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.

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 Modular forms of degree n and representation by quadratic forms

1979 ◽  
Vol 74 ◽  
pp. 95-122 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Circle Method ◽  
Positive Definite ◽  
Siegel Modular Forms ◽  
Partial Result ◽  
Definite Integral ◽  
Integral Matrices

Let A(m), B(n) be positive definite integral matrices and suppose that B is represented by A over each p-adic integers ring Zp. Using the circle method or theory of modular forms in case of n = 1, B, if sufficiently large, is represented by A provided that m ≥ 5. The approach via the theory of modular forms has been extended by [7] to Siegel modular forms to obtain a partial result in the particular case when n = 2, m ≥ 7.

scholarly journals Modular forms of degree n and representation by quadratic forms II

1982 ◽  
Vol 87 ◽  
pp. 127-146 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Asymptotic Formula ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Definite Integral ◽  
Analytic Methods ◽  
Integral Matrices ◽  
Integer Ring

Let S(m), T(n) be positive definite integral matrices and suppose that T is represented by S over each p-adic integer ring Zp. We proved arithmetically in [3] that T is represented by S over Z provided that m ≥ 2n + 3 and the minimum of T is sufficiently large. This guarantees the existence of at least one representation but does not give any asymptotic formula for the number of representations. To get an asymptotic formula we must employ analytic methods.

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 FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

2016 ◽  
Vol 234 ◽  
pp. 1-16
Author(s):  
SIEGFRIED BÖCHERER ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Main Tool ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Growth Properties

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

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.

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