Attach Jacobians to function fields and curves

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

@@ -1,5 +1,11 @@
+Plane and Space Curves
+======================
+
+Sage enables computations with curves in affine and projective ambient spaces,
+curves over `\CC` as Riemann surfaces, and Jacobians of projective curves.
+
 Curves
-======
+------
 
 .. toctree::
    :maxdepth: 1
@@ -12,10 +18,8 @@ Curves
    sage/schemes/curves/closed_point
    sage/schemes/curves/zariski_vankampen
 
-   sage/schemes/jacobians/abstract_jacobian
-
 Plane conics
-============
+------------
 
 .. toctree::
    :maxdepth: 1
@@ -28,7 +32,7 @@ Plane conics
    sage/schemes/plane_conics/con_rational_function_field
 
 Plane quartics
-=========================
+--------------
 
 .. toctree::
    :maxdepth: 1
@@ -37,11 +41,19 @@ Plane quartics
    sage/schemes/plane_quartics/quartic_generic
 
 Riemann surfaces
-================
+----------------
 
 .. toctree::
    :maxdepth: 1
 
    sage/schemes/riemann_surfaces/riemann_surface
 
+Jacobians
+---------
+
+.. toctree::
+   :maxdepth: 1
+
+   sage/schemes/jacobians/abstract_jacobian
+
 .. include:: ../footer.txt

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

@@ -38,6 +38,19 @@ algebraic closure of `\QQ`.
 
 A basic reference for the theory of algebraic function fields is [Stich2009]_.
 
+Jacobians of function fields
+----------------------------
+
+Arithmetic in Jacobians of function fields are available in two flavors.
+
+.. toctree::
+   :maxdepth: 1
+
+   sage/rings/function_field/jacobian_base
+   sage/rings/function_field/jacobian_hess
+   sage/rings/function_field/jacobian_khuri_makdisi
+   sage/rings/function_field/khuri_makdisi
+
 A Support Module
 ----------------
 

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

@@ -3181,6 +3181,9 @@ REFERENCES:
             Cryptanalysis* ; 2002' available at
             http://www.engr.mun.ca/~howard/PAPERS/ldc_tutorial.pdf
 
+.. [Hes2004] Florian Hess, "Computing relations in divisor class groups of
+              algebraic curves over finite fields," Preprint, 2004.
+
 .. [Hes2002] Florian Hess, "Computing Riemann-Roch spaces in algebraic
              function fields and related topics," J. Symbolic
              Comput. 33 (2002), no. 4, 425--445.
@@ -3776,6 +3779,9 @@ REFERENCES:
             block cipher*, Lightweight Cryptography Workshop, 2016.
             https://www.nist.gov/sites/default/files/documents/2016/10/18/karpman-paper-lwc2016.pdf
 
+.. [Khu2004] \K. Khuri-Makdisi. *Linear algebra algorithms for divisors on an algebraic curve*,
+             Mathematics of Computation 73, no. 245 (2004) pp. 333-357.
+
 .. [Kin1992] Nancy G. Kinnersley, *The vertex separation number of a graph
              equals its path-width*, Information Processing Letters
              42(6):345-350, 1992. :doi:`10.1016/0020-0190(92)90234-M`.

src/sage/categories/all.py

@@ -105,7 +105,7 @@
 
 # schemes and varieties
 from sage.categories.modular_abelian_varieties import ModularAbelianVarieties
-from sage.categories.schemes import Schemes, AbelianVarieties
+from sage.categories.schemes import Schemes, AbelianVarieties, Jacobians
 
 # * with basis
 from sage.categories.modules_with_basis import ModulesWithBasis

src/sage/categories/category.py

@@ -2598,6 +2598,7 @@ def category_sample():
          Category of Hecke modules over Rational Field,
          Category of Hopf algebras over Rational Field,
          Category of Hopf algebras with basis over Rational Field,
+         Category of Jacobians over Rational Field,
          Category of Lie algebras over Rational Field,
          Category of Weyl groups,
          Category of abelian varieties over Rational Field,

src/sage/categories/schemes.py

@@ -2,14 +2,18 @@
 r"""
 Schemes
 """
+
 # ****************************************************************************
-#  Copyright (C) 2005      David Kohel <kohel@maths.usyd.edu>
+#       Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu>
 #                          William Stein <wstein@math.ucsd.edu>
-#                2008-2012 Nicolas M. Thiery <nthiery at users.sf.net>
+#                2008-2012 Nicolas M. Thiery <nthiery@users.sf.net>
 #
-#  Distributed under the terms of the GNU General Public License (GPL)
+# 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.categories.category import Category
 from sage.categories.category_types import Category_over_base
@@ -19,6 +23,7 @@
 from sage.categories.rings import Rings
 from sage.categories.fields import Fields
 from sage.categories.homsets import HomsetsCategory
+from sage.misc.abstract_method import abstract_method
 
 
 class Schemes(Category):
@@ -56,7 +61,6 @@ class Schemes(Category):
         sage: Schemes().Homsets().super_categories()
         [Category of homsets]
     """
-
     @staticmethod
     def __classcall_private__(cls, X=None):
         """
@@ -74,8 +78,7 @@ def __classcall_private__(cls, X=None):
             Category of schemes over Integer Ring
         """
         if X is not None:
-            from sage.schemes.generic.scheme import is_Scheme
-            if not is_Scheme(X):
+            if X not in Schemes():
                 X = Schemes()(X)
             return Schemes_over_base(X)
         return super().__classcall__(cls)
@@ -156,9 +159,6 @@ def _call_(self, x):
             raise TypeError("No way to create an object or morphism in %s from %s" % (self, x))
 
