Shimura sums related to imaginary quadratic fields
type: Item
instance of: scholarly article
title: Shimura sums related to imaginary quadratic fields
zbMATH Open document ID: 0813.11031
author: Ken Ono
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-12-19T00:00:00Z
review text: Motivated by the relation of the Fourier coefficients of modular forms and their Shimura lift, the author defines Shimura sums as follows: Let \(g(m)\) be an arithmetical function and \(c\) a positive integer. Then for \(m \geq 1\) the Shimura sum \(Sh(m,g,c)\) is defined by: \[ Sh(m,g,c) = \sum^{m-1}_{k=1} g \left( {m^ 2 - k^ 2 \over c} \right). \] In the special case when \(K = \mathbb{Q} (\sqrt {-2})\), the author proves that: \(P = \sqrt {{-Sh(4p,a,4) \over 2}}\), if \(p\) is inert in \(K\), and \(P = \sqrt {a^ 2(p) + {Sh(4p,a,4) \over 2}}\), if \(p\) splits or ramifies in \(K\). Here \(a(n)\) is the \(n\)-th coefficient of the Fourier expansion of some modular form of weight \(k = 3\) on \(\Gamma_ 0 (8)\) with Nebentypus \(\chi (d) = \left( {-2 \over d} \right)\).
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 701470
zbMATH Keywords: Shimura sums
zbMATH Keywords: arithmetical function
zbMATH Keywords: Fourier expansion
zbMATH Keywords: modular form
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
full work available at URL: pjaa.70.146
OpenAlex ID: W1977713362
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DOI: 10.3792/PJAA.70.146
description: scientific article; zbMATH DE number 701470