On component groups of Jacobians of Drinfeld modular curves.
type: Item
instance of: scholarly article
title: On component groups of Jacobians of Drinfeld modular curves.
zbMATH Open document ID: 1071.11034
author: Mihran Papikian
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116171
review text: Given an abelian variety \(A\) over a local field \(K\) with Neron model \({\mathcal A}\), the component group \(\Phi_A\) of the special fiber of \({\mathcal A}\) is an important invariant. In number theoretical situations, if \(A\) is the Jacobian variety of a modular curve, important arithmetical information is encoded in \(\Phi_{A,p}\), the component group at a prime \(p\), and in its relation with other data, e.g., the cuspidal divisor class group \(C\) of \(A\). Let now \(A\) be the Jacobian \(J_0(N)\) of the Hecke modular curve \(X_0(N)\) with prime conductor \(N\). In this situation, there are three basic results: (1) (B. Mazur) The canonical maps \(C \to \Phi_{J_0(N),N}\) and \(C \to J_0(N)({\mathbb Q})_{\text{tor}}\) are bijective. (2) (Former conjecture of W. Stein, proved by M. Emerton) Let \(A_f\) be the optimal abelian variety quotient of \(J_0(N)\) corresponding to a normalized weight-2 newform of level \(N\). Then the specialization mappings from the cuspidal group \(C_f\) of \(A_f\) to the component group \(\Phi_{A_f,N}\) and to \(A_f({\mathbb Q})_{\text{tor}}\) are isomorphisms. (3) (Emerton) The natural map on component groups \(\Phi_{J_0(N),N} \to \Phi_{A_f,N}\) is surjective. The author first discusses these results (and their failure if \(N\) is no longer assumed prime) and then considers the analogous situation where \(J_0(N)\) is replaced by the Jacobian of a Drinfeld modular curve. Here, as the author writes, ``some effort has been made to transfer Mazur's results on the Eisenstein ideal to the context of Drinfeld modular curves \dots ''; ``One encounters very non-trivial technical difficulties while doing this, so the theory over function fields is not yet as satisfactory as over \({\mathbb Q}\). There is an analogue of the cuspidal group \(C \subset J_0(N)({\mathbb Q})\), and Gekeler proved \dots that the specialization map \(C_{\mathfrak n}:\: C \to \Phi_{J_0(\mathfrak n),\mathfrak n}\) is an isomorphism.'' ``Even though one should expect …
reviewed by: Ernst-Ulrich Gekeler
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 2162453
zbMATH Keywords: component groups
zbMATH Keywords: Drinfeld modular curves
zbMATH Keywords: monodromy pairing
MaRDI profile type: Publication
OpenAlex ID: W2313532629
cites work: Degenerating abelian varieties
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Néron Models
cites work: Irreducible components of rigid spaces
cites work: Component groups of purely toric quotients.
cites work: Formes Modulaires et Representations De GL(2)
cites work: ELLIPTIC MODULES
cites work: Optimal quotients of modular Jacobians
cites work: Géométrie analytique rigide et applications. (Ridgid analytic geometry and applications)
cites work: Automorphe Formen über \({\mathbb{F}}_ q(T)\) mit kleinem Führer. (Automorphic forms over \({\mathbb{F}}_ q(T)\) with small conductor)
cites work: Q3734575
cites work: Analytical construction of Weil curves over function fields
cites work: Improper Eisenstein series on Bruhat-Tits trees
cites work: On the cuspidal divisor class group of a Drinfeld modular curve
cites work: FUNDAMENTAL DOMAINS OF SOME ARITHMETIC GROUPS OVER FUNCTION FIELDS
cites work: Jacobians of Drinfeld modular curves.
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: Schottky groups and Mumford curves
cites work: Modeles de neron et monodromie
cites work: Éléments de géométrie algébrique. I: Le langage des schémas. II: Étude globale élémentaire de quelques classe de morphismes. III: Étude cohomologique des faisceaux cohérents (première partie)
cites work: Éléments de géométrie algébrique. III: Étude cohomologique des faisceaux cohérents. (Séconde partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Q4026192
cites work: Modular curves and the Eisenstein ideal
cites work: Q4090463
cites work: Q5635548
cites work: On Schottky coverings of Drinfeld modular curves
cites work: On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms
cites work: Q3341041
cites work: The Eisenstein quotient of the Jacobian variety of a Drin'feld modular curve.
cites work: Q3762615
cites work: Discrete groups, Mumford curves and theta functions
cites work: Modular Parametrizations of Elliptic Curves
DOI: 10.5802/AIF.2078
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162453