Singular moduli and supersingular moduli of Drinfeld modules
type: Item
instance of: scholarly article
title: Singular moduli and supersingular moduli of Drinfeld modules
zbMATH Open document ID: 0767.11028
DOI: 10.1007/BF01231341
author: Martin Brown
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144059
review text: Let \({\mathcal C}\) be a smooth, projective curve over the finite field \(\mathbb{F}_ r\), \(r=p^ n\) where we assume that \(\mathbb{F}_ r\) is the full field of constants on \({\mathcal C}\). Let \(\infty\) be a fixed place of \({\mathcal C}\) and let \({\mathbf A}\) be the affine ring of functions regular on \({\mathcal C}-\infty\). So \({\mathbf A}\) is a Dedekind ring with finite class group and unit group \(\simeq\mathbb{F}_ r^*\); we denote its field of fractions by \({\mathbf k}\). By its very construction, the ring \({\mathbf A}\) is analogous to \(\mathbb{Z}\), or the integers \({\mathfrak O}\) in a complex quadratic field, in that it has a unique prime at \(\infty\). The prototypical example of such \({\mathbf A}\) is the polynomial algebra \(\mathbb{F}_ r[T]\). For \(\mathbb{Z}\) one has lattices \((\subset\mathbb{C})\) of rank 1 or 2; for a complex quadratic field one has \({\mathfrak O}\)-lattices (again \(\subset\mathbb{C})\) of \({\mathfrak O}\)-rank 1 (and thus of \(\mathbb{Z}\)-rank 2). It is therefore reasonable to look for ``\({\mathbf A}\)-lattices'' contained in the algebraic closure of \({\mathbf k}_ \infty\); such objects are easy to find and lead to algebraic entities called ``elliptic \({\mathbf A}\)-modules''. The theory of elliptic \({\mathbf A}\)-modules (or, more commonly, ``Drinfeld modules'') was inaugurated in 1973 in \textit{V. G. Drinfel'd}'s paper of the same name [Math. USSR, Sb. 23(1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)]; however a very important specific rank 1 example was worked out by \textit{L. Carlitz} in the 1930's [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904)]. This example is called, naturally, the ``Carlitz module'' and it acts as a very suitable analog of \(\mathbb{G}_ m\) for \(\mathbb{F}_ r[T]\). In general, a Drinfeld module \(\psi\) over a field equips the field with a new exotic action of \({\mathbf A}\); it thus makes sense to discuss division points…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 167403
zbMATH Keywords: exceptional \(j\)-invariant
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Carlitz module
zbMATH Keywords: rank 2 Drinfeld modules
zbMATH Keywords: \(q\)-expansions
zbMATH Keywords: supersingular reduction
zbMATH Keywords: modular forms
zbMATH Keywords: quadratic reciprocity
zbMATH Keywords: Chebotarev density theorem
reviewed by: David Goss
MaRDI profile type: Publication
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OpenAlex ID: W2092230331
description: scientific article; zbMATH DE number 167403