Drinfeld modular forms of weight one
type: Item
instance of: scholarly article
title: Drinfeld modular forms of weight one
zbMATH Open document ID: 0894.11028
author: Gunther Cornelissen
published in: Journal of Number Theory
publication date: 1998-09-08T00:00:00Z
review text: Let \(A=\mathbb{F}_q[T]\) be the polynomial ring over the finite field \(\mathbb{F}_q\), \(K= \mathbb{F}_q(T)\) its quotient field, \(K_\infty= \mathbb{F}_q((T^{-1}))\) the \(\infty\)-adic completion, \(C\) the completed algebraic closure of \(K_\infty\), and \(\Omega= C-K_\infty\) the Drinfeld upper half-plane. The modular group \(\Gamma(1)= GL(2,A)\) acts on \(\Omega\), and the associated \(C\)-algebras \(M(\Gamma(1))\) of Drinfeld modular forms is a polynomial ring \(C[g,\Delta]\) in two modular forms \(g\) and \(\Delta\) of respective weights \(q-1\) and \(q^2-1\). It is a basic albeit unsolved problem to determine the structure of the algebra of modular forms for convergence subgroups \(\Gamma\) of \(\Gamma(1)\), especially for full congruence subgroups \(\Gamma(N)= \{\gamma\in\Gamma\mid \gamma\equiv 1\pmod N\}\) of \(\Gamma(1)\). Fix such a \(\Gamma=\Gamma(N)\) with \(N\in A-\mathbb{F}_q\), and let \(M(\Gamma)=\oplus M_k(\Gamma)\) be its algebra of modular forms (\(k\)= weight). \(M_k\) certainly contains the Eisenstein series \(E_u^{(k)}\) \((u\in(N^{-1}/ A)^2)\), where \[ E_u^{(k)}(z)= \mathop{{\sum}'}_{\substack{ (a,b)\in A^2\\ (a,b)\equiv u\pmod A}} \frac{1} {(az+b)^k}. \tag \(*\) \] Using an argument of Hecke, the author first shows that the \(E_u^{(k)}\) generate a vector space complement in \(M_k\) of the cusp forms of weight \(k\) (Prop. 1.12). Since there are no cusp forms of weight one, the result of Mumford may be applied to conclude that \(M(\Gamma)\) is generated by the \(E_u^{(1)}\) and the cusp forms of weight two (Prop. 1.15). Two natural questions arise: (1) Do we already have \[ \begin{aligned} M(\Gamma) & = C\left[E_u^{(1)}\mid u \text{ as in (*)}\right]\\ & =: E(\Gamma)?\end{aligned} \] (2) What are the relations between the different \(E_u^{(1)}\)? At least, \(M(\Gamma)\) is the integral closure of \(E(\Gamma)\) in their common field of fractions (Prop. 1.17). Now the main result of the paper is Theorem 2.2, too complicated to be stated he…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 1109648
zbMATH Keywords: projective normality
zbMATH Keywords: modular forms for full convergence subgroups
zbMATH Keywords: Drinfeld upper half-plane
zbMATH Keywords: Drinfeld modular forms
zbMATH Keywords: Eisenstein series
zbMATH Keywords: cusp forms
reviewed by: Ernst-Ulrich Gekeler
MaRDI profile type: Publication
full work available at URL: jnth.1997.2185
OpenAlex ID: W1985530425
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DOI: 10.1006/JNTH.1997.2185
description: scientific article; zbMATH DE number 1109648