On the Dimension of Some Spaces of Generalized Ternary Theta-Series

2002 ◽  
Vol 9 (1) ◽  
pp. 167-178
Author(s):  
K. Shavgulidze
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Theta Series ◽  
Vector Spaces ◽  
Ternary Quadratic Forms ◽  
Generalized Theta Series

Abstract The upper bound of dimension of vector spaces of generalized theta-series corresponding to some ternary quadratic forms is established. In a number of cases, the dimension of vector spaces of generalized theta-series is established and bases of these spaces are constructed.

On the space of generalized theta-series for certain quadratic forms in any number of variables

2019 ◽  
Vol 69 (1) ◽  
pp. 87-98
Author(s):  
Ketevan Shavgulidze
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Theta Series ◽  
Vector Spaces ◽  
Generalized Theta Series

Abstract An upper bound of the dimension of vector spaces of generalized theta-series corresponding to some nondiagonal quadratic forms in any number of variables is established. In a number of cases, an upper bound of the dimension of the space of theta-series with respect to the quadratic forms of five variables is improved and the basis of this space is constructed.

scholarly journals REPRESENTATIONS OF INTEGERS BY TERNARY QUADRATIC FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 127-158 ◽  
Author(s):  
BEN KANE
Keyword(s):  
Quadratic Forms ◽  
Theta Series ◽  
Implied Constant ◽  
Cusp Forms ◽  
Local Conditions ◽  
Full List ◽  
Ternary Quadratic Forms ◽  
Representation Of Integers ◽  
Representations Of Integers

We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen's plus space [Formula: see text], where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for L-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan–Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree.

scholarly journals Non-negative values of quadratic forms

1971 ◽  
Vol 12 (2) ◽  
pp. 224-238 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Simple Consequence ◽  
Ternary Quadratic Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic

In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimum m+(f) of the non-negative values1 of an indefinite quadratic form f in n variables, of given determinant det(f) ≠ 0 and of signature s. In particular it was announced (and later proved — see [2]) that m+(f) ≦ (16/5)+ for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is that m+(f) ≦ (256/135)+ for quaternary quadratic forms of determinant — 1 and signature — 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].

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