scholarly journals Spinor representations of positive definite ternary quadratic forms

2018 ◽  
Vol 14 (02) ◽  
pp. 581-594 ◽  
Author(s):  
Jangwon Ju ◽  
Kyoungmin Kim ◽  
Byeong-Kweon Oh
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Ternary Quadratic Form ◽  
Definite Integral ◽  
Constant Multiple ◽  
Ternary Quadratic Forms ◽  
Spinor Genus ◽  
Local Densities ◽  
Spinor Representations

For a positive definite integral ternary quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. The famous Minkowski–Siegel formula implies that if the class number of [Formula: see text] is one, then [Formula: see text] can be written as a constant multiple of a product of local densities which are easily computable. In this paper, we consider the case when the spinor genus of [Formula: see text] contains only one class. In this case the above also holds if [Formula: see text] is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the recent results on both the representations of generalized Bell ternary forms and the representations of ternary quadratic forms with some congruence conditions.

The number of representations of integers by generalized Bell ternary quadratic forms

2020 ◽  
pp. 1-15
Author(s):  
Kyoungmin Kim ◽  
Yeong-Wook Kwon
Keyword(s):  
Quadratic Form ◽  
Class Number ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Cusp Forms ◽  
Ternary Quadratic Form ◽  
Closed Formula ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Representations Of Integers

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

REPRESENTATIONS OF ARITHMETIC PROGRESSIONS BY POSITIVE DEFINITE QUADRATIC FORMS

2011 ◽  
Vol 07 (06) ◽  
pp. 1603-1614 ◽  
Author(s):  
BYEONG-KWEON OH
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Negative Integer ◽  
Definite Integral ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

scholarly journals Ternary quadratic forms and Brandt matrices

1986 ◽  
Vol 102 ◽  
pp. 117-126 ◽  
Author(s):  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Main Tool ◽  
Positive Definite ◽  
Cusp Forms ◽  
Definite Integral ◽  
Ternary Quadratic Forms ◽  
Spinor Genus ◽  
Genus One ◽  
Space Of Cusp Forms

In a recent paper [9] the author showed (among other results) estimates on the asymptotic behaviour of the representation numbers of positive definite integral ternary quadratic forms, in particular, that for n in a fixed square class tZ2 and lattices L, K in the same spinor genus one has . The main tool utilized for the proof was the theory of modular forms of weight 3/2, especially Shimura’s lifting from the space of cusp forms of weight 3/2 to the space of modular forms of weight 2.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

On the existence of integral ternary quadratic forms without automorphs of finite order

2017 ◽  
Vol 26 (14) ◽  
pp. 1750102 ◽  
Author(s):  
José María Montesinos-Amilibia
Keyword(s):  
Quadratic Form ◽  
Finite Order ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms

An example of an integral ternary quadratic form [Formula: see text] such that its associated orbifold [Formula: see text] is a manifold is given. Hence, the title is proved.

scholarly journals New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms

2006 ◽  
Vol 13 (4) ◽  
pp. 687-691
Author(s):  
Guram Gogishvili
Keyword(s):  
Quadratic Form ◽  
General Result ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Precise Estimate ◽  
Definite Integral ◽  
Singular Series ◽  
Prime Factors ◽  
Quaternary Quadratic Form ◽  
Best Estimates

Abstract Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑0𝑑1, if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

scholarly journals On indefinite ternary quadratic forms

1974 ◽  
Vol 18 (4) ◽  
pp. 388-401 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms ◽  
Indefinite Ternary Quadratic Form ◽  
Image Position

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

The positive values of inhomogeneous ternary quadratic forms

1961 ◽  
Vol 2 (2) ◽  
pp. 127-132 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Equality Sign ◽  
Ternary Quadratic Forms ◽  
Positive Multiple ◽  
Indefinite Ternary Quadratic Form ◽  
Image Position

Let f(x, y, z) be an indefinite ternary quadratic form of signature (2, 1) and determinant d ≠ 0. Davenport [3] has shown that there exist integral x, y, z with, the equality sign being necessary if and only if f is a positive multiple of f1(x, y, z) = x2 + yz.

scholarly journals PRIME-UNIVERSAL DIAGONAL QUADRATIC FORMS

2020 ◽  
pp. 1-15
Author(s):  
JANGWON JU ◽  
DAEJUN KIM ◽  
KYOUNGMIN KIM ◽  
MINGYU KIM ◽  
BYEONG-KWEON OH
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms

Abstract A (positive definite and integral) quadratic form is said to be prime-universal if it represents all primes. Recently, Doyle and Williams [‘Prime-universal quadratic forms $ax^2+by^2+cz^2$ and $ax^2+by^2+cz^2+dw^2$ ’, Bull. Aust. Math. Soc.101 (2020), 1–12] classified all prime-universal diagonal ternary quadratic forms and all prime-universal diagonal quaternary quadratic forms under two conjectures. We classify all prime-universal diagonal quadratic forms regardless of rank, and prove the so-called 67-theorem for a diagonal quadratic form to be prime-universal.

scholarly journals Inequalities for inert primes and their applications

2020 ◽  
Vol 16 (08) ◽  
pp. 1819-1832
Author(s):  
Zilong He
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Kronecker Symbol ◽  
Ternary Quadratic Forms ◽  
New Criterion

For any given non-square integer [Formula: see text], we prove Euclid’s type inequalities for the sequence [Formula: see text] of all primes satisfying the Kronecker symbol [Formula: see text], [Formula: see text] and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones’s argument in the classification of regular ternary quadratic forms to some extent.

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