# Fermat Curves X^m+Y^m=1 for higher genus g=(m-1)(m-2)/2
# Computing the number of points N(Fp^s) over higher dimensional finite fields
#
def NsFCm(p,n,m): # X^m+Y^m=1 over Fp^n, where p prime and n integer
    N=0;
    q=p^n; K=GF(q,'a')
    for x in K:
        for y in K:
            f=x^m+y^m-1
            # print f
            if f==0:
                N+=1; # print "N=", N
    return N # do NOT add points at infinity here!
#  End procedure

# test Fermat Curve X^3+Y^3=1
p=7; m=3; max=3 # prime p, expeonent m, s<=max
N=[0 for k in range(max)]
for k in range(max):
    s=k+1 # dimension of Fp^s over Fp
    N[k]=NsFCm(p,s,m) # number of finite solutions
print "p=", p, " max s=", max, " N(Fp^s)=", N

# Checking the formula for AFFINE points N_aff(s)=N_proj-m
# since m is odd, so -1=(-1)^m and (X/Y)^m=-1 has m|p-1 solutions
#
#             N_aff(s)=p^s+(1-m) - w^s - w*^s
#

# For m=3 N1=p-2+A, A=2 Re(J(c,c))
# For p=7: N1=6 => A=N1-p+2=1 => Re(J)=1/2

# Predictions of number of affine points: 
#       with w=-J, N(s)=p^s-2-w^s-w*^s 
# ... but we need J!

# Computing Jacobi sum J(c,c) for character of order m
G=DirichletGroup(p); G
gen=G[1]; gen # generator of group of multiplicative characters
c=gen^ZZ((p-1)/m); c
print "Order:", c.order()
J=c.jacobi_sum(c); J
A=J+conjugate(J); print "A=",A

# A is "ok" ... Now raise w to power s ...
w1=-J; w2=conjugate(w1) # J from above!
print "Weil zeros: w1=", w1, " w2=", w2
for k in range(max):
    s=k+1 # s=1 ... max
    print "s=", s, " N(s)=", p^s-2-w1^s-w2^s

# Success! Counting the number of points and computing N(s) coincide.

# The Zeta Function= P(t) / Q(t) for the Fermat Cubic x^3+y^3=1
# Whatever K was defined above ...
p=7;
L=CyclotomicField(p-1) # Where Jacobi sums and Weil zeros "live" 
R.<t>= L["t"]; R
WBP=(1-w1*t)*(1-w2*t); WBP # Weil-Betti polynomial (for m=3)
WBQ=(1-t)*(1-p*t); WBQ     # Typical denominator for genus g=1
print "Zeta function=", WBP/WBQ

# Success! recovering p & A=2 Re(J) from Weil zeros from the formula giving N(s)
# ******************************************************************************
︠b8eefe3c-496f-44d7-8bb1-049e65600b56︠

# Other primes: p=61 & 67 (p.95 & 97)
#


# Other exponents: m=5, 7, etc & p=me+1 ...