On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials
type: Item
instance of: scholarly article
title: On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials
zbMATH Open document ID: 0842.11014
DOI: 10.1007/BF02570464
author: I. Kiming
published in: Manuscripta Mathematica
publication date: 1996-07-28T00:00:00Z
full work available at URL: https://eudml.org/doc/156077
review text: Let \(F_1\) and \(F_2\) be two integral, primitive, positive-definite, binary quadratic forms of discriminant \({\mathcal D}\), which are in the same genus of classes, and consider the quaternary form \(f= F_1\oplus F_2\) and associated theta series \[ \theta_f (z)= \theta_{F_1} (z) \theta_{F_2} (z)= \sum_{x,y, u,v\in \mathbb{Z}} e^{2\pi i (F_1 (x,y)+ F_2 (u, v))z} \] for \(\text{Im} (z)>0\), which is a modular form of weight 2 on \(\Gamma_0 (|{\mathcal D}|)\). The paper studies certain linear combinations \(E({\mathcal D})\) of Eisenstein series of the same type and proves that \(\theta_f (z)- E({\mathcal D})\) for each \(E({\mathcal D})\) is a cusp form of the mentioned type. This result leads to an asymptotic formula for the number \(r(f,n)\) of representations of natural numbers \(n\) by the form \(f\). The paper gives a convenient explicit formula for the main term of it. For the error term the paper gives \(O(p^{1/2})\), as was obtained earlier by Eichler. \textit{T. Vepkhvadze} [Tr. Tbilis. Math. Inst. Razmadze 45, 41-59 (1974; Zbl 0317.10028)]\ discussed the problem in the case \(F_1\) and \(F_2\) are arbitrary binary forms. The formulae for the evolution of the main term there was reduced to the main term of a corresponding diagonal quaternary form, a convenient formulae for which was obtained by the reviewer [Tr. Tbilis. Math. Inst. Razmadze 38, 5-30 (1970; Zbl 0218.10033)].
reviewed by: Guram Gogishvili
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 784064
zbMATH Keywords: representations of natural numbers by forms
zbMATH Keywords: binary quadratic forms
zbMATH Keywords: theta series
zbMATH Keywords: modular form
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
cites work: Q4858021
cites work: Q3972631
cites work: Heegner points and derivatives of \(L\)-series
cites work: The number of curves of genus two with elliptic differentials.
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description: scientific article; zbMATH DE number 784064