Invariants of some algebraic curves related to Drinfeld modular curves
type: Item
instance of: scholarly article
title: Invariants of some algebraic curves related to Drinfeld modular curves
zbMATH Open document ID: 0979.11036
author: Ernst-Ulrich Gekeler
published in: Journal of Number Theory
publication date: 2001-12-05T00:00:00Z
reviewed by: Marc Reversat
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 1678871
zbMATH Keywords: Drinfeld modular curves
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: genera
zbMATH Keywords: number of cusps
zbMATH Keywords: function fields
describes a project that uses: manYPoints
MaRDI profile type: Publication
full work available at URL: jnth.2000.2642
OpenAlex ID: W2060964184
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DOI: 10.22028/D291-26165
review text: This paper is about properties of Drinfeld modular curves, classifying Drinfeld modules defined on \(\mathbb{F}_q[T]\) (where \(\mathbb{F}_q\) is a finite field), especially formulae for the genera, formulae for the number of cusps, descriptions of function fields\dots.NEWLINENEWLINENEWLINEThe main and important result, which takes place in the long story of the comparison over a finite field between genus and number of rational points of a curve, is the following: let \((N_k)_k\) be a sequence of elements of \(\mathbb{F}_q[T]\) coprime with \(T\), such that \(\text{deg} N_k\) tends to \(\infty\), let \(X_0(N_k)_{/\mathbb{F}_q}\) be the reduction modulo \(T\) of the Drinfeld modular curve \(X_0(N_k)\) (this is a smooth curve over \(\mathbb{F}_q = \mathbb{F}_q[T]/(T))\). Then the sequence \((X_0(N_k)_{/\mathbb{F}_q})_k\) is asymptotically optimal over \(\mathbb{F}_{q^2}\) (viewed as a quadratic extension of \(\mathbb{F}_q[T]/(T))\), meaning that, when \(k\) tends to \(\infty\) NEWLINE\[NEWLINE\frac{\#(X_0(N_k)_{/\mathbb{F}_q}(\mathbb{F}_{q^2}))}{\text{genus of }X_0(N_k)}\to q-1.NEWLINE\]
description: scientific article; zbMATH DE number 1678871