On a congruence prime criterion for cusp forms on \(\mathrm{GL}_2\) over number fields
type: Item
instance of: scholarly article
title: On a congruence prime criterion for cusp forms on \(\mathrm{GL}_2\) over number fields
zbMATH Open document ID: 1331.11037
author: Kenichi Namikawa
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2015-11-11T00:00:00Z
review text: Let \(f\) be a normalized Hecke eigenform on GL\(_2\) over a number field \(F\), and \(\mathfrak{P}\) be a prime ideal of a number field \(K\) which contains the Galois closure of the number field generated by all Fourier coefficients of \(f\) over \(F\). The aim of the paper under review is to give a sufficient condition for \(\mathfrak{P}\) to be a congruence prime for \(f\). The main result of the paper, whose precise statements are rather technical and will not be given here, extends the congruence prime criteria known previously for cusp forms on GL\(_2\) over \(\mathbb{Q}\) by \textit{H. Hida} [Invent. Math. 63, 225--261 (1981; Zbl 0459.10018)], the case of cusp forms on GL\(_2\) over imaginary quadratic fields by \textit{E. Urban} [Compos. Math. 99, No. 3, 283--324 (1995; Zbl 0846.11029)], and also the case of cusp forms on GL\(_2\) over totally real number fields by \textit{E. Ghate} [Compos. Math. 132, No. 3, 243--281 (2002; Zbl 1004.11019)] and \textit{M. Dimitrov} [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 4, 505--551 (2005; Zbl 1160.11325)]. \ The strategy of the proof of the main theorem is similar to that of Hida's theorem for cusp forms on GL\(_2\) over \(\mathbb{Q}\). The author first computes the special value at \(s=1\) of the adjoint \(L\)-function associated with a cusp form \(f\), by using the cup product of the cuspidal cohomologies of the local system \(\mathscr{L}(n,\chi;\mathbb{C})\) on \(Y_U\), where \(\chi\) is the central character of \(f\) and \(Y_U\) is a manifold associated to certain subgroup \(U\) of GL\(_2(\widehat{\mathcal{O}}_F)\) (see section 4.1). The author extends Shimura's computation for the Hilbert modular case and that by Urban for the imaginary quadratic fields case to arbitrary number fields. Secondly, the author proves that the special values of the \(L\)-functions are contained in \(\mathcal{O}_{K,\mathfrak{P}}\) by using a nondegenerate bilinear form on an \(\mathcal{O}_{K,\mathfrak{P}}\)-lattice in the space of cusp for…
reviewed by: Dongwen Liu
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 6507151
zbMATH Keywords: cusp form
zbMATH Keywords: congruence prime
zbMATH Keywords: Eichler-Shimura isomorphism
zbMATH Keywords: cuspidal cohomology
MaRDI profile type: Publication
full work available at URL: crelle-2013-0076
OpenAlex ID: W2320268831
DOI: 10.1515/CRELLE-2013-0076
description: scientific article; zbMATH DE number 6507151