A remark on Serre's example of \(p\)-adic Eisenstein series
type: Item
instance of: scholarly article
title: A remark on Serre's example of \(p\)-adic Eisenstein series
zbMATH Open document ID: 0982.11021
author: Shoyu Nagaoka
published in: Mathematische Zeitschrift
publication date: 2002-03-27T00:00:00Z
review text: Let \(p>3, p\equiv 3 \pmod 4\) be a prime number. Building on ideas of Serre, the author evaluates the \(p\)-adic limit of the Siegel-Eisenstein series of arbitrary (fixed) degree \(n\) and weight \(1+{p-1\over 2}p^m, m\to\infty,\) and expresses the constant term as half a class number and the other coefficients by arithmetic expressions. The main input is information on Bernoulli numbers and the Fourier coefficients of the Siegel-Eisenstein series. In degree 2, the author also derives a formula for the mod \(p\) Fourier coefficients of Siegel-Eisenstein series of weight \({p+1\over 2}\). As a last result, it is shown that the \(p\)-adic Eisenstein series under consideration coincides with a genus theta series, and, up to a factor, also with a certain weight-1-Eisenstein series.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F46
zbMATH DE Number: 1546901
zbMATH Keywords: congruences for modular forms
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Siegel-Eisenstein series
zbMATH Keywords: \(p\)-adic modular forms
MaRDI profile type: Publication
full work available at URL: s002090000134
OpenAlex ID: W2028361750
DOI: 10.1007/S002090000134
description: scientific article; zbMATH DE number 1546901