Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids

1990 ◽  
Vol 99 (1) ◽  
pp. 49-57 ◽  
Author(s):  
William Duke ◽  
Rainer Schulze-Pillot
Keyword(s):  
Quadratic Forms ◽  
Lattice Points ◽  
Ternary Quadratic Forms ◽  
Representation Of Integers

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.

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