quadratic_form__genus.py

"""
Local and Global Genus Symbols

"""


#############################################################
##                                                         ##
##  Wrappers for the Genus/Genus Symbol Code in ../genera/ ##
##                                                         ##
#############################################################

from sage.quadratic_forms.genera.genus import Genus, LocalGenusSymbol, \
        is_GlobalGenus, is_2_adic_genus, canonical_2_adic_compartments, \
        canonical_2_adic_trains, canonical_2_adic_reduction, \
        basis_complement, p_adic_symbol, is_even_matrix, \
        split_odd,  trace_diag_mod_8, two_adic_symbol 
        #is_trivial_symbol
        #GenusSymbol_p_adic_ring, GenusSymbol_global_ring

## Removed signature_pair_of_matrix due to a circular import issue.

## NOTE: Removed the signature routine here... and rewrote it for now.


from sage.rings.integer_ring import IntegerRing
from sage.rings.arith import is_prime, prime_divisors



def global_genus_symbol(self):
    """
    Returns the genus of a two times a quadratic form over ZZ.  These
    are defined by a collection of local genus symbols (a la Chapter
    15 of Conway-Sloane), and a signature.

    EXAMPLES::
        
        sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
        sage: Q.global_genus_symbol()
        Genus of [2 0 0 0]
        [0 4 0 0]
        [0 0 6 0]
        [0 0 0 8]

    ::
    
        sage: Q = QuadraticForm(ZZ, 4, range(10))
        sage: Q.global_genus_symbol()
        Genus of [ 0  1  2  3]
        [ 1  8  5  6]
        [ 2  5 14  8]
        [ 3  6  8 18]

    """
    ## Check that the form is defined over ZZ
    if not self.base_ring() == IntegerRing():
        raise TypeError, "Oops!  The quadratic form is not defined over the integers."

    ## Return the result
    try:
        return Genus(self.Hessian_matrix())      
    except:
        raise TypeError, "Oops!  There is a problem computing the genus symbols for this form."



def local_genus_symbol(self, p):
    """
    Returns the Conway-Sloane genus symbol of 2 times a quadratic form
    defined over ZZ at a prime number p.  This is defined (in the
    Genus_Symbol_p_adic_ring() class in the quadratic_forms/genera
    subfolder) to be a list of tuples (one for each Jordan component
    p^m*A at p, where A is a unimodular symmetric matrix with
    coefficients the p-adic integers) of the following form:

        1. If p>2 then return triples of the form [`m`, `n`, `d`] where
            `m` = valuation of the component
            
            `n` = rank of A
            
            `d` = det(A) in {1,u} for normalized quadratic non-residue u.

        2. If p=2 then return quintuples of the form [`m`,`n`,`s`, `d`, `o`] where
            `m` = valuation of the component
            
            `n` = rank of A
            
            `d` = det(A) in {1,3,5,7}
            
            `s` = 0 (or 1) if A is even (or odd)
            
            `o` = oddity of A (= 0 if s = 0) in Z/8Z
              = the trace of the diagonalization of A            

    NOTE: The Conway-Sloane convention for describing the prime 'p =
    -1' is not supported here, and neither is the convention for
    including the 'prime' Infinity.  See note on p370 of Conway-Sloane
    (3rd ed) for a discussion of this convention.

    INPUT:
    
        -`p` -- a prime number > 0

    OUTPUT:
        Returns a Conway-Sloane genus symbol at p, which is an
        instance of the Genus_Symbol_p_adic_ring class.

    EXAMPLES::
    
        sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
        sage: Q.local_genus_symbol(2)
        Genus symbol at 2 : [[1, 2, 3, 1, 4], [2, 1, 1, 1, 1], [3, 1, 1, 1, 1]]
        sage: Q.local_genus_symbol(3)
        Genus symbol at 3 : [[0, 3, 1], [1, 1, -1]]
        sage: Q.local_genus_symbol(5)
        Genus symbol at 5 : [[0, 4, 1]]
        
    """
    ## Check that p is prime and that the form is defined over ZZ.
    if not is_prime(p):
        raise TypeError, "Oops!  The number " + str(p) + " isn't prime."
    if not self.base_ring() == IntegerRing():
        raise TypeError, "Oops!  The quadratic form is not defined over the integers."

    ## Return the result
    try:
        M = self.Hessian_matrix()
        return LocalGenusSymbol(M, p)      
    except:
        raise TypeError, "Oops!  There is a problem computing the local genus symbol at the prime " + str(p) + " for this form."






def CS_genus_symbol_list(self, force_recomputation=False):
    """
    Returns the list of Conway-Sloane genus symbols in increasing order of primes dividing 2*det.

    EXAMPLES::
    
        sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
        sage: Q.CS_genus_symbol_list()
        [Genus symbol at 2 : [[1, 2, 3, 1, 4], [2, 1, 1, 1, 1], [3, 1, 1, 1, 1]],
        Genus symbol at 3 : [[0, 3, 1], [1, 1, -1]]]
        
    """    
    ## Try to use the cached list
    if force_recomputation == False:
        try:
            return self.__CS_genus_symbol_list
        except:
            pass

    ## Otherwise recompute and cache the list
    list_of_CS_genus_symbols = [ ]

    for p in prime_divisors(2 * self.det()):
        list_of_CS_genus_symbols.append(self.local_genus_symbol(p))

    self.__CS_genus_symbol_list = list_of_CS_genus_symbols
    return list_of_CS_genus_symbols