On Gauss sum characters of finite groups and generalized Bernoulli numbers
type: Item
instance of: scholarly article
title: On Gauss sum characters of finite groups and generalized Bernoulli numbers
zbMATH Open document ID: 0848.11052
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-03-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93617
full work available at URL: jtnb7-1.html
review text: Let \(f: X\to Y\) be a Galois covering of compact Riemann surfaces with \(G= \text{Gal}(X/Y)\) (finite) and \(V= H^0(X, \Omega^1_X)\) denote the space of holomorphic differentials on \(X\). The problem addressed in this paper is the decomposition of \(V\) into irreducible characters of \(G\). Previously, results were obtained by several mathematicians (e.g., Hecke, Shih, Weintrab, the author and others) for some specific groups: \(G= \text{PSL}_2(\mathbb{F}_p)\), or when \(G\) has a pair of characters whose values generate an imaginary quadratic field. In this paper, the problem is solved in most generality, that is, for a group \(G\) without any assumptions. The result is formulated in terms of an explicit description of certain inner products associated to Gauss sum characters and the specific character, \(\mu\). Let \(G\) be a finite group. For a character \(\chi\) of \(G\), let \(\mathbb{Q}(\chi)= \mathbb{Q}(\chi(g)\mid g\in G)\subset \mathbb{C}\) be the value field of \(\chi\), let \(\Gamma_\chi= \text{Gal}(\mathbb{Q}(\chi)/ \mathbb{Q})\), and \((\Gamma_\chi)^\wedge= \text{Him}(\Gamma_\chi, \mathbb{C}^\times)\). For a character \(\chi\) of \(G\) and \(\lambda\in (\Gamma_\chi)^\wedge\), define a Gauss sum character \(\alpha(\chi, \lambda)\) of \(G\) by \(\alpha(\chi, \lambda)= \sum_{\gamma\in \Gamma_\chi} \lambda(\gamma)\chi^\gamma\). Now let \(X\), \(Y\) be connected compact Riemann surfaces of genus \(g_X\) and \(g_Y\), respectively. Denote by \(\mathcal F\) a locally free \({\mathcal O}_Y\)-module of finite rank. If \(f: X\to Y\) is a finite Galois cover with Galois group \(G\), let \(f^* {\mathcal F}\) denote the pull-back of \(\mathcal F\) to \(X\). Define the character \(\mu= \text{ch}(H^0(X, f^* {\mathcal F}))- \text{ch}(H^1(X, f^* {\mathcal F}))\), where \(\text{ch}(V)\) stands for the character of \(G\) determined by \(V\). Further, define the inner product \[ m(\chi, \lambda)= \langle \alpha(\chi, \lambda), \mu\rangle_G= \sum_{\gamma\in \Gamma_\chi} \l…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 858021
zbMATH Keywords: Galois covering
zbMATH Keywords: compact Riemann surfaces
zbMATH Keywords: holomorphic differentials
zbMATH Keywords: Gauss sum characters
zbMATH Keywords: generalized Bernoulli number
author: Shoichi Nakajima
MaRDI profile type: Publication
OpenAlex ID: W2053270173
cites work: Q4143433
cites work: Q3720671
cites work: Q5735288
cites work: Q5605180
cites work: On the evaluation of Gaussian sums for non-primitive Dirichlet characters
cites work: A Generalization of a Theorem of Hecke
cites work: Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties
cites work: On the representation of \(\text{SL}_2(F_q)\) in the space of Hilbert modular forms
cites work: Q4126536
cites work: On the construction of Galois extensions of function fields and number fields
cites work: Q3943047
cites work: Q3912660
DOI: 10.5802/JTNB.137
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description: scientific article; zbMATH DE number 858021