On the coefficients of Drinfeld modular forms
type: Item
instance of: scholarly article
title: On the coefficients of Drinfeld modular forms
zbMATH Open document ID: 0653.14012
DOI: 10.1007/BF01410204
author: Ernst-Ulrich Gekeler
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143613
review text: Let \(\mathbb A=\mathbb F_ r[T]\) and \(k=\mathbb F_ r(T)\). Let \(K=k_{\infty}\) be the completion of \(k\) at the prime \(\infty\), and let \(K^{\text{ac}}\) be an algebraic closure of \(K\). We denote the space \(K^{\text{ac}}-K\) by \(\Omega\). One knows that \(\Omega\) inherits a rigid-analytic structure as a subspace of the affine line and that it is connected as a rigid space. The group \(\text{GL}_ 2(\mathbb A)]\), and its congruence subgroups, act discretely on \(\Omega\) and the quotient spaces can also be given a rigid structure. In fact, they are the underlying rigid spaces of certain algebraic curves which are the moduli spaces of Drinfeld modules [of rank two and with, possibly, ``level structure'']. A ``Drinfeld module'' is obtained by looking at the \(\mathbb A\)-module structure that is obtained on \(K^{\text{ac}}\) when one divides \(K^{\text{ac}}\) by a discrete finitely generated \(\mathbb A\)-submodule (i.e., a ``lattice''). Drinfeld modules thus are the correct \(\mathbb A\)-analogs of elliptic curves. On \(\Omega\) one can talk about ``modular forms'' with the straightforward definition. Through the use of rigid analysis one can show that such forms have ``\(q\)-expansions'' (called ``\(t\)-expansions'' by the author). Such forms have been around for a number of years and much of the recent work on them has been carried out by the author. The paper under review contains very interesting results in a number of directions: (1) Given a lattice \(\Lambda\) one forms the rigid-meromorphic function \(S_{k,\Lambda}\) defined by \(S_{k,\Lambda}=\sum (Z+\lambda)^{- k}\), where one sums over all \(\lambda\) in \(\Lambda\). One knows that \(S_{k,\Lambda}\) is a polynomial of degree \(k\), \(G_{k,\Lambda}\), in the function \(S_{1,\Omega}\). The author presents a lovely formula for the generating function of the \(G_{k,\Lambda}\). This allows him to e.g., make some progress in understanding the action of the Hecke operators on modular forms. (2) By defini…
reviewed by: David Goss
Mathematics Subject Classification ID: 11F52
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F25
zbMATH DE Number: 4065157
zbMATH Keywords: rigid space
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: modular forms
zbMATH Keywords: rigid-meromorphic function
zbMATH Keywords: Hecke operators
zbMATH Keywords: Swinnerton-Dyer theorem
MaRDI profile type: Publication
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OpenAlex ID: W2047335221
description: scientific article; zbMATH DE number 4065157