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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 303492, 10 pages
http://dx.doi.org/10.1155/2012/303492
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Cusp Forms in and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic Forms with Discriminant

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

Received 20 August 2011; Accepted 15 September 2011

Copyright © 2012 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

A basis of is given and the formulas for the number of representations of positive integers by some direct sum of the quadratic forms , , are determined.

1. Introduction

This paper is the correction of the paper [1]. (1) It is stated that at page 643 in [1]. But this dimension is 11 as stated at page 299 in [2]. Therefore, the coefficients of power series in , , , , , , , , , , , and have to be calculated up to , and Theorem 2.4 and consequently Theorem 2.7 are false since 5 vectors cannot be a basis of 11-dimensional vector space.(2) The notations and are both used as if they are different like at line 11 at page 638.(3)The definitions of have never been used in the paper.(4) The class number of is 3; therefore, only , , and their combinations have been examined and a basis of could be obtained. The authors in [1] also examined only two quadratic forms and their combinations. But, by simple calculations, it is possible to see that these quadratic forms are not enough to get a basis of . The class number of is 5; therefore, and their combinations have to be examined. Only in that case, it is possible to obtain a basis of as we have done in the following.

2. Determination of a Basis of

We can calculate all reduced forms of a positive definite quadratic form with discriminant as follows:

Here is the inverse of , and they represent the same integers. Similarly, is the inverse of and they represent the same integers. Therefore, the theta series of and are the same with the theta series of and , respectively. is the identity element. It can be seen easily that, the group of these quadratic forms is a group of order 5 and can be described as We can easily see that for the quadratic forms the determinant, the discriminant, and the character are Consequently, their theta series are in Hence by Theorem 2.1 in [3], , , , , , and are quadratic forms whose theta series are in

We immediately obtain the following Corollary by Theorem 2.2 in [3].

Corollary 2.1. Let be a positive definite form of 8 variables whose theta series is in Then the Eisenstein part of is where Here

Now we will determine the sum of quadratic forms , , and and select 11 spherical functions such that the corresponding cusp forms are linearly independent.(1) For quadratic form the determinant and a cofactor are By putting , , and appropriate in Theorem 2.1 in [3], we get which will be spherical functions of second order with respect to .(2)Similarly, for the determinant and some cofactors are By putting , , and appropriate in Theorem 2.1 in [3], we get which will be spherical functions of second order with respect to .(3) Similarly, for quadratic form the determinant and some cofactors are By putting , , and appropriate in Theorem 2.1 in [3], we get which will be spherical functions of second order with respect to .(4) Similarly, for quadratic form the determinant and some cofactors are By putting , , and appropriate in Theorem 2.1 in [3], we get which will be spherical functions of second order with respect to .

Now we can determine a basis of whose dimension is 11, see [2].

Theorem 2.2. The following generalized 11 theta series: are a basis of .

Proof. The series are cusp forms because of Theorem 2.1 in [3]. Moreover, by simple calculations, we have The determinant of the coefficients matrix is . So, the set of theta series in the Theorem is a basis of .

3. Representation Numbers of

Proposition 3.1. The theta series of the quadratic forms are and the substraction of the any one of these theta series by the Eisenstein series is a linear combinations of the theta series in the preceding theorem. The coefficients are given in table [4].

Proof. By determination of solutions of we easily calculate the theta series For the second part, now let us look at the case: By equating the coefficients of in both sides for , we get an equation in coefficients We repeat the same procedure for the other cases. At the end, by solving 11 linear equations in 11 variables, we get the coefficients in table [4].

Corollary 3.2. The representation numbers for the quadratic forms are The coefficients corresponding to the quadratic form are given in [4].

Proof. It follows from the preceding theorem.

References

  1. A. Tekcan and O. Bizim, β€œOn the number of representations of positive integers by quadratic forms as the basis of the space S4(Γ0(47),1),” International Journal of Mathematics and Mathematical Sciences, no. 9–12, pp. 637–646, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. T. Miyake, Modular Forms, Springer, Berlin, Germany, 1989.
  3. B. Kendirli, β€œCusp Forms in S4(Γ0(79)) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant −79,” Bulletin of the Korean Mathematical Society. In press.
  4. http://math.fatih.edu.tr/~bkendirli/Announcements.htm.