Galois representations into \(\text{GL}_2(\mathbb Z_pX)\) attached to ordinary cusp forms.
type: Item
instance of: scholarly article
title: Galois representations into \(\text{GL}_2(\mathbb Z_pX)\) attached to ordinary cusp forms.
zbMATH Open document ID: 0612.10021
DOI: 10.1007/BF01390329
author: Haruzo Hida
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143381
review text: Let \(p\) be a prime \(\geq 5\), \(N\) an integer prime to \(p\), \(\Gamma\) the topological group \(1+p{\mathbb Z}_ p\), and \(\Omega\) a \(p\)-adic completion of an algebraic closure of \({\mathbb Q}_ p\). The author uses the study of the parabolic cohomology groups of \(\Gamma_ 1(Np^ r)\) to obtain results about Galois representations attached to cusp forms. Beginning with a study of the ordinary part \(h^ 0(N,{\mathbb Z}_ p)\) of the universal Hecke algebra (generated by the Hecke operators regarded as endomorphisms of the space of \(p\)-adic cusp forms of level \(N\)) the author proves the following theorem concerning \(q\)-expansions: Let \(\Lambda ={\mathbb Z}_ pX\) be the one variable Iwasawa algebra, and fix a non-trivial \(\Lambda\)-algebra homomorphism \(\lambda: h^ 0(N,{\mathbb Z})\to \Lambda\). The image under \(\lambda\) of the formal power series \(\sum^{\infty}_{n=1}T(n)q^ n\) is, when evaluated at \(X=\varepsilon (u)u^ k-1\) \((u=1+p)\), the complex \(q\)-expansion of a common eigenform \(f_{k,\varepsilon}\) for all \(T(n)\) in \(S_ k(\Gamma_ 1(p^ r))\) where \(\varepsilon: \Gamma\to \Omega^*\) is of finite order, \(\text{ker}\;\varepsilon=1+p^ r {\mathbb Z}_ p\), and \(k\) is an integer \(\geq 2\). Moreover, \(f_{k,\varepsilon}\) is not the twist by any Dirichlet character of a form of lower level. The author then shows that one can associate to \(\lambda\) a unique Galois representation \(\pi(\lambda): \text{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \text{GL}_ 2(\Lambda)\) such that the reduction of \(\pi(\lambda)\) is equivalent to the irreducible \(\text{Gal}(\bar{\mathbb Q}/\mathbb Q)\)-representation \(\pi(f_{k,\varepsilon})\) into \(\text{GL}_ 2(\Omega)\) associated to \(f_{k,\varepsilon}\) (by Eichler-Shimura for \(f=2\), and by Deligne for \(k>2\)). The paper ends with a study of the connections between the special values of the \(L\)-function associated to the 3-dimensional subrepresentation contained in \(\pi (f_{k,\varepsilon})\otimes {\til…
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3989449
zbMATH Keywords: parabolic cohomology groups
zbMATH Keywords: Galois representations
zbMATH Keywords: cusp forms
zbMATH Keywords: Hecke algebra
zbMATH Keywords: Hecke operators
zbMATH Keywords: \(p\)-adic cusp forms
zbMATH Keywords: \(q\)-expansions
zbMATH Keywords: Iwasawa algebra
zbMATH Keywords: special values of the \(L\)-function
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
Wikidata QID: Q105611569
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description: scientific article; zbMATH DE number 3989449