Iwasawa invariants for the false-Tate extension and congruences between modular forms
type: Item
instance of: scholarly article
title: Iwasawa invariants for the false-Tate extension and congruences between modular forms
zbMATH Open document ID: 1193.11105
author: Aribam Chandrakant Sharma
published in: Journal of Number Theory
publication date: 2009-06-23T00:00:00Z
review text: Let \(F=\mathbb{Q}(\mu_p)\) and let \(F^{\text{cyc}}\) be its cyclotomic \(\mathbb{Z}_p\)-extension (\(p\) an odd prime). Let \(f\) be a newform of weight \(k\geq 2\), level \(Np^r\) (with \((N,p)=1\)) and nebentypus character \(\chi\). Associated to \(f\) one has a two dimensional representation \(\rho_f: G_{\mathbb{Q}}\rightarrow \text{GL}_2(K_{f,\mathfrak{p}})\) (where \(K_{f,\mathfrak{p}}\) is the completion at a prime \(\mathfrak{p}|p\) of a number field \(K_f\) containing all the Fourier coefficients of \(f\)). Let \(T_f\) be a lattice in the representation space \(V_f\) of \(\rho_f\) and define \(A_f:=V_f/T_f\,\). For any pro-\(p\), \(p\)-adic Lie extension \(L/F\) one defines Selmer groups \(\text{Sel}(L,A_f)\) as subgroups of \(H^1(L,A_f)\) satisfying certain local conditions (as usual) and the goal of the paper is to study their structure as modules over the Iwasawa algebra \(\mathbb{Z}_p[[\text{Gal}(L/F)]]\) as \(f\) varies in a Hida family \({\mathbf H}(\overline{\rho})\) (where \(\overline{\rho}: G_{\mathbb{Q}}\rightarrow\text{GL}_2(\mathbb{F})\) is an absolutely irreducible residual representation into a finite field, note that in particular one has that \(f\in {\mathbf H}(\overline{\rho})\) yields \(\overline{\rho}_f=\overline{\rho}\)). For \(L=F^{\text{cyc}}\) using a deep result of \textit{K. Kato} [Astérisque 295, 117--290 (2004; Zbl 1142.11336)] which shows that the \(\text{Sel}(F^{\text{cyc}},A_f)\) are cotorsion modules, the author proves that if \(f,g\in {\mathbf H}(\overline{\rho})\) then \(\mu_f=0\) implies \(\mu_g=0\) and that \(\lambda_f=\lambda_g\) if they are in the same branch (where the \(\mu_*\) and \(\lambda_*\) are the classical Iwasawa invariants associated to the module \(\text{Sel}(F^{\text{cyc}},A_*)\,\)). For \(L=F_\infty:=\bigcup \mathbb{Q}(\mu_{p^n},m^{1/p^m})\) (i.e. the false-Tate extension) under the assumption that \(\text{Sel}(F_\infty,A_f)\) is cofinitely generated for some \(f\in {\mathbf H}(\overline{\rho})\), the auth…
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F85
zbMATH DE Number: 5569030
zbMATH Keywords: modular forms
zbMATH Keywords: Hida theory
zbMATH Keywords: Galois representations
zbMATH Keywords: Selmer groups
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.03.001
OpenAlex ID: W2076894797
cites work: Q4097036
cites work: Q3665216
cites work: Q4314335
cites work: Euler characteristics and elliptic curves. II.
cites work: Links between cyclotomic and \(\text{GL}_2\) Iwasawa theory
cites work: Root numbers of non-abelian twists of elliptic curves
cites work: Variation of Iwasawa invariants in Hida families
cites work: Q3972988
cites work: Q4702588
cites work: Introduction to Iwasawa theory for elliptic curves
cites work: \(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms
cites work: On the Iwasawa invariants of elliptic curves
cites work: EULER CHARACTERISTICS AS INVARIANTS OF IWASAWA MODULES
cites work: Iwasawa modules attached to congruences of cusp forms
cites work: Galois representations into \(\text{GL}_2(\mathbb Z_pX)\) attached to ordinary cusp forms.
cites work: Q4490714
cites work: Q4699507
cites work: Completely faithful Selmer groups over Kummer extensions
cites work: Q4831110
cites work: Modular Forms and ℓ-Adic Representations
cites work: Q5651322
cites work: Q4044702
cites work: Arithmetic of Weil curves
cites work: On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer
cites work: Q4935965
cites work: Euler systems and exact formulas in number theory
cites work: Q4249392
cites work: NON-ARCHIMEDEAN MEASURES CONNECTED WITH DIRICHLET SERIES
cites work: Iwasawa invariants of Galois deformations
cites work: On ordinary \(\lambda\)-adic representations associated to modular forms
DOI: 10.1016/J.JNT.2009.03.001
description: scientific article; zbMATH DE number 5569030