Transcendence of the Carlitz-Goss gamma function at rational arguments
type: Item
instance of: scholarly article
title: Transcendence of the Carlitz-Goss gamma function at rational arguments
zbMATH Open document ID: 0862.11039
published in: Journal of Number Theory
publication date: 1997-05-26T00:00:00Z
review text: Let \({\mathbf A} = \mathbb{F}_q [T]\) where \(q= p^{m_0}\) is a prime power and set \({\mathbf k} = \mathbb{F}_q (T)\) and \({\mathbf K} = \mathbb{F}_q ((1/T))\). One checks easily that \({\mathbf A}\) is contained in \({\mathbf K}\) discretely (the topology on \({\mathbf K}\) is given by measuring the order of zero of a function at \(\infty)\), and that \({\mathbf K} / {\mathbf A}\) is compact. Of course this is similar to \(\mathbb{Z} \subset \mathbb{R}\). It is by now well known that one can construct a theory of ``objects with \({\mathbf A}\)-multiplication'' in analogy with the theory of ``objects with \(\mathbb{Z}\)-multiplication'' (i.e., \(\mathbb{G}_m\), elliptic curves, abelian varieties, etc.); these objects with \({\mathbf A}\)-multiplication are named Drinfeld modules after V. G. Drinfeld who initiated their general theory in 1973 (we note that more general objects with \({\mathbf A}\)-multiplication, called \({\mathbf A}\)-modules, were constructed by G. Anderson in the 1980's). However, back in the 1930's L. Carlitz had actually worked out the details of a particularly simple Drinfeld module called the Carlitz module. The Carlitz module, like all Drinfeld modules defined over extensions of \({\mathbf k}\), has both an algebraic and analytic side to it; in particular there is an entire analytic function called the Carlitz exponential which uniformizes the Carlitz module just as \(e^z\) uniformizes \(\mathbb{G}_m\) classically. The Carlitz exponential, \(e_C(z)\), may be expressed as \(e_C(z) = \sum^\infty_{i=0} (z^{q^i}/D_i)\), where \(D_0 = 1\) and for \(i>0\), \(D_i = (T^{q^i} -T) (T^{q^i} -T^q) \cdots (T^{q^i} - T^{q^{i -1}})\). With a little work, one can also see that \(D_i\) is the product of all monic polynomials of degree \(i\); thus, the polynomials \(D_i\) look very much like they are a ``factorial.'' However, as Carlitz realized, the elements \(D_i\) alone cannot be a good factorial for \({\mathbf A}\) as \(D_i\) is the coefficient of \(z^{…
reviewed by: David Goss
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 955262
zbMATH Keywords: transcendence
zbMATH Keywords: Carlitz-Goss gamma functions
zbMATH Keywords: Carlitz module
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: Carlitz exponential
zbMATH Keywords: factorial
author: Jean-Paul Allouche
MaRDI profile type: Publication
full work available at URL: jnth.1996.0126
OpenAlex ID: W2034928777
DOI: 10.1006/JNTH.1996.0126
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description: scientific article; zbMATH DE number 955262