counting-points.ipynb

Kernel: SageMath 10.2

def test(q):
    k = GF(q)
    V = VectorSpace(k,4)
    R.<X,Y,Z,W> = PolynomialRing(k)
    F = lambda a: X*Y + Z^2 - a*W^2

    def discSquare(a):
        if a in [ b^2 for b in k ]:
            return True
        else:
            return False    
    def eval(F,v):
        return F(X=v[0],Y=v[1],Z=v[2],W=v[3])
    def expect(a):
        if discSquare(a): return q^2 + 2*q + 1
        else: return q^2 + 1
    def count_sols(F):
        pts = [ v for v in V if eval(F,v)==0 and v != 0]
        return len(pts)/(q-1)
    return [ (discSquare(a),expect(a),count_sols(F(a))) for a in k if a !=0 ]

test(9)

[(False, 82, 82),
 (True, 100, 100),
 (False, 82, 82),
 (True, 100, 100),
 (False, 82, 82),
 (True, 100, 100),
 (False, 82, 82),
 (True, 100, 100)]

ell.<z> = GF(7^4)

def trace(x):
    return sum([x^(p^i) for i in range(n)])

def is_square(x,k):
    if x in [ a^2 for a in k]:
        return True
    else:
        return False

def M(p,n):
    ell.<z> = GF(p^n)
    M = MatrixSpace(ell,n).matrix([[trace(z^(i+j),p,n) for i in range(n) ] for j in range(n)])
    return is_square(M.determinant(),ell)
    
[ M(31,i) for i in range(1,5) ]

[True, True, True, True]

M

[3 1 1]
[1 1 3]
[1 3 6]

is_square(4,k)

True