Add hyperelliptic curves using the smooth model

src/doc/en/reference/arithmetic_curves/index.rst

@@ -128,4 +128,41 @@ Hyperelliptic curves
 
    sage/interfaces/genus2reduction
 
-.. include:: ../footer.txt
+Hyperelliptic curves (smooth model)
+===================================
+
+.. toctree::
+   :maxdepth: 1
+
+   sage/schemes/hyperelliptic_curves_smooth_model/hyperelliptic_constructor
+   sage/schemes/hyperelliptic_curves_smooth_model/hyperelliptic_generic
+   sage/schemes/hyperelliptic_curves_smooth_model/hyperelliptic_finite_field
+   sage/schemes/hyperelliptic_curves_smooth_model/hyperelliptic_padic_field
+   sage/schemes/hyperelliptic_curves_smooth_model/hyperelliptic_rational_field
+   sage/schemes/hyperelliptic_curves_smooth_model/hyperelliptic_g2
+
+   sage/schemes/hyperelliptic_curves_smooth_model/invariants
+   sage/schemes/hyperelliptic_curves_smooth_model/mestre
+   sage/schemes/hyperelliptic_curves_smooth_model/monsky_washnitzer
+
+Jacobians of hyperelliptic curves (smooth model)
+------------------------------------------------
+
+.. toctree::
+   :maxdepth: 1
+
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_generic
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_homset_generic
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_homset_ramified
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_homset_split
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_homset_inert
+
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_g2_generic
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_g2_homset_ramified
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_g2_homset_split
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_g2_homset_inert
+
+   sage/schemes/hyperelliptic_curves_smooth_model/jacobian_morphism
+
+
+.. include:: ../footer.txt

src/doc/en/reference/references/index.rst

@@ -2980,6 +2980,10 @@ REFERENCES:
 .. [GHJV1994] \E. Gamma, R. Helm, R. Johnson, J. Vlissides, *Design
               Patterns: Elements of Reusable Object-Oriented
               Software*. Addison-Wesley (1994). ISBN 0-201-63361-2.
+
+.. [GHM2008] \S. Galbraith, M. Harrison, D. Mireles Morales, 
+             *Efficient hyperelliptic arithmetic using balanced representation for divisors*,
+             Algorithmic Number Theory: 8th International Symposium, ANTS-VIII Banff, Canada, May 17-22, 2008 Proceedings 8.
 
 .. [Gil1959] Edgar Nelson Gilbert. *Random Graphs*, Annals of Mathematical
              Statistics. 30 (4): 1141-1144, 1959.

src/sage/schemes/all.py

@@ -42,6 +42,12 @@
 
 from sage.schemes.product_projective.all import *
 
+from sage.schemes.weighted_projective.all import *
+
 from sage.schemes.cyclic_covers.all import *
 
 from sage.schemes.berkovich.all import *
+
+from sage.schemes.hyperelliptic_curves_smooth_model.all import *
+
+from sage.schemes.weighted_projective.all import *

src/sage/schemes/curves/constructor.py

@@ -18,6 +18,12 @@
     Projective Plane Curve over Finite Field of size 5
      defined by -x^9 + y^2*z^7 - x*z^8
 
+Here, we construct a hyperelliptic curve manually::
+
+    sage: WP.<x,y,z> = WeightedProjectiveSpace([1, 3, 1], GF(103))
+    sage: Curve(y^2 - (x^5*z + 17*x^2*z^4 + 92*z^6), WP)
+    Weighted Projective Curve over Finite Field of size 103 defined by y^2 - x^5*z - 17*x^2*z^4 + 11*z^6
+
 AUTHORS:
 
 - William Stein (2005-11-13)
@@ -49,6 +55,7 @@
 from sage.schemes.generic.algebraic_scheme import AlgebraicScheme
 from sage.schemes.affine.affine_space import AffineSpace, AffineSpace_generic
 from sage.schemes.projective.projective_space import ProjectiveSpace, ProjectiveSpace_ring
+from sage.schemes.weighted_projective.weighted_projective_space import WeightedProjectiveSpace_ring
 from sage.schemes.plane_conics.constructor import Conic
 
 from .projective_curve import (ProjectiveCurve,
@@ -71,6 +78,8 @@
                            IntegralAffinePlaneCurve,
                            IntegralAffinePlaneCurve_finite_field)
 
