Pic⁰ of a curve of genus one #
Let X be a proper integral curve over a field k whose codimension-one local rings are discrete
valuation rings, with k integrally closed in the function field k(X), and suppose that X has
genus one, dim_k H¹(X, 𝒪_X) = 1. Fix a codimension-one point x₀ of residue degree one, for
instance the image of a k-rational point. This file proves that the Abel–Jacobi map
x ↦ [𝒪_X(x - x₀)]
is a bijection from the codimension-one points of residue degree one onto the degree-zero part
Pic⁰ X of the Picard group, sending x₀ to zero. Under it the group law of Pic⁰ X is read
on points by linear equivalence: the images of x and y add up to the image of z exactly when
the divisors x + y and z + x₀ are linearly equivalent.
The two inputs are consequences of the Riemann–Roch theorem in genus one: a divisor of degree one
is linearly equivalent to exactly one point of residue degree one. They are transported from the
corresponding facts for elliptic function fields along the identification of codimension-one
points of X with the places of k(X) (SchemeWeilDivisor.equivFunctionFieldDivisor), which
preserves degrees and linear equivalence; the genus of X is the genus of k(X) by
SchemeWeilDivisor.genus_eq_genus_functionField.
Main declarations #
SchemeWeilDivisor.exists_linearlyEquivalent_ofPoint_of_genus_eq_oneandSchemeWeilDivisor.eq_of_linearlyEquivalent_ofPoint_of_genus_eq_one: in genus one a divisor of degree one is linearly equivalent to exactly one point of residue degree one;SchemeWeilDivisor.degreeOneEquivPicZero: the bijectionx ↦ [𝒪_X(x - x₀)]from the points of residue degree one ontoPic⁰ X, withSchemeWeilDivisor.coe_degreeOneEquivPicZero_apply;SchemeWeilDivisor.degreeOneEquivPicZero_baseandSchemeWeilDivisor.degreeOneEquivPicZero_add_eq_iff: the base point goes to zero, and the group law ofPic⁰ Xread on points.
References #
- J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., GTM 106, Springer, 2009, Proposition III.3.4.
- R. Hartshorne, Algebraic Geometry, Chapter IV, Section 4.
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Propositions 6.1.6 and 6.1.7.
Divisors of degree one in genus one #
A divisor of degree one on a curve of genus one is linearly equivalent to a point of residue
degree one. Here X is a proper integral curve over k whose codimension-one local rings are
discrete valuation rings, with k integrally closed in k(X).
Linearly equivalent points of residue degree one on a curve of genus one are equal. Here
X is a proper integral curve over k whose codimension-one local rings are discrete valuation
rings, with k integrally closed in k(X), and only the point x is required to have residue
degree one.
The points of residue degree one as Pic⁰ X #
The points of residue degree one of a curve of genus one form Pic⁰. On a proper integral
curve of genus one over k whose codimension-one local rings are discrete valuation rings, with
k integrally closed in k(X) and a base point x₀ of residue degree one, the Abel–Jacobi map
x ↦ [𝒪_X(x - x₀)] is a bijection from the codimension-one points of residue degree one onto
Pic⁰ X.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The class attached to a point x by SchemeWeilDivisor.degreeOneEquivPicZero is the class of
the line bundle 𝒪_X(x - x₀).
The base point goes to zero in Pic⁰ X.
The group law of Pic⁰ X on the points of residue degree one. The classes of x and y
add up to the class of z exactly when the divisors x + y and z + x₀ are linearly
equivalent.