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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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