Congruence module for \(\text{GL}(2)\) of an imaginary quadratic field and Iwasawa theory of biquadratic CM field
type: Item
instance of: scholarly article
title: Congruence module for \(\text{GL}(2)\) of an imaginary quadratic field and Iwasawa theory of biquadratic CM field
zbMATH Open document ID: 0973.11095
DOI: 10.1215/S0012-7094-98-09204-3
author: Eric Urban
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: Let \(M\) be a biquadratic CM-field and \(K\) one of its imaginary quadratic subfields. Let \(\tau\) be the nontrivial \(K\)-automorphism of \(M\) and \(p\geq 3\) be a prime number which is totally split in \(M/K\). Let \(M_\infty\) be the union of all ray class fields of conductor a \(p\)-power over \(M\) and \(M_\infty '\) the subfield of \(M_\infty\) which is fixed by the ``centralizer'' of \(\tau\) in the Galois group of \(M_\infty\) over \(M\). One wants to study a certain Galois group \textbf{X} (attached to \(M_\infty '\)) as a module under the Iwasawa-algebra \({\mathbb Z}_pW'\), where \(W'\) is a complement to the torsion subgroup of \(\text{ Gal}(M_\infty '/M)\). For every character \(v\) of this torsion subgroup there exists a \(\Lambda\)-adic modular form \(\Theta_v\), leading to a congruence module \(C_0\) for the ordinary Hecke-algebra. One of he main issues of the paper is to show that the characteristic series of \(C_0\) divides the characteristic series of \textbf{X}. The methods build on those developed by \textit{H. Hida} and \textit{J. Tilouine} in [Invent. Math. 117, No. 1, 89-147 (1994; Zbl 0819.11047)]. As a preparation, a good deal of results from the theory of automorphic forms for \(\text{GL}(2,K)\) is explained. Then \(\Lambda\)-adic forms are introduced, following Taylor and Hida. To \(\Theta_v\) there corresponds a Galois-representation \(\rho_v\), and a Selmer group for \(\text{Ad}(\rho_v)\) is linked to \textbf{X} by Mazur's deformation theory and this gives the final clue to the main theorem. The comparison of his results with loc.\ cit.\ lead the author to conjecture a relation between the congruence modules for \(K\) and for the real quadratic subfield of \(M\).
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F75
zbMATH DE Number: 1425092
zbMATH Keywords: congruence modules
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: CM-field
zbMATH Keywords: \(\Lambda\)-adic modular form
zbMATH Keywords: Hecke-algebras
MaRDI profile type: Publication
cites work: Q3476889
cites work: Q4200278
cites work: Corners and arithmetic groups (Appendice: Arrondissement des varietes a coins par A. Douady et L. Herault)
cites work: Q5541099
cites work: Q4314335
cites work: On some results of Atkin and Lehner
cites work: Q5340151
cites work: Nonvanishing theorems for automorphic \(L\)-functions on \(\text{GL} (2)\)
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: Q4297081
cites work: Eisenstein cohomology of arithmetic groups. The case \(GL_ 2\)
cites work: \(l\)-adic representations associated to modular forms over imaginary quadratic fields. I: Lifting to \(GSp_ 4(\mathbb{Q})\)
cites work: A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I
cites work: \(p\)-ordinary cohomology groups for \(\text{SL}(2)\) over number fields
cites work: On the critical values of \(L\)-functions of \(GL(2)\) and \(GL(2)\times GL(2)\)
cites work: On p-adic Hecke algebras for \(GL_ 2\) over totally real fields
cites work: Control theorems of $p$-nearly ordinary cohomology groups for $\text {SL} (n)$
cites work: \(p\)-adic Hecke algebras for \(GL(2)\)
cites work: Q3792754
cites work: Q3972777
cites work: On the anticyclotomic main conjecture for CM fields
cites work: Automorphic forms on GL (2)
cites work: \(p\)-adic \(L\)-functions for CM fields
cites work: Modular Forms and ℓ-Adic Representations
cites work: Galois representations, Kähler differentials and ``main conjectures''
cites work: On automorphic forms on \(mathrm{GL}_2\) and Hecke operators
cites work: Pseudo representations
cites work: Q3936841
cites work: Q4325444
cites work: Appendix: A refinement of the strong multiplicity one theorem for \(GL(2)\)
cites work: Q5625265
cites work: \(l\)-adic representations associated to modular forms over imaginary quadratic fields. II
cites work: Galois representations associated to Siegel modular forms of low weight
cites work: On the \(l\)-adic cohomology of Siegel threefolds
cites work: Q3813887
cites work: Q4230881
cites work: Q4863558
cites work: Several-variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations
cites work: Q4863731
cites work: Dirichlet series and automorphic forms
cites work: On ordinary \(\lambda\)-adic representations associated to modular forms
description: scientific article; zbMATH DE number 1425092