Growth order and congruences of coefficients of the Drinfeld discriminant function
type: Item
instance of: scholarly article
title: Growth order and congruences of coefficients of the Drinfeld discriminant function
zbMATH Open document ID: 0978.11026
author: Ernst-Ulrich Gekeler
published in: Journal of Number Theory
publication date: 2000-02-14T00:00:00Z
review text: The analogue of the discriminant of an elliptic curve in the arithmetic of function fields is given by the so-called Drinfeld discriminant function \(\Delta\), which is the leading coefficient of a rank two Drinfeld module over \(A={\mathbb F}_q[T]\), which can be seen as a modular form on \(GL(2,A)\) depending on a rank-two lattice in an algebraically closed complete field containing the completion of the quotient field of \(A\) w.r.t.\ the valuation \(-\deg\). As such, it admits a series expansion in the parameter \(t\) at the cusp, and this paper contains some properties of the (largely mysterious) expansion coefficients of \(\Delta\). It turns out that \(\Delta\) admits a \((q-1)\)-th root \(h\); and \(h/t\) admits a \((q^2-1)\)-th root \(U\). Let \([i]:=T^{q^i}-T\). The main results of this paper are as follows: (a) it is shown that the coefficients \(c_k\) in the \(t\)-expansion of \(U\) have degree \(\leq k\) in \(T\), and the exact coefficient of \(c_k\) in \(T^k\) is determined (in particular, \(\deg(c_k)=k\) occurs infinitely often); (b) \(U\) is ``balanced'', i.e., each \(c_k\) is a polynomial in \([1]\) and the coefficient of \(c_k\) in \([1]^j\) is only non-zero if \(j=k\) mod \(q-1\); (c) The coefficients \(b_k\) of \(-h/t\) in \(T\) satisfy \(b_{k q^d} = b_k\) mod \([d]\). The starting point for the proofs is an expansion of \(\Delta\) as an infinite product previously obtained by the author [J. Number Theory 21, 135-140 (1985; Zbl 0572.10021)]. For (c), Hecke operators are used. The paper finishes with some further conjectures about the coefficients.
reviewed by: Gunther Cornelissen
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11F52
zbMATH DE Number: 1375112
zbMATH Keywords: Drinfeld modular form
zbMATH Keywords: congruence
zbMATH Keywords: discriminant
MaRDI profile type: Publication
full work available at URL: jnth.1999.2387
OpenAlex ID: W2091441220
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DOI: 10.1006/JNTH.1999.2387
description: scientific article; zbMATH DE number 1375112