Genus of abstract modular curves with level-\(\ell\) structures
type: Item
instance of: scholarly article
title: Genus of abstract modular curves with level-\(\ell\) structures
zbMATH Open document ID: 1471.14058
author: Anna Cadoret
author: Akio Tamagawa
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2019-07-03T00:00:00Z
review text: Let \(X\) be a smooth geometrically connected curve over a finitely generated field \(k\) of characteristic \(p\ge 0\). Let \(H_\ell\) be a continuous \(\pi_1(X)\)-module of finite rank \(r_\ell\) over \(\mathbb F_\ell\) or a field \(F_\ell\) contained in \(\bar {\mathbb F}_\ell\), and let \(G_\ell\) be the image of the Galois representation \(\rho_\ell: \pi_1(X)\to\mathrm{GL}(H_\ell)\simeq \mathrm{GL}_{r_\ell}(F_\ell)\). An abstract modular curve of level-\(\ell\) structure referred in the title is the étale cover \(X_U\) over \(X\) corresponding to the open subgroup \(\rho_\ell^{-1}(U)\) for a subgroup \(U\subset G_\ell\). Let \(|X|\) denote the set of closed points of \(X\), and for any \(x\in |X|\), denote by \(G_{\ell,x}\) the image of the local Galois representation at \(x\). The aim of the present paper is to describe the local Galois image \(G_{\ell,x}\) as \(x\) varies in \(|X|\). The authors consider a family of \(F_\ell \pi_1(X)\)-modules \(H_\ell\) of rank \(r_\ell\le r\) (\(r\ge 1\) fixed) for \(\ell\) in an infinite set \(L\) of primes \(\neq p\), and fix a set \(\mathcal F_\ell\) (= the ``moduli data'') of subgroups \(U\) of \(G_\ell\) which does not contain the image \(\overline G_\ell\) of the geometric fundamental group \(\pi_1(\overline X)\). The geometric cover \(\overline X_U \to \overline X\) corresponds to the subgroup \(\overline U:=U\cap \overline G_\ell\), and the above condition ensures that the cover is nontrivial. The paper under review mainly studies the problem whether \[ g^{\rho_\ell}_{X_{\mathcal F_\ell}}:=\min \{g(X_U): U\in \mathcal F_\ell\} \to +\infty? \] For a finite subgroup \(G\subset\mathrm{GL}(H_\ell)\), let \(G^+\) denote the (normal) subgroup generated by the elements of order \(\ell\). Let \(\mathcal F_{\ell,+}\) be the set of all subgroups \(U\) of \(G_\ell\) such that \({\overline G}^+_\ell\not \subset \overline U\). and set \(X_+^{\rho_\ell}:=X_{\mathcal F_{\ell,+}}^{\rho_\ell}\). In [Int. Math. Res. Not. 2019, No. 9, …
reviewed by: Chia-Fu Yu
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 7075965
zbMATH Keywords: abstract modular curves
zbMATH Keywords: genus
MaRDI profile type: Publication
full work available at URL: crelle-2016-0057
OpenAlex ID: W2552697523
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DOI: 10.1515/CRELLE-2016-0057
description: scientific article; zbMATH DE number 7075965