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# *************************************************************
# Computes Jacobi Sums: J(c,c')=Sum c(t) c'(1-t)
# Syntax: e.jacobi_sum(f) where e and f are Dirichlet characters from
#         the same group DirichletGroup(p)
# *************************************************************
# Referece:
# https://mvngu.googlecode.com/hg/onepage/sage/modular/dirichlet/sage.modular.dirichlet.DirichletCharacter.jacobi_sum.html

# Example
p=7
DP = DirichletGroup(p); DP
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2 
c1=DP[1]; c1
c2=DP[2]; c2
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1 
c1.jacobi_sum(c2)
2*zeta6 + 1 
# Example 2 and check by hand
p=3; print "p=",p
DP=DirichletGroup(p); DP
p= 3 Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 2 and degree 1 
e=DP[0]; e
f=DP[1]; f
Dirichlet character modulo 3 of conductor 1 mapping 2 |--> 1 Dirichlet character modulo 3 of conductor 3 mapping 2 |--> -1 
# DirichletGroup(3) has only 2 characters
e.jacobi_sum(f)
e.jacobi_sum(e)
f.jacobi_sum(f)
-1 1 1 
# Check by hand ...

# *****************************************************************
# Jacobi sums J(c,c)

# Getting used to Dirichlet characters ...
p=19; DP = DirichletGroup(p); DP
for k in range(len(list(DP))):
    DP[k]
print "\n"
c=DP[1] # is a generator
for k in range(p-1):
    c^k
Group of Dirichlet characters of modulus 19 over Cyclotomic Field of order 18 and degree 6 Dirichlet character modulo 19 of conductor 1 mapping 2 |--> 1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^2 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^3 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^4 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^5 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^3 - 1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^4 - zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^5 - zeta18^2 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^2 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^3 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^4 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^5 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^3 + 1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^4 + zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^5 + zeta18^2 Dirichlet character modulo 19 of conductor 1 mapping 2 |--> 1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^2 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^3 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^4 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^5 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^3 - 1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^4 - zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^5 - zeta18^2 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^2 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^3 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^4 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^5 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^3 + 1 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^4 + zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^5 + zeta18^2 
# **************************************************************************
#                  Part II: Comparing Jacobi sum with Wen Wang 
#                           EC y^2=x^3+D, p=19, D=5
# **************************************************************************
# Constructing c1: a Dirichlet Character of order 2 ("rho" in WW-paper)
# and computing J(c,c)
m=2;  n=ZZ((p-1)/m); print m,n
c=DP[1]; c # generator of Dirichlet characters
c1=c^n; c1 # character of order m=2 (Legendre)
if c1^m==DP[0]:
    print "char of order:", m
print "J(c,c)=", c1.jacobi_sum(c1) # J(c,c), ord(c)=2
2 9 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18 Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -1 char of order: 2 J(c,c)= 1 
# Constructing c2: a Dirichlet Cjaracter of order 3 
# and computing its Jacobi sum J(c2,c2)
#
m=3; c2=DP[1]^ZZ((p-1)/m); c2 # character of order m=3
J=c2.jacobi_sum(c2)
print "ord(c)=",m, " J(c,c)=", J # J(c,c), ord(c)=3
Dirichlet character modulo 19 of conductor 19 mapping 2 |--> zeta18^3 - 1 ord(c)= 3 J(c,c)= -3*zeta18^3 + 5 
# In WW paper J(c,c)=5+3*zeta3 ... we got 5-3*zeta6, but -zeta6=zeta3^2=conj(zeta3)
# so our J=5+3*zeta3^2
# Now in Weil zeros for x^3+D file we got w=-(5+3zeta3^2), so w=-J ... close enough :)
D=5
cc=c1*c2; cc
d=(c1*c2)(4*D)
print c1(4*D), c2(4*D), d
Dirichlet character modulo 19 of conductor 19 mapping 2 |--> -zeta18^3 + 1 1 1 1 
# Checking N=p+1+2*Re[cc(4D) J(c2,c2)]
p+1+d*J+conjugate(d*J)
27 
# As expected! N=p+1-a, a=-7 (see Weil zeros file)