Congruences between cusp forms: The \((p,p)\) case
type: Item
instance of: scholarly article
title: Congruences between cusp forms: The \((p,p)\) case
zbMATH Open document ID: 0857.11021
DOI: 10.1215/S0012-7094-95-08022-3
published in: Duke Mathematical Journal
publication date: 1996-06-18T00:00:00Z
review text: Let \(p\) be a prime. Let \(f\) be a newform of weight 2 for the group \(\Gamma_0 (N) \cap \Gamma_1(p^t)\) such that the associated \(\text{mod }p\) representation is absolutely irreducible. The main result of this paper (Theorem 5) determines all the various \(p\)-power levels and nebentypes with which \(f\) occurs \(\text{mod }p\). This provides an exhaustive analysis of congruences in the \((p,p)\) case (i.e. when \(p\) divides \(N)\). Let us stress that one cannot use the theorem of Langlands and Carayol to guess the congruences in this case -- in the non-\((p,p)\) case it is possible [see: \textit{H. Carayol}, Duke Math. J. 59, 785-801 (1989; Zbl 0703.11027)]. The results on the congruences between forms of weight 2 in the \((p,p)\) case are applied to study the ring of modular deformations of an absolutely irreducible \(\text{mod }p\) representation which is modular.
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 871840
zbMATH Keywords: cusp forms
zbMATH Keywords: \(p\)-power levels
zbMATH Keywords: newform
zbMATH Keywords: nebentypes
zbMATH Keywords: congruences
zbMATH Keywords: modular deformations
author: Chandrashekhar Khare
describes a project that uses: ecdata
MaRDI profile type: Publication
cites work: Hecke operators on \(\Gamma_0(m)\)
cites work: Twists of newforms and pseudo-eigenvalues of \(W\)-operators
cites work: Modular forms in characteristic \(\ell\) and special values of their \(L\)-functions
cites work: Q5749421
cites work: Sur les représentations galoisiennes modulo \(\ell\) attachées aux formes modulaires. (On Galois representations modulo \(\ell\) attached to modular forms)
cites work: Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert
cites work: Q4003857
cites work: Non-optimal levels of \(\text{mod }l\) modular representations
cites work: The weight in Serre's conjectures on modular forms
cites work: Arithmetic of p-adic modular forms
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: A tameness criterion for Galois representations associated to modular forms (mod \(p\))
cites work: Galois representations into \(\text{GL}_2(\mathbb Z_pX)\) attached to ordinary cusp forms.
cites work: Modular 𝑝-adic 𝐿-functions and 𝑝-adic Hecke algebras
cites work: Q4516131
cites work: A Study of the Local Components of the Hecke Algebra mod l
cites work: Higher congruences between modular forms
cites work: Q4111656
cites work: Congruences between cusp forms and the geometry of Jacobians of modular curves
cites work: Q4026195
cites work: Q3199524
cites work: Modular curves and the Eisenstein ideal
cites work: Q4026197
cites work: Class fields of abelian extensions of \(\mathbb Q\)
cites work: Q3754095
cites work: Q3695338
cites work: On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms
cites work: Q4297093
cites work: Q3484925
cites work: Schémas en groupes de type $(p,\ldots,p)$
cites work: Q5625265
cites work: On the modular representations of degree two of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\)
cites work: Galois properties of points of finite order of elliptic curves
cites work: Q3341041
cites work: Le problème des groupes de congruence pour \(SL_2\)
cites work: Modular elliptic curves and Fermat's Last Theorem
full work available at URL: s0012-7094-95-08022-3
OpenAlex ID: W1997796210
description: scientific article; zbMATH DE number 871840