Modular forms and \(p\)-adic Hodge theory
type: Item
instance of: scholarly article
title: Modular forms and \(p\)-adic Hodge theory
zbMATH Open document ID: 0877.11034
author: Takeshi Saito
published in: Inventiones Mathematicae
publication date: 1997-12-08T00:00:00Z
review text: For an elliptic modular form, \textit{P. Deligne} constructs an associated \(\ell\)-adic representation of the absolute Galois group \(\text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\) [Sémin. Bourbaki 1968/69, No. 355, 139-172 (1971; Zbl 0206.49901)]. In the paper, it is shown that the construction is compatible with the local Langlands correspondence at \(p= \ell\) in the sense of Fontaine using a result of Carayol for \(p\neq\ell\). More precisely, the restriction to the local Galois group \(\text{Gal} (\overline \mathbb{Q}_p/ \mathbb{Q}_p)\) of the \(p\)-adic representation \(V_p(f)\) associated to a modular form \(f\) is potentially semi-stable as follows from a theorem of \textit{T. Tsuji} [Invent. Math. 137, No. 2, 233--411 (1999; Zbl 0945.14008)]. \textit{J.-M. Fontaine} has associated a representation of the Weil-Deligne group to a potentially semi-stable representation by using the ring \(B_{st}\) [Représentations \(\ell\)-adiques potentiellement semi-stables. Périodes \(p\)-adiques, Astérisque 223, 321-347 (1994)]. On the other hand, an automorphic representation \(\pi_f\) of the adele group \(GL_2 (\mathbb{A})\) is associated to a modular form \(f\). By factoring the tensor product \(\pi_f= \bigotimes \pi_{p,f}\), we obtain an irreducible admissible representation \(\pi_{p,f}\) of \(GL_2 (\mathbb{Q}_p)\). The main result asserts that the representation of the Weil-Deligne group associated to \(V_p(f)\) corresponds to the irreducible admissible representation \(\pi_{p,f}\) of \(GL_2 (\mathbb{Q}_p)\) in the sense of the local Langlands correspondence. The proof goes as follows. Since the correspondence is established for \(\ell \neq p\) by \textit{H. Carayol} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 409-468 (1986; Zbl 0616.10025)], it suffices to compare \(p\) and \(\ell\). The comparison is made by reducing to the Lefschetz trace formula, which is the same for \(p\) and \(\ell\), by using the weight spectral sequences of Steenbrink, \textit{M. Rapoport}, an…
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 1054512
zbMATH Keywords: \(p\)-adic Hodge theory
zbMATH Keywords: \(p\)-adic representation
zbMATH Keywords: elliptic modular form
zbMATH Keywords: absolute Galois group
zbMATH Keywords: local Langlands correspondence
zbMATH Keywords: representation of the Weil-Deligne group
MaRDI profile type: Publication
full work available at URL: s002220050175
OpenAlex ID: W1967042767
DOI: 10.1007/S002220050175
description: scientific article; zbMATH DE number 1054512