On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials
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)].
representations of natural numbers by forms
scientific article; zbMATH DE number 784064
scientific article