ell_padic_field.py

"""
Elliptic curves over padic fields
"""

#*****************************************************************************
#       Copyright (C) 2007 Robert Bradshaw <[email protected]>
#                          William Stein   <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
#*****************************************************************************


import sage.rings.ring as ring
from ell_field import EllipticCurve_field
import ell_point
from sage.rings.all import PolynomialRing

# Elliptic curves are very different than genus > 1 hyperelliptic curves, 
# there is an "is a" relationship here, and common implementation with regard 
# Coleman integration. 
from sage.schemes.hyperelliptic_curves.hyperelliptic_padic_field import HyperellipticCurve_padic_field

import sage.databases.cremona


class EllipticCurve_padic_field(EllipticCurve_field, HyperellipticCurve_padic_field):
    """
    Elliptic curve over a padic field.
    """
    def __init__(self, x, y=None):
        """
        Constructor from [a1,a2,a3,a4,a6] or [a4,a6].
        EXAMPLES:
        sage: Qp=pAdicField(17)
        sage: E=EllipticCurve(Qp,[2,3]); E
        Elliptic Curve defined by y^2  = x^3 + (2+O(17^20))*x + (3+O(17^20)) over 17-adic Field with capped relative precision 20
        sage: E == loads(dumps(E))
        True
        """
        if y is None:
            if isinstance(x, list):
                ainvs = x
                field = ainvs[0].parent()
        else:
            if isinstance(y, str):
                field = x
                X = sage.databases.cremona.CremonaDatabase()[y]
                ainvs = [field(a) for a in X.a_invariants()]
            else:
                field = x
                ainvs = y
        if not (isinstance(field, ring.Ring) and isinstance(ainvs,list)):
            raise TypeError
        
        EllipticCurve_field.__init__(self, [field(x) for x in ainvs])

        self._point = ell_point.EllipticCurvePoint_field
        self._genus = 1

    def frobenius(self, P=None):
        """
        Returns the Frobenius as a function on the group of points of
        this elliptic curve.

        EXAMPLE:
            sage: Qp=pAdicField(13)
            sage: E=EllipticCurve(Qp,[1,1])
            sage: type(E.frobenius())
            <type 'function'>
            sage: point=E(0,1)
            sage: E.frobenius(point)
            (0 : 1 + O(13^20) : 1 + O(13^20))
        """
        try:
            _frob = self._frob
        except AttributeError:
            K = self.base_field()
            p = K.prime()
            x = PolynomialRing(K, 'x').gen(0)
            
            a1, a2, a3, a4, a6 = self.a_invariants()
            if a1 != 0 or a2 != 0:
                raise NotImplementedError, "Curve must be in weierstrass normal form."
            
            f = x*x*x + a2*x*x + a4*x + a6
            h = (f(x**p) - f**p)

            # internal function: I don't know how to doctest it...
            def _frob(P):
                x0 = P[0]
                y0 = P[1]
                uN = (1 + h(x0)/y0**(2*p)).sqrt()
                yres=y0**p * uN
                xres=x0**p
                if (yres-y0).valuation() == 0:
                    yres=-yres
                return self.point([xres,yres, K(1)])
                
            self._frob = _frob
            
        if P is None:
            return _frob
        else:
            return _frob(P)