\(\mathcal L\)-invariants of Tate curves
type: Item
instance of: scholarly article
title: \(\mathcal L\)-invariants of Tate curves
zbMATH Open document ID: 1244.11056
author: Haruzo Hida
published in: Pure and Applied Mathematics Quarterly
publication date: 2009-11-18T00:00:00Z
review text: Let \(p\) be an odd prime and let \(F\) be a totally real number field. Let \(E\) be a Hilbert modular elliptic curve, without complex multiplication, defined over \(F\), and assume that \(E\) has multiplicative or good ordinary reduction at all primes dividing \(p\). Let \(\rho\) be the representation of \(\text{Gal}(\overline{F}/F)\) on the \(p\)-adic Tate module of \(E\). Under the assumption that a certain universal deformation ring is a power series ring, the author gives a formula for \({\mathcal L}(\text{Ind}_F^{\mathbb Q}\text{Ad}(\rho_E))\), where \({\mathcal L}\) is Greenberg's \({\mathcal L}\)-invariant, and he conjectures a formula for \({\mathcal L}(\text{Ind}_F^{\mathbb Q} (\text{Sym}^{\otimes n}(\rho_E)(-m))\), where \(n\geq 1\) and \(m\) is an integer such that the motive \(\text{Sym}^{\otimes n}(H^1(E))(-m)\) is critical at 1. The conjecture implies that the \({\mathcal L}\)-invariant should be essentially independent of \(n\). The present paper generalizes work of \textit{R. Greenberg} [Contemp. Math. 165, 149--174 (1994; Zbl 0838.11070)], of \textit{R. Greenberg} and \textit{G. Stevens} [Contemp. Math. 165, 183--211 (1994; Zbl 0846.11030)], and of \textit{B. Mazur, J. Tate} and \textit{J. Teitelbaum} [Invent. Math. 84, 1--48 (1986; Zbl 0699.14028)]. A different treatment of the present material, one that is more Galois cohomological and less automorphic, is given in subsequent works of the author [Publ. Res. Inst. Math. Sci. 45, No. 1, 1--24 (2009; Zbl 1173.11033)], [Int. Math. Res. Not. 2007, No. 23, Article ID rnm102, 49 p. (2007; Zbl 1183.11028)].
reviewed by: Lawrence C. Washington
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5635148
zbMATH Keywords: Greenberg L-invariant
zbMATH Keywords: adjoint representation
zbMATH Keywords: Tate curves
zbMATH Keywords: p-adic L-functions
zbMATH Keywords: Galois representations
zbMATH Keywords: elliptic curves
MaRDI profile type: Publication
DOI: 10.4310/PAMQ.2009.V5.N4.A6
description: scientific article; zbMATH DE number 5635148