 
-#############################################################
-# Schemes over a given base scheme.
-#############################################################
 class Schemes_over_base(Category_over_base):
     """
     The category of schemes over a given base scheme.
@@ -173,7 +173,6 @@ class Schemes_over_base(Category_over_base):
         sage: C = Schemes(ZZ)
         sage: TestSuite(C).run()
     """
-
     def base_scheme(self):
         """
         EXAMPLES::
@@ -199,12 +198,12 @@ def _repr_object_names(self):
             sage: Schemes(Spec(ZZ)) # indirect doctest
             Category of schemes over Integer Ring
         """
-        # To work around the name of the class (schemes_over_base)
         from sage.schemes.generic.scheme import is_AffineScheme
-        if is_AffineScheme(self.base_scheme()):
-            return "schemes over %s" % self.base_scheme().coordinate_ring()
-        else:
-            return "schemes over %s" % self.base_scheme()
+        base = self.base()
+        if is_AffineScheme(base):
+            base = base.coordinate_ring()
+        return f"schemes over {base}"
+
 
 class AbelianVarieties(Schemes_over_base):
     r"""
@@ -237,6 +236,18 @@ def __init__(self, base):
             raise ValueError('category of abelian varieties is only defined over fields')
         super().__init__(base)
 
+    def base_scheme(self):
+        """
+        EXAMPLES::
+
+            sage: Schemes(Spec(ZZ)).base_scheme()
+            Spectrum of Integer Ring
+        """
+        base = self.base()
+        if base not in Schemes():
+            base = Schemes()(base)
+        return base
+
     def super_categories(self):
         """
         EXAMPLES::
@@ -254,7 +265,7 @@ def _repr_object_names(self):
             sage: AbelianVarieties(Spec(QQ))  # indirect doctest
             Category of abelian varieties over Rational Field
         """
-        return "abelian varieties over %s" % self.base_scheme()
+        return "abelian varieties over %s" % self.base()
 
     class Homsets(HomsetsCategory):
         r"""
@@ -295,3 +306,87 @@ def extra_super_categories(self):
                     True
                 """
                 return [Rings()]
+
+
+class Jacobians(Schemes_over_base):
+    """
+    The category of Jacobians attached to curves or function fields.
+
+    EXAMPLES::
+
+        sage: Jacobians(QQ)
+        Category of Jacobians over Rational Field
+
+    TESTS::
+
+        sage: TestSuite(Jacobians(QQ)).run()
+    """
+    def __init__(self, base):
+        r"""
+        Constructor of this category.
+
+        EXAMPLES::
+
+            sage: Jacobians(QQ)
+            Category of Jacobians over Rational Field
+            sage: Jacobians(Spec(QQ))
+            Category of Jacobians over Rational Field
+        """
+        from sage.schemes.generic.scheme import is_AffineScheme
+        if is_AffineScheme(base):
+            base = base.coordinate_ring()
+        if base not in Fields():
+            raise ValueError('category of Jacobians is only defined over fields')
+        super().__init__(base)
+
+    def base_scheme(self):
+        """
+        Return the base scheme of this Jacobians category.
+
+        EXAMPLES::
+
+            sage: Jacobians(QQ).base_scheme()
+            Spectrum of Rational Field
+        """
+        base = self.base()
+        if base not in Schemes():
+            base = Schemes()(base)
+        return base
+
+    def super_categories(self):
+        """
+        Return the super categories of this Jacobians category.
+
+        EXAMPLES::
+
+            sage: Jacobians(QQ).super_categories()
+            [Category of abelian varieties over Rational Field]
+        """
+        return [AbelianVarieties(self.base_scheme())]
+
+    def _repr_object_names(self):
+        """
+        Return the string representation of this category.
+
+        EXAMPLES::
+
+            sage: Jacobians(Spec(QQ))  # indirect doctest
+            Category of Jacobians over Rational Field
+        """
+        return "Jacobians over %s" % self.base()
+
+    class ParentMethods:
+
+        @abstract_method
+        def base_curve(self):
+            """
+            Return the curve to which this Jacobian is attached.
+
+            EXAMPLES::
+
+                sage: # needs sage.rings.function_field
+                sage: K.<x> = FunctionField(GF(2))
+                sage: J = K.jacobian()
+                sage: J.base_curve()
+                Rational function field in x over Finite Field of size 2
+            """

src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py

@@ -2066,7 +2066,7 @@ def scalar_multiplication(self, x):
             sage: phi = DrinfeldModule(A, [z, 0, 1, z])
             sage: phi
             Drinfeld module defined by T |--> z*t^3 + t^2 + z
-            sage: phi.hom(T)
+            sage: phi.hom(T)  # indirect doctest
             Endomorphism of Drinfeld module defined by T |--> z*t^3 + t^2 + z
               Defn: z*t^3 + t^2 + z