner Ash
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 4089671
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: algebraicity of special values of L-functions
zbMATH Keywords: Hecke characters
zbMATH Keywords: Kronecker-Eisenstein series
MaRDI profile type: Publication
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OpenAlex ID: W2042930695
description: scientific article; zbMATH DE number 4089671