Implement computations in picard groups via global sections of line bundles #15113
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Attachment: picard_group.sage.gz first draft version |
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comment:1
An example script: this computes on an elliptic curve The We tell the system about a rational point. This allows the determination of unique representatives of divisor classes, at a small computational cost (which could be eliminated with more careful programming) We define a point. Note that computing in H0(O(6*Z)) (which is what we do) doesn't quite give enough room to compute in all of Pic (the strategies implemented have trouble with some degree 2 classes), but we can compute in Pic0: Check what the order of the point should be: And verify that our arithmetic agrees: We are about 100 times slower than normal elliptic curve arithmetic, which I think is quite a promising start. A genus 3 example, with 42-torsion The current script has a constructor for projective plane curves. Note that:
The timings above used #15104, which is quite necessary to get linear algebra to perform reasonably. Finally, the code does not require k to be a finite field, although there are probably some matrix features that would have to get implemented in some other matrix classes too, but those are very straightforward. It is rather important to do something about divisor class representation for non-finite fields (hence |
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comment:2
Todo:
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Changed keywords from none to sd86.5 |
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Commit: |
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comment:9
referenced in #34232. |
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comment:10
The commit is not by me. Trac seems to have been confused. Anyway, #34232 sort of supersedes the commit. |
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<!-- 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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<!-- 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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<!-- 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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<!-- 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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<!-- 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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In a bunch of papers, Kamal Khuri Makdisi outlines how standard techniques to compute with coherent sheafs via global sections of their twists can be used to obtain (at least asymptotically) efficient algorthims to compute in the Picard group of an algebraic curve (he outlines
Pic^0, but the ideas readily generalize to all of Pic).A little experimentation shows that these techniques can be fairly efficient in practice as well (and certainly usable!). We'll have to see if the method can be truly competitive with the Hess-type "function field as finite extension of k(x)" approach.
In any case, Kamal's approach is much easier to implement and, thanks to its uniformity, much easier to trust, so at least as a stepping stone, it's useful to have an implementation available.
CC: @pjbruin
Component: algebraic geometry
Keywords: sd86.5
Branch/Commit: u/roed/implement_computations_in_picard_groups_via_global_sections_of_line_bundles @
1ea07ddIssue created by migration from https://trac.sagemath.org/ticket/15113
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