Zeros of Eisenstein series, quadratic class numbers and supersingularity for rational function fields
type: Item
instance of: scholarly article
title: Zeros of Eisenstein series, quadratic class numbers and supersingularity for rational function fields
zbMATH Open document ID: 0979.11029
author: Gunther Cornelissen
published in: Mathematische Annalen
publication date: 2002-02-24T00:00:00Z
review text: Let \(\mathbb{F}_q\) be the finite field with \(q\)-elements and let \(A= \mathbb{F}_q[t]\) be the 1-variable polynomial ring in one indeterminate with quotient field \(k = \mathbb{F}_q(t)\). The completion of the field \(k\) at the place given by infinity is \(k_\infty = \mathbb{F}_q((1/t))\); one lets \(C\) denote the completion of a fixed algebraic closure of \(k_\infty\) equipped with its canonical valuation. The set \(C - k_\infty\) is a connected rigid analytic space with an action of \(Gl_2(A)\) on it in the standard fashion. The Eisenstein series \(E_k(z)\) are defined, in analogy with classical theory, by \(E_k(z) :=\sum_{(0,0)\neq (c,d)\in A^2}(cz + d)^{-k}\); they are non-trivial precisely when \(k\) is divisible by \(q -1\). In the paper being reviewed, the author establishes a number of interesting results about the zeros of such functions. Such results are not known in the classical setting, but the author does believe that much of what he has discovered will carry over.
Mathematics Subject Classification ID: 11F52
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11M38
zbMATH DE Number: 1337030
zbMATH Keywords: class numbers
zbMATH Keywords: connected rigid analytic space
zbMATH Keywords: Eisenstein series
zbMATH Keywords: zeros
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: s002080050291
OpenAlex ID: W2021713377
DOI: 10.1007/S002080050291
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description: scientific article; zbMATH DE number 1337030