On Drinfeld modular curves with many rational points over finite fields
type: Item
instance of: scholarly article
title: On Drinfeld modular curves with many rational points over finite fields
zbMATH Open document ID: 1031.11036
author: Andreas Schweizer
published in: Finite Fields and their Applications
publication date: 2003-04-02T00:00:00Z
review text: Given a finite field \({\mathbb F}_q\) and a curve \(X/{\mathbb F}_q\) (where a ``curve'' is a smooth projective geometrically connected algebraic curve over \({\mathbb F}_q\)), the number \(N = \# X(F_q)\) of rational points is roughly \(q+1\), where the difference \(|N-(q+1)|\) is subject to various bounds due to A. Weil and others. These depend on \(q\) and the genus \(g\) of \(X\), and are in general not sharp. For several reasons stemming from combinatorics, but also from genuine arithmetical interest, one would like to know the precise bounds. In particular, there is a lively contest (the results of which are documented and updated in \url{http://www.wins.uva.nl/~geer}) in constructing curves that, for \((q,g)\) given, realize the maximum number of rational points. These constructions are arithmetical in nature, and in many cases make use of modular curves of some sort: classical elliptic, Shimura, or Drinfeld modular curves. The reason is twofold: modular curves are well understood and their points easy to describe, and the underlying arithmetic produces enough of the symmetry required. Pursuing the strategy described below, the author of the present paper is able to find a series of new record curves for \(q = 4\) and 9. The main idea is as follows: Take a Drinfeld modular curve \(X_0(n)\) of Hecke type, reduced at a suitable place \((p)\) of degree 1 such that the reduction has a large number of rational points over the residue field extension \(L\) of degree 2 (which as an abstract field is isomorphic with \({\mathbb F}_{q^2}\)). The relevant invariants of \(Y:= X_0(n)/L\) (genus, number of supersingular points, and of cusps) are given through complicated but explicit formulas, which involve only the splitting type of \(n\). Then choose an Atkin-Lehner involution \(w\) (which corresponds to a divisor \(n'\) of \(n\)) with the number of fixed points as large as possible. (The last condition depends on ``good'' choices of \(n\) and \(n'\).) Now the number \(…
reviewed by: Ernst-Ulrich Gekeler
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1891456
zbMATH Keywords: curve over finite field
zbMATH Keywords: many rational points
zbMATH Keywords: asymptotically optimal
zbMATH Keywords: Drinfeld modular curve
zbMATH Keywords: Atkin-Lehner involution
describes a project that uses: manYPoints
MaRDI profile type: Publication
full work available at URL: ffta.2001.0351
OpenAlex ID: W2059555273
cites work: Tables of curves with many points
cites work: Invariants of some algebraic curves related to Drinfeld modular curves
cites work: Rational points on curves over finite fields. Theory and Applications
cites work: Q4234809
cites work: On elliptic curves over function fields of characteristic two
DOI: 10.1006/FFTA.2001.0351
description: scientific article; zbMATH DE number 1891456