+from .weighted_projective_curve import WeightedProjectiveCurve
+
 
 def _is_irreducible_and_reduced(F) -> bool:
     """
@@ -376,5 +385,12 @@ def Curve(F, A=None):
             return ProjectivePlaneCurve_field(A, F)
         return ProjectivePlaneCurve(A, F)
 
+    elif isinstance(A, WeightedProjectiveSpace_ring):
+        # currently, we only support curves in a weighted projective plane
+        if n != 2:
+            raise NotImplementedError("ambient space has to be a weighted projective plane")
+        # currently, we do not perform checks on weighted projective curves
+        return WeightedProjectiveCurve(A, F)
+
     else:
         raise TypeError('ambient space neither affine nor projective')

src/sage/schemes/curves/weighted_projective_curve.py

@@ -0,0 +1,91 @@
+# sage.doctest: needs sage.libs.singular
+r"""
+Weighted projective curves
+
+Weighted projective curves in Sage are curves in a weighted projective space or
+a weighted projective plane.
+
+EXAMPLES:
+
+For now, only curves in weighted projective plane is supported::
+
+    sage: WP.<x, y, z> = WeightedProjectiveSpace([1, 3, 1], QQ)
+    sage: C1 = WP.curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6); C1
+    Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+    sage: C2 = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C2
+    Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+    sage: C1 == C2
+    True
+"""
+
+from sage.schemes.curves.curve import Curve_generic
+from sage.schemes.weighted_projective.weighted_projective_space import WeightedProjectiveSpace_ring
+
+
+class WeightedProjectiveCurve(Curve_generic):
+    """
+    Curves in weighted projective spaces.
+
+    EXAMPLES:
+
+    We construct a hyperelliptic curve manually::
+
+        sage: WP.<x, y, z> = WeightedProjectiveSpace([1, 3, 1], QQ)
+        sage: C = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C
+        Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+    """
+    def __init__(self, A, X, *kwargs):
+        if not isinstance(A, WeightedProjectiveSpace_ring):
+            raise TypeError(f"A(={A}) is not a weighted projective space")
+        self._weights = A._weights
+        super().__init__(A, X, *kwargs)
+
+    def _repr_type(self):
+        r"""
+        Return a string representation of the type of this curve.
+
+        EXAMPLES::
+
+            sage: WP.<x,y,z> = WeightedProjectiveSpace([1, 3, 1], QQ)
+            sage: C = Curve(y^2 - x^5 * z - 3 * x^2 * z^4 - 2 * z^6, WP); C
+            Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 - 2*z^6
+            sage: C._repr_type()
+            'Weighted Projective'
+        """
+        return "Weighted Projective"
+
+    def projective_curve(self):
+        r"""
+        Return this weighted projective curve as a projective curve.
+
+        A weighted homogeneous polynomial `f(x_1, \ldots, x_n)`, where `x_i` has
+        weight `w_i`, can be viewed as an unweighted homogeneous polynomial
+        `f(y_1^{w_1}, \ldots, y_n^{w_n})`. This correspondence extends to
+        varieties.
+
+        .. TODO:
+
+            Implement homsets for weighted projective spaces and implement this
+            as a ``projective_embedding`` method instead.
+
+        EXAMPLES::
+
+            sage: WP = WeightedProjectiveSpace([1, 3, 1], QQ, "x, y, z")
+            sage: x, y, z = WP.gens()
+            sage: C = WP.curve(y^2 - (x^5*z + 3*x^2*z^4 - 2*x*z^5 + 4*z^6)); C
+            Weighted Projective Curve over Rational Field defined by y^2 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6
+            sage: C.projective_curve()
+            Projective Plane Curve over Rational Field defined by y^6 - x^5*z - 3*x^2*z^4 + 2*x*z^5 - 4*z^6
+        """
+        from sage.schemes.projective.projective_space import ProjectiveSpace
+
+        WP = self.ambient_space()
+        PP = ProjectiveSpace(WP.dimension_relative(), WP.base_ring(), WP.variable_names())
+        PP_ring = PP.coordinate_ring()
+        subs_dict = {name: var**weight for (name, var), weight in
+                     zip(WP.gens_dict().items(), WP.weights())}
+
+        wp_polys = self.defining_polynomials()
+        pp_polys = [PP_ring(poly.subs(**subs_dict)) for poly in wp_polys]
+
+        return PP.curve(pp_polys)

src/sage/schemes/hyperelliptic_curves_smooth_model/all.py

@@ -0,0 +1,3 @@
+from sage.schemes.hyperelliptic_curves_smooth_model.hyperelliptic_constructor import (
+    HyperellipticCurveSmoothModel,
+)