add a specialized parent object for elliptic-curve morphisms

src/sage/schemes/elliptic_curves/ell_field.py

@@ -1113,6 +1113,24 @@ def division_field(self, n, names='t', map=False, **kwds):
             L = L, F_to_K.post_compose(K_to_L)
         return L
 
+    def _Hom_(*args, **kwds):
+        r"""
+        Hook to make :class:`~sage.categories.homset.Hom`
+        set the correct parent
+        :class:`~sage.schemes.elliptic_curves.homset.EllipticCurveHomset`
+        for
+        :class:`~sage.schemes.elliptic_curves.hom.EllipticCurveHom`
+        objects.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve(GF(19), [1,0])
+            sage: type(E._Hom_(E))
+            <class 'sage.schemes.elliptic_curves.homset.EllipticCurveHomset_with_category'>
+        """
+        from . import homset
+        return homset.EllipticCurveHomset(*args, **kwds)
+
     def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True, algorithm=None):
         r"""
         Return an elliptic-curve isogeny from this elliptic curve.

src/sage/schemes/elliptic_curves/homset.py

@@ -0,0 +1,151 @@
+r"""
+Homsets and endomorphism rings of elliptic curves
+
+The set of homomorphisms between two elliptic curves (:class:`EllipticCurveHom`)
+forms an abelian group under addition. Moreover, if the two curves are the same,
+it even forms a (not always commutative) ring under composition.
+
+This module encapsulates the set of homomorphisms between two given elliptic
+curves as a Sage object.
+
+.. NOTE::
+
+    Currently only little nontrivial functionality is available, but this will
+    hopefully change in the future.
+
+EXAMPLES:
+
+The only useful thing this class does at the moment is coercing integers into
+the endomorphism ring as scalar multiplications::
+
+    sage: E = EllipticCurve([1,2,3,4,5])
+    sage: f = End(E)(7); f
+    Scalar-multiplication endomorphism [7] of Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
+    sage: f == E.scalar_multiplication(7)
+    True
+
+::
+
+    sage: E = EllipticCurve(GF(431^2), [0,1])
+    sage: E.automorphisms()[0] == 1
+    True
+    sage: E.automorphisms()[1] == -1
+    True
+    sage: omega = E.automorphisms()[2]
+    sage: omega == 1
+    False
+    sage: omega^3 == 1
+    True
+    sage: (1 + omega + omega^2) == 0
+    True
+    sage: (2*omega + 1)^2 == -3
+    True
+
+AUTHORS:
+
+- Lorenz Panny (2023)
+"""
+
+# ****************************************************************************
+#       Copyright (C) 2023 Lorenz Panny
+#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
+
+from sage.rings.integer_ring import ZZ
+from sage.categories.morphism import Morphism
+from sage.schemes.generic.homset import SchemeHomset_generic
+
+
+class EllipticCurveHomset(SchemeHomset_generic):
+    r"""
+    This class represents the set of all homomorphisms between two fixed
+    elliptic curves.
+
+    EXAMPLES::
+
+        sage: E = EllipticCurve(GF(419^2), [1,0])
+        sage: E.frobenius_isogeny() in End(E)
+        True
+        sage: phi = E.isogenies_prime_degree(7)[0]
+        sage: phi in End(E)
+        False
+        sage: phi in Hom(E, phi.codomain())
+        True
+
+    Note that domain and codomain are *not* taken up to isomorphism::
+
+        sage: iso = E.isomorphism_to(EllipticCurve(GF(419^2), [2,0]))
+        sage: iso in End(E)
+        False
+    """
+    def __init__(self, *args, **kwds):
+        r"""
+        Construct the homset for a given pair of curves.
+
+        TESTS::
+
+            sage: E = EllipticCurve('37a1')                                             # needs sage.schemes
+            sage: Hom(E, E).__class__                                                   # needs sage.schemes
+            <class 'sage.schemes.elliptic_curves.homset.EllipticCurveHomset_with_category'>
+
+        ::
+
+            sage: E1 = EllipticCurve(j=42)
+            sage: E2 = EllipticCurve(j=43)
+            sage: Hom(E1, E2)
+            Group of elliptic-curve morphisms
+              From: Elliptic Curve defined by y^2 = x^3 + 5901*x + 1105454 over Rational Field
+              To:   Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 1510*x - 140675 over Rational Field
+        """
+        super().__init__(*args, **kwds)
+
+        if self.domain() == self.codomain():
+
+            # set up automated coercion of integers to scalar multiplications
+            from sage.schemes.elliptic_curves.hom_scalar import EllipticCurveHom_scalar
+
+            class ScalarMultiplicationEmbedding(Morphism):
+
+                def __init__(self, End):
+                    assert End.domain() is End.codomain()
+                    super().__init__(ZZ, End)
+
+                def _call_(self, m):
+                    return EllipticCurveHom_scalar(self.codomain().domain(), m)
+
+            self.register_coercion(ScalarMultiplicationEmbedding(self))
+
+    def _repr_(self):
+        r"""
+        Output a description of this homset, with special formatting
+        for endomorphism rings.
+
+        EXAMPLES::
+
+            sage: E1 = EllipticCurve([1,1])
+            sage: E2 = EllipticCurve([2,2])
+            sage: End(E1)
+            Ring of elliptic-curve endomorphisms
+              From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
+              To:   Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
+            sage: Hom(E1, E1)
+            Ring of elliptic-curve endomorphisms
+              From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
+              To:   Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
+            sage: Hom(E1, E2)
+            Group of elliptic-curve morphisms
+              From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
+              To:   Elliptic Curve defined by y^2 = x^3 + 2*x + 2 over Rational Field
+        """
+        if self.domain() == self.codomain():
+            s = 'Ring of elliptic-curve endomorphisms'
+        else:
+            s = 'Group of elliptic-curve morphisms'
+        s += f'\n  From: {self.domain()}'
+        s += f'\n  To:   {self.codomain()}'
+        return s

src/sage/schemes/generic/scheme.py

@@ -645,10 +645,6 @@ def _Hom_(self, Y, category=None, check=True):
             sage: S._Hom_(P).__class__
             <class 'sage.schemes.generic.homset.SchemeHomset_generic_with_category'>
 
-            sage: E = EllipticCurve('37a1')                                             # needs sage.schemes
-            sage: Hom(E, E).__class__                                                   # needs sage.schemes
-            <class 'sage.schemes.projective.projective_homset.SchemeHomset_polynomial_projective_space_with_category'>
-
             sage: Hom(Spec(ZZ), Spec(ZZ)).__class__
             <class 'sage.schemes.generic.homset.SchemeHomset_generic_with_category_with_equality_by_id'>
         """