Changeset 5447:cd60620ddfe5

Timestamp:

06/05/07 01:49:06 (6 years ago)

Author:

David Kohel <kohel@…>

Branch:

default

Children:

5448:a711eb87b846, 5449:bb432d7a85b0

Message:

Fixes to left multiplication and coercion of 0 in Jacobians and point sets.

Location:

sage/schemes

Files:

• generic/scheme.py (modified) (1 diff)
• hyperelliptic_curves/jacobian_generic.py (modified) (1 diff)
• hyperelliptic_curves/jacobian_homset.py (modified) (3 diffs)
• hyperelliptic_curves/jacobian_morphism.py (modified) (4 diffs)

sage/schemes/generic/scheme.py

@@ -76 +76 @@
         """
         raise NotImplementedError
-
 
     def __call__(self, *args):

sage/schemes/hyperelliptic_curves/jacobian_generic.py

@@ -18 +18 @@
 
 class HyperellipticJacobian_generic(Jacobian_generic):
-    def __call__(self, *args, **kwds):
-        """
-        EXAMPLES:
-            sage: FF = FiniteField(2003)
-            sage: R.<x> = PolynomialRing(FF)
-            sage: f = x**5 + 1184*x**3 + 1846*x**2 + 956*x + 560
-            sage: C = HyperellipticCurve(f)
-            sage: J = Jacobian(C)
-            sage: a = x**2 + 376*x + 245; b = 1015*x + 1368
-            sage: X = J(FF)
-            sage: D = X([a,b])
-            sage: D
-            (x^2 + 376*x + 245, y + 988*x + 635)
-            sage: J(0)
-            (1)
-            sage: D == J([a,b])
-            True
-            sage: D == D + J(0) 
-            True
-        """     
-        if len(args) == 1: 
-            if is_Ring(args[0]):
-                return jacobian_homset.JacobianHomset_divisor_classes(self, args[0])
-            return jacobian_homset.JacobianHomset_divisor_classes(self, self.base_ring())(args[0])
-        raise TypeError, "Arguments must be a coefficient ring or Mumford divisor."
+    """
+    EXAMPLES:
+        sage: FF = FiniteField(2003)
+        sage: R.<x> = PolynomialRing(FF)
+        sage: f = x**5 + 1184*x**3 + 1846*x**2 + 956*x + 560
+        sage: C = HyperellipticCurve(f)
+        sage: J = C.jacobian()
+        sage: a = x**2 + 376*x + 245; b = 1015*x + 1368
+        sage: X = J(FF)
+        sage: D = X([a,b])
+        sage: D
+        (x^2 + 376*x + 245, y + 988*x + 635)
+        sage: J(0)
+        (1)
+        sage: D == J([a,b])
+        True
+        sage: D == D + J(0) 
+        True
+
+    An more extended example, demonstrating arithmetic in J(QQ) and J(K)
+    for a number field K/QQ.
+
+        sage: P.<x> = PolynomialRing(QQ)                     
+        sage: f = x^5 - x + 1; h = x                         
+        sage: C = HyperellipticCurve(f,h,'u,v')              
+        sage: C
+        Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
+        sage: PP = C.ambient_space()                         
+        sage: PP
+        Projective Space of dimension 2 over Rational Field
+        sage: C.defining_polynomial()                        
+        -x0^5 + x0*x1*x2^3 + x1^2*x2^3 + x0*x2^4 - x2^5
+        sage: C(QQ)
+        Set of Rational Points of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
+        sage: K.<t> = NumberField(x^2-2)                     
+        sage: C(K)    
+        Set of Rational Points over Number Field in t with defining polynomial x^2 - 2 of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
+        sage: P = C(QQ)(0,1,1); P                            
+        (0 : 1 : 1) 
+        sage: P == C(0,1,1)                                  
+        True
+        sage: C(0,1,1).parent()                              
+        Set of Rational Points of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
+        sage: P1 = C(K)(P)                                   
+        sage: P2 = C(K)([2,4*t-1,1])                         
+        sage: P3 = C(K)([-1/2,1/8*(7*t+2),1])                
+        sage: P1, P2, P3                                     
+        ((0 : 1 : 1), (2 : 4*t - 1 : 1), (-1/2 : 7/8*t + 1/4 : 1))
+        sage: J = C.jacobian()                               
+        sage: J       
+        Jacobian of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
+        sage: Q = J(QQ)(P); Q
+        (u, v + -1)
+        sage: for i in range(6): Q*i
+        (1)
+        (u, v + -1)
+        (u^2, v + u - 1)
+        (u^2, v + 1)
+        (u, v + 1)
+        (1)
+        sage: Q1 = J(K)(P1); print "%s -> %s"%( P1, Q1 )     
+        (0 : 1 : 1) -> (u, v + -1)
+        sage: Q2 = J(K)(P2); print "%s -> %s"%( P2, Q2 )
+        (2 : 4*t - 1 : 1) -> (u + -2, v + -4*t + 1)
+        sage: Q3 = J(K)(P3); print "%s -> %s"%( P3, Q3 )
+        (-1/2 : 7/8*t + 1/4 : 1) -> (u + 1/2, v + -7/8*t - 1/4)
+        sage: R.<x> = PolynomialRing(K)                      
+        sage: Q4 = J(K)([x^2-t,R(1)])                        
+        sage: for i in range(4): Q4*i
+        (1)
+        (u^2 + -t, v + -1)
+        (u^2 + (-3/4*t - 9/16)*u + 1/2*t + 1/4, v + (-1/32*t - 57/64)*u + 1/2*t + 9/16)
+        (u^2 + (1352416/247009*t - 1636930/247009)*u + -1156544/247009*t + 1900544/247009, v + (-2326345442/122763473*t + 3233153137/122763473)*u + 2439343104/122763473*t - 3350862929/122763473)
+        sage: R2 = Q2*5; R2             
+        (u^2 + (-3789465233/116983808)*u + -267915823/58491904, v + (-233827256513849/1789384327168*t + 1/2)*u + -15782925357447/894692163584*t)
+        sage: R3 = Q3*5; R3             
+        (u^2 + 5663300808399913890623/14426454798950909645952*u + -26531814176395676231273/28852909597901819291904, v + (253155440321645614070860868199103/2450498420175733688903836378159104*t + 1/2)*u + 2427708505064902611513563431764311/4900996840351467377807672756318208*t)
+        sage: R4 = Q4*5; R4             
+        (u^2 + (-3789465233/116983808)*u + -267915823/58491904, v + (233827256513849/1789384327168*t + 1/2)*u + 15782925357447/894692163584*t)
+        sage: # Thus we find the following identity:
+        sage: 5*Q2 + 5*Q4 
+        (1)
+        sage: # Moreover the following relation holds in the 5-torsion subgroup:
+        sage: Q2 + Q4 == 2*Q1                     
+        True
+    """     
     
