def mass__by_Siegel_densities(self, odd_algorithm="Pall", even_algorithm="Watson"): """ Gives the mass of transformations (det 1 and -1). WARNING: THIS IS BROKEN RIGHT NOW... =( Optional Arguments: - When p > 2 -- odd_algorithm = "Pall" (only one choice for now) - When p = 2 -- even_algorithm = "Kitaoka" or "Watson" REFERENCES: - Nipp's Book "Tables of Quaternary Quadratic Forms". - Papers of Pall (only for p>2) and Watson (for `p=2` -- tricky!). - Siegel, Milnor-Hussemoller, Conway-Sloane Paper IV, Kitoaka (all of which have problems...) EXAMPLES:: sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) sage: m = Q.mass__by_Siegel_densities(); m 1/384 sage: m - (2^Q.dim() * factorial(Q.dim()))^(-1) 0 :: sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1]) sage: m = Q.mass__by_Siegel_densities(); m 1/48 sage: m - (2^Q.dim() * factorial(Q.dim()))^(-1) 0 """ ## Setup n = self.dim() s = (n - 1) // 2 if n % 2 != 0: char_d = squarefree_part(2 * self.det()) ## Accounts for the det as a QF else: char_d = squarefree_part(self.det()) ## Form the generic zeta product generic_prod = ZZ(2) * (pi)**(-ZZ(n) * (n + 1) / 4) ########################################## generic_prod *= (self.det())**( ZZ(n + 1) / 2) ## ***** This uses the Hessian Determinant ******** ########################################## #print "gp1 = ", generic_prod generic_prod *= prod([gamma__exact(ZZ(j) / 2) for j in range(1, n + 1)]) #print "\n---", [(ZZ(j)/2, gamma__exact(ZZ(j)/2)) for j in range(1,n+1)] #print "\n---", prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)]) #print "gp2 = ", generic_prod generic_prod *= prod([zeta__exact(ZZ(j)) for j in range(2, 2 * s + 1, 2)]) #print "\n---", [zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)] #print "\n---", prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)]) #print "gp3 = ", generic_prod if (n % 2 == 0): generic_prod *= quadratic_L_function__exact(n // 2, ZZ(-1)**(n // 2) * char_d) #print " NEW = ", ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d) #print #print "gp4 = ", generic_prod #print "generic_prod =", generic_prod ## Determine the adjustment factors adj_prod = ZZ.one() for p in prime_divisors(2 * self.det()): ## Cancel out the generic factors p_adjustment = prod([1 - ZZ(p)**(-j) for j in range(2, 2 * s + 1, 2)]) if (n % 2 == 0): p_adjustment *= (1 - kronecker( (-1)**(n // 2) * char_d, p) * ZZ(p)**(-n // 2)) #print " EXTRA = ", ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2)) #print "Factor to cancel the generic one:", p_adjustment ## Insert the new mass factors if p == 2: if even_algorithm == "Kitaoka": p_adjustment = p_adjustment / self.Kitaoka_mass_at_2() elif even_algorithm == "Watson": p_adjustment = p_adjustment / self.Watson_mass_at_2() else: raise TypeError( "There is a problem -- your even_algorithm argument is invalid. Try again. =(" ) else: if odd_algorithm == "Pall": p_adjustment = p_adjustment / self.Pall_mass_density_at_odd_prime( p) else: raise TypeError( "There is a problem -- your optional arguments are invalid. Try again. =(" ) #print "p_adjustment for p =", p, "is", p_adjustment ## Put them together (cumulatively) adj_prod *= p_adjustment #print "Cumulative adj_prod =", adj_prod ## Extra adjustment for the case of a 2-dimensional form. #if (n == 2): # generic_prod *= 2 ## Return the mass mass = generic_prod * adj_prod return mass
def mass__by_Siegel_densities(self, odd_algorithm="Pall", even_algorithm="Watson"): """ Gives the mass of transformations (det 1 and -1). WARNING: THIS IS BROKEN RIGHT NOW... =( Optional Arguments: - When p > 2 -- odd_algorithm = "Pall" (only one choice for now) - When p = 2 -- even_algorithm = "Kitaoka" or "Watson" REFERENCES: - Nipp's Book "Tables of Quaternary Quadratic Forms". - Papers of Pall (only for p>2) and Watson (for `p=2` -- tricky!). - Siegel, Milnor-Hussemoller, Conway-Sloane Paper IV, Kitoaka (all of which have problems...) EXAMPLES:: sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) sage: Q.mass__by_Siegel_densities() 1/384 sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1) 0 :: sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1]) sage: Q.mass__by_Siegel_densities() 1/48 sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1) 0 """ ## Setup n = self.dim() s = (n-1) // 2 if n % 2 != 0: char_d = squarefree_part(2*self.det()) ## Accounts for the det as a QF else: char_d = squarefree_part(self.det()) ## Form the generic zeta product generic_prod = ZZ(2) * (pi)**(-ZZ(n) * (n+1) / 4) ########################################## generic_prod *= (self.det())**(ZZ(n+1)/2) ## ***** This uses the Hessian Determinant ******** ########################################## #print "gp1 = ", generic_prod generic_prod *= prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)]) #print "\n---", [(ZZ(j)/2, gamma__exact(ZZ(j)/2)) for j in range(1,n+1)] #print "\n---", prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)]) #print "gp2 = ", generic_prod generic_prod *= prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)]) #print "\n---", [zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)] #print "\n---", prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)]) #print "gp3 = ", generic_prod if (n % 2 == 0): generic_prod *= ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d) #print " NEW = ", ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d) #print #print "gp4 = ", generic_prod #print "generic_prod =", generic_prod ## Determine the adjustment factors adj_prod = 1 for p in prime_divisors(2 * self.det()): ## Cancel out the generic factors p_adjustment = prod([1 - ZZ(p)**(-j) for j in range(2, 2*s+1, 2)]) if (n % 2 == 0): p_adjustment *= ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2)) #print " EXTRA = ", ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2)) #print "Factor to cancel the generic one:", p_adjustment ## Insert the new mass factors if p == 2: if even_algorithm == "Kitaoka": p_adjustment = p_adjustment / self.Kitaoka_mass_at_2() elif even_algorithm == "Watson": p_adjustment = p_adjustment / self.Watson_mass_at_2() else: raise TypeError("There is a problem -- your even_algorithm argument is invalid. Try again. =(") else: if odd_algorithm == "Pall": p_adjustment = p_adjustment / self.Pall_mass_density_at_odd_prime(p) else: raise TypeError("There is a problem -- your optional arguments are invalid. Try again. =(") #print "p_adjustment for p =", p, "is", p_adjustment ## Put them together (cumulatively) adj_prod *= p_adjustment #print "Cumulative adj_prod =", adj_prod ## Extra adjustment for the case of a 2-dimensional form. #if (n == 2): # generic_prod *= 2 ## Return the mass mass = generic_prod * adj_prod return mass