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sagemathgh-35467: Attach Jacobians to function fields and curves
<!-- Please provide a concise, informative and self-explanatory title. --> <!-- Don't put issue numbers in the title. Put it in the Description below. --> <!-- For example, instead of "Fixes sagemath#12345", use "Add a new method to multiply two integers" --> ### 📚 Description We attach Jacobians to function fields and curves, enabling arithmetic with the points of the Jacobian. Fixes sagemath#34232. A point of Jacobian is represented by an effective divisor `D` such that the point is the divisor class of `D - B` (of degree 0) with a fixed base divisor `B`. There are two models for Jacobian arithmetic: - Hess model: `D` is internally represented by a pair of certain ideals and arithmetic relies on divisor reduction using Riemann-Roch space computation by Hess' algorithm. - Khuri-Makdisi model: `D` is internally represented by a linear subspace `W_D` of a linear space `V` and arithmetic uses Khuri-Makdisi's linear algebra algorithms. For implementation, sagemath#15113 was referenced. An example with non-hyperelliptic genus 3 curve: ```sage sage: A2.<x,y> = AffineSpace(QQ, 2) sage: f = y^3 + x^4 - 5*x^2*y + 2*x*y - x^2 - 5*y - 4*x + 1 sage: C = Curve(f, A2) sage: X = C.projective_closure() sage: X.genus() 3 sage: X.rational_points(bound=5) [(0 : 0 : 1), (1/3 : 1/3 : 1)] sage: Q = X(0,0,1).place() sage: P = X(1,1,3).place() sage: D = P - Q sage: D.degree() 0 sage: J = X.jacobian(model='hess', base_div=3*Q) sage: G = J.group() sage: p = G.point(D) sage: 2*p + 3*p == 5*p True ``` An example with elliptic curve: ```sage sage: k.<a> = GF((5,2)) sage: E = EllipticCurve(k,[1,0]); E Elliptic Curve defined by y^2 = x^3 + x over Finite Field in a of size 5^2 sage: E.order() 32 sage: P = E([a, 2*a + 4]) sage: P (a : 2*a + 4 : 1) sage: P.order() 8 sage: p = P.point_of_jacobian_of_curve() sage: p [Place (x + 4*a, y + 3*a + 1)] sage: p.order() 8 sage: Q = 3*P sage: q = Q.point_of_jacobian_of_curve() sage: q == 3*p True sage: G = p.parent() sage: G.order() 32 sage: G Group of rational points of Jacobian over Finite Field in a of size 5^2 (Hess model) sage: J = G.parent(); J Jacobian of Projective Plane Curve over Finite Field in a of size 5^2 defined by x^2*y + y^3 - x*z^2 (Hess model) sage: J.curve() == E.affine_patch(2).projective_closure() True ``` An example with hyperelliptic curve: ```sage sage: R.<x> = PolynomialRing(GF(11)) sage: f = x^6 + x + 1 sage: H = HyperellipticCurve(f) sage: J = H.jacobian() sage: D = J(H.lift_x(1)) sage: D # divisor in Mumford representation (x + 10, y + 6) sage: jacobian_order = sum(H.frobenius_polynomial()) sage: jacobian_order 234 sage: p = D.point_of_jacobian_of_curve(); p sage: p # Jacobian point represented by an effective divisor [Place (1/x0, 1/x0^3*x1 + 1) + Place (x0 + 10, x1 + 6)] sage: p.order() 39 sage: 234*p == 0 True sage: G = p.parent() sage: G Group of rational points of Jacobian over Finite Field of size 11 (Hess model) sage: J = G.parent() sage: J Jacobian of Projective Plane Curve over Finite Field of size 11 defined by x0^6 + x0^5*x1 + x1^6 - x0^4*x2^2 (Hess model) sage: C = J.curve() sage: C Projective Plane Curve over Finite Field of size 11 defined by x0^6 + x0^5*x1 + x1^6 - x0^4*x2^2 sage: C.affine_patch(0) == H.affine_patch(2) True ``` [](https://mybinder.org/v2 /gh/kwankyu/sage/p/35467/add-jacobian-groups-notebook-binder) prepared with sagemath#36245 <!-- Describe your changes here in detail. --> <!-- Why is this change required? What problem does it solve? --> <!-- If this PR resolves an open issue, please link to it here. For example "Fixes sagemath#12345". --> <!-- If your change requires a documentation PR, please link it appropriately. --> ### 📝 Checklist <!-- Put an `x` in all the boxes that apply. It should be `[x]` not `[x ]`. --> - [x] The title is concise, informative, and self-explanatory. - [x] The description explains in detail what this PR is about. - [x] I have linked a relevant issue or discussion. - [x] I have created tests covering the changes. - [x] I have updated the documentation accordingly. ### ⌛ Dependencies <!-- List all open PRs that this PR logically depends on - sagemath#12345: short description why this is a dependency - sagemath#34567: ... --> <!-- If you're unsure about any of these, don't hesitate to ask. We're here to help! --> URL: sagemath#35467 Reported by: Kwankyu Lee Reviewer(s): Kwankyu Lee, Matthias Köppe
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,14 +1,18 @@ | ||
| r""" | ||
| Schemes | ||
| """ | ||
|
|
||
| # **************************************************************************** | ||
| # Copyright (C) 2005 David Kohel <[email protected]> | ||
| # Copyright (C) 2005 David Kohel <[email protected]> | ||
| # William Stein <[email protected]> | ||
| # 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 | ||
|
|
@@ -18,6 +22,7 @@ | |
| 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 | ||
|
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||
|
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| class Schemes(Category): | ||
|
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@@ -55,7 +60,6 @@ class Schemes(Category): | |
| sage: Schemes().Homsets().super_categories() | ||
| [Category of homsets] | ||
| """ | ||
|
|
||
| @staticmethod | ||
| def __classcall_private__(cls, X=None): | ||
| """ | ||
|
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@@ -73,8 +77,7 @@ 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) | ||
|
|
@@ -155,9 +158,6 @@ def _call_(self, x): | |
| raise TypeError("No way to create an object or morphism in %s from %s" % (self, x)) | ||
|
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||
|
|
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| ############################################################# | ||
| # Schemes over a given base scheme. | ||
| ############################################################# | ||
| class Schemes_over_base(Category_over_base): | ||
| """ | ||
| The category of schemes over a given base scheme. | ||
|
|
@@ -172,7 +172,6 @@ class Schemes_over_base(Category_over_base): | |
| sage: C = Schemes(ZZ) | ||
| sage: TestSuite(C).run() | ||
| """ | ||
|
|
||
| def base_scheme(self): | ||
| """ | ||
| EXAMPLES:: | ||
|
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@@ -198,12 +197,12 @@ 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}" | ||
|
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| class AbelianVarieties(Schemes_over_base): | ||
| r""" | ||
|
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@@ -236,6 +235,18 @@ 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:: | ||
|
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@@ -253,7 +264,7 @@ 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() | ||
|
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||
| class Homsets(HomsetsCategory): | ||
| r""" | ||
|
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@@ -294,3 +305,87 @@ def extra_super_categories(self): | |
| True | ||
| """ | ||
| return [Rings()] | ||
|
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| 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())] | ||
|
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||
| 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() | ||
|
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||
| 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 | ||
| """ | ||
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