     def dimension(self):
         return self.__curve.genus()
+
+    def point(self, mumford, check=True):
+        try: 
+            return jacobian_homset.JacobianHomset_divisor_classes(self, self.base_ring())(mumford)
+        except AttributeError:
+            raise ValueError, "Arguments must determine a valid Mumford divisor."
 
     def _homset_class(self, *args, **kwds):

sage/schemes/hyperelliptic_curves/jacobian_homset.py

@@ -27 +27 @@
 
 import sage.schemes.generic.spec as spec
-from sage.rings.all import is_Polynomial, PolynomialRing, ZZ
+from sage.rings.all import is_Polynomial, PolynomialRing, Integer, ZZ
 from sage.schemes.generic.homset import SchemeHomset_generic
 from sage.schemes.generic.morphism import is_SchemeMorphism
@@ -65 +65 @@
             sage: J = C.jacobian()
             sage: Q = J(QQ)(P)
-            sage: for i in range(6): Q*i
+            sage: for i in range(6): i*Q
             (1)
             (u, v + -1)
@@ -73 +73 @@
             (1)
         """
-        if P == 0:
+        if isinstance(P,(int,long,Integer)) and P == 0:
             R = PolynomialRing(self.value_ring(), 'x')
             return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
+        elif isinstance(P,(list,tuple)):
+            if len(P) == 1 and P[0] == 0:
+                R = PolynomialRing(self.value_ring(), 'x')
+                return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
+            elif len(P) == 2:
+                P1 = P[0]; P2 = P[1]
+                if is_Polynomial(P1) and is_Polynomial(P2):
+                    return JacobianMorphism_divisor_class_field(self, P)
+                if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
+                    return self(P1) - self(P2)
+            raise TypeError, "Argument P (= %s) must have length 2."%P
         elif isinstance(P,JacobianMorphism_divisor_class_field) and self == P.parent():
             return P
-        elif isinstance(P,(list,tuple)):
-            if len(P) != 2:
-                raise TypeError, "Argument P (= %s) must have length 2"%P
-            P1 = P[0]; P2 = P[1]
-            if is_Polynomial(P1) and is_Polynomial(P2):
-                return JacobianMorphism_divisor_class_field(self, P)
-            if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
-                return self(P1) - self(P2)
         elif is_SchemeMorphism(P):
             x0 = P[0]; y0 = P[1]

sage/schemes/hyperelliptic_curves/jacobian_morphism.py

@@ -125 +125 @@
         return "(%s, %s)"%(a(x), y - b(x))
 
+    def scheme(self):
+        return self.codomain()
+
     def list(self):
         return self.__polys
@@ -138 +141 @@
             D = cantor_composition(self.__polys,other.__polys,f,h,C.genus())
         return JacobianMorphism_divisor_class_field(X, D, reduce=False, check=False)
+
+    def __cmp__(self, other):
+        if not isinstance(other, JacobianMorphism_divisor_class_field):
+            try:
+                other = self.parent()(other)
+            except TypeError:
+                return -1
+        return cmp(self.__polys, other.__polys)
+
+    def __nonzero__(self):
+        return self.__polys[0] != 1
 
     def __sub__(self, other):
@@ -162 +176 @@
         X = self.parent()
         if n < 0:
-            return self * (-n)
+            return - self * (-n)
         elif n == 0:
             return self.parent()(0)
         elif n == 1:
             return self
-        D = self.__mul__(n//2)
+        elif n < 4:
+            D = self
+        else:       
+            D = self.__mul__(n//2)
         if n % 2 == 0:
             return D + D
@@ -173 +190 @@
             return D + D + self
 
-    def __nonzero__(self):
-        return self.__polys[0] != 1
+    def _rmul_(self, n):
+        return self.__mul__(n)
 
-    def scheme(self):
-        return self.codomain()
 
+