A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)

2013 ◽  
Vol 61 (1) ◽  
pp. 23-26 ◽  
Author(s):  
A. Schinzel
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Linear Forms

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

On the Distribution of Integer Solutions of f(x, y) = z2 for a Definite Binary Quadratic Form f

1967 ◽  
Vol 10 (5) ◽  
pp. 755-756
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Integer Solutions

Let f be a positive definite binary quadratic form with rational coefficients. We shall call a point (x, y) in E2 with integers x and y a Pythagorean point of f when f(x, y) = z2 is satisfied with some integer z, and shall prove the following theorem.

scholarly journals Sums of squares of integral linear forms

2000 ◽  
Vol 69 (3) ◽  
pp. 298-302 ◽  
Author(s):  
Hideyo Sasaki
Keyword(s):  
Quadratic Form ◽  
Positive Definite ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Linear Forms ◽  
Integral Quadratic Form

AbstractIn this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

scholarly journals Petersson inner products of weight-one modular forms

2019 ◽  
Vol 2019 (749) ◽  
pp. 133-159
Author(s):  
Maryna Viazovska
Keyword(s):  
Quadratic Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Theta Series ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Arithmetic Properties ◽  
Factorization Formula ◽  
Weight One

Abstract In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

On a problem about the Epstein zeta-function

1964 ◽  
Vol 60 (4) ◽  
pp. 855-875 ◽  
Author(s):  
Veikko Ennola
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Special Form ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Simple Pole ◽  
Epstein Zeta Function ◽  
Image Position

1. Letbe a positive definite binary quadratic form with determinant αβ − δ2 = 1. A special form of this kind isWe consider the Epstein zeta-functionthe series converging for . The function Zh(s) can be analytically continued over the whole s-plane and it is regular except for a simple pole with residue π at s = 1.

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