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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 695265, 34 pages
http://dx.doi.org/10.1155/2013/695265
Research Article

The Bases of and the Number of Representation of Integers

Department of Mathematics, Faculty of Arts and Sciences, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey

Received 17 January 2013; Accepted 6 February 2013

Academic Editor: Safa Bozkurt Coskun

Copyright © 2013 Barış Kendirli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Following a fundamental theorem of Hecke, some bases of and are determined, and explicit formulas are obtained for the number of representations of positive integers by all possible direct sums (111 different combinations) of seven quadratic forms from the class group of equivalence classes of quadratic forms with discriminant −71 whose representatives are .

1. Introduction

In this work, we will obtain the representation number of certain positive definite quadratic forms by means of the deep theorems of Hecke [1] and Schoeneberg [2]. Lomadze [3] and Vepkhvadze [4] have studied this subject for several quadratic forms. Also the works of Kani [5], Sun and Williams [6] clarify and contribute the subject. Of couse, the work of Hijikata et al. [7] for the construction of a general bases of modular forms of [7] is very important work. Here, we have found bases in special cases by extending our work on [8, 9] to the case of discriminant −71 including weight 6 case.

There exist 7 equivalence classes of binary quadratic forms of discriminant whose reduced forms are

Here, is the inverse of , and they represent the same integers. Similarly, is the inverse of , and they represent the same integers, and is the inverse of , and they represent the same integers. The group of these quadratic forms is a group of order 7 and can be described, see [10], easily as Therefore, the theta series of , and , and and are the same, respectively. is the identity element. Since 71 is prime, there is only one genus, that is, principal genus.

Let Be   direct sums of the quadratic forms. In this paper, we will obtain the formulas of for the quadratic forms of 8 variables and quadratic form of 12 variables

In these formulas one can replace ,, and by , and , respectively.

2. The Positive Definite Quadratic Forms

Now we will give some definitions, an important theorem and evaluation of our quadratic forms.

Definition 1. Let be a positive definite integer-valued form of variables Let be the determinant of the quadratic form And let be the cofactors of for . If , then is the smallest positive integer, called the level of , for which    is again an even integral matrix like . Then,
is called the discriminant of the form .

Theorem 2. Let be a positive definite integer-valued form of variables of level and discriminant . Then the theta function is a modular form on of weight and Kronecker character that is, . Consider the following:
The homogeneous quadratic polynomials in variables are spherical functions of second order with respect to . Consider the following:
The theta series is a cusp form in .
If two quadratic forms have the same level and the characters respectively, then the direct sum of the quadratic forms has the same level and the character .

Proof. See [1, 2].

Now, let us look at the positive definite quadratic forms of discriminant .

For the quadratic form , the determinant and a cofactor are So, , and the discriminant is The character of is the Kronecker symbol

Similarly, for the quadratic form , the determinant and a cofactor are So, , and the character of is (15).

For the quadratic form , the determinant and a cofactor are So, , and the character of is (15).

For the quadratic form , the determinant and some cofactors are So, , and the character of is (15).

Consequently, are quadratic forms whose theta series are in ; hence, by Theorem 2,   and are quadratic forms whose theta series are in . Moreover, the theta series of quadratic forms in (4) are in , and the theta series of quadratic forms in (5) are in .

We have the following important theorem for the Eisenstein part of theta series associated with the quadratic form.

Theorem 3. Let be a positive definite form of or variables whose theta series are in and respectively. Then, the Eisenstein parts of are respectively, where

Proof. See [1].

3. The Selection of Spherical Functions for Weight 4 and for Weight 6

First, we will be able to select 17 spherical functions such that the corresponding cusp forms become a basis of since its dimension is 17; see [11].

For the quadratic form the determinant and a cofactor are . By putting , and appropriate in Theorem 2, we get which will be a spherical function of second order with respect to . Similarly,

Similarly, we will be able to select 29 spherical functions such that the corresponding cusp forms become a basis of since its dimension is 29; see [11],

4. The Solutions of and the Theta Series Associated with the Quadratic Forms

has the following solutions:

Consider the following: There is no integral solutions for .

Consider the following:

has the following solutions: There is no integral solutions for .

Consider the following:

has the following solutions: There is no integral solutions for .

has the following solutions:

Consider the following: There is no integral solutions for .

Hence, for quadratic forms of 8 variables, we have