scalar-multiplication endomorphisms

sagemath · Nov 5, 2021 · 64865d2 · 64865d2
1 parent 72ab0b8
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diff --git a/src/doc/en/reference/arithmetic_curves/index.rst b/src/doc/en/reference/arithmetic_curves/index.rst
@@ -22,6 +22,7 @@ Maps between them
    sage/schemes/elliptic_curves/ell_curve_isogeny
    sage/schemes/elliptic_curves/isogeny_small_degree
    sage/schemes/elliptic_curves/hom_composite
+   sage/schemes/elliptic_curves/hom_scalar
 
 
 Elliptic curves over number fields

diff --git a/src/sage/schemes/elliptic_curves/ell_generic.py b/src/sage/schemes/elliptic_curves/ell_generic.py
@@ -2215,6 +2215,35 @@ def multiplication_by_m_isogeny(self, m):
 
         assert False, 'bug in multiplication_by_m_isogeny()'
 
+    def scalar_multiplication(self, m):
+        r"""
+        Return the scalar-multiplication map `[m]` on this elliptic
+        curve as a
+        :class:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar`
+        object.
+
+        .. WARNING::
+
+            This method is currently experimental. It is intended to
+            eventually supersede :meth:`multiplication_by_m_isogeny`.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve('77a1')
+            sage: m = E.scalar_multiplication(-7); m
+            doctest:warning ...
+            Scalar-multiplication endomorphism [-7] of Elliptic Curve defined by y^2 + y = x^3 + 2*x over Rational Field
+            sage: m.degree()
+            49
+            sage: P = E(2,3)
+            sage: m(P)
+            (-26/225 : -2132/3375 : 1)
+            sage: m.rational_maps() == E.multiplication_by_m(-7)
+            True
+        """
+        from sage.schemes.elliptic_curves.hom_scalar import EllipticCurveHom_scalar
+        return EllipticCurveHom_scalar(self, m)
+
     def isomorphism_to(self, other):
         """
         Given another weierstrass model ``other`` of self, return an

diff --git a/src/sage/schemes/elliptic_curves/hom.py b/src/sage/schemes/elliptic_curves/hom.py
@@ -169,6 +169,7 @@ def degree(self):
         - :meth:`EllipticCurveIsogeny.degree`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.degree`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.degree`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.degree`
 
         TESTS::
 
@@ -189,6 +190,7 @@ def kernel_polynomial(self):
         - :meth:`EllipticCurveIsogeny.kernel_polynomial`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.kernel_polynomial`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.kernel_polynomial`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.kernel_polynomial`
 
         TESTS::
 
@@ -209,6 +211,7 @@ def dual(self):
         - :meth:`EllipticCurveIsogeny.dual`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.dual`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.dual`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.dual`
 
         TESTS::
 
@@ -231,6 +234,7 @@ def rational_maps(self):
         - :meth:`EllipticCurveIsogeny.rational_maps`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.rational_maps`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.rational_maps`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.rational_maps`
 
         TESTS::
 
@@ -252,6 +256,7 @@ def x_rational_map(self):
         - :meth:`EllipticCurveIsogeny.x_rational_map`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.x_rational_map`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.x_rational_map`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.x_rational_map`
 
         TESTS::
 
@@ -529,8 +534,7 @@ def __neg__(self):
             sage: psi.rational_maps() == (f, -g)
             True
         """
-        a1,_,a3,_,_ = self.codomain().a_invariants()
-        return wm.WeierstrassIsomorphism(self.codomain(), (-1,0,-a1,-a3)) * self
+        return wm.negation_morphism(self.codomain()) * self
 
     @cached_method
     def __hash__(self):