In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Jacobian variety ''J''(''C'') of a non-singular
algebraic curve ''C'' of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''g'' is the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
of degree 0
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
s. It is the connected component of the identity in the
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of ''C'', hence an
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
.
Introduction
The Jacobian variety is named after
Carl Gustav Jacobi, who proved the complete version of the
Abel–Jacobi theorem, making the injectivity statement of
Niels Abel into an isomorphism. It is a principally polarized
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
, of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
''g'', and hence, over the complex numbers, it is a
complex torus. If ''p'' is a point of ''C'', then the curve ''C'' can be mapped to a
subvariety
A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms.
Plant taxonomy
Subvariety is ranked:
*below that of variety (''varietas'')
*above that of form (''forma'').
Subva ...
of ''J'' with the given point ''p'' mapping to the identity of ''J'', and ''C'' generates ''J'' as a
group.
Construction for complex curves
Over the complex numbers, the Jacobian variety can be realized as the
quotient space ''V''/''L'', where ''V'' is the dual of the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of all global holomorphic differentials on ''C'' and ''L'' is the
lattice of all elements of ''V'' of the form
:
where ''γ'' is a closed
path in ''C''. In other words,
:
with
embedded in
via the above map. This can be done explicitly with the use of
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
s.
The Jacobian of a curve over an arbitrary field was constructed by as part of his proof of the Riemann hypothesis for curves over a finite field.
The
Abel–Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be identified with its
Picard variety
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ...
of degree 0 divisors modulo linear equivalence.
Algebraic structure
As a group, the Jacobian variety of a curve is isomorphic to the quotient of the group of divisors of degree zero by the subgroup of principal divisors, i.e., divisors of rational functions. This holds for fields that are not algebraically closed, provided one considers divisors and functions defined over that field.
Further notions
Torelli's theorem states that a complex curve is determined by its Jacobian (with its polarization).
The
Schottky problem asks which principally polarized abelian varieties are the Jacobians of curves.
The
Picard variety
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ...
, the
Albanese variety,
generalized Jacobian, and
intermediate Jacobians are generalizations of the Jacobian for higher-dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the
Albanese variety, but in general this need not be isomorphic to the Picard variety.
See also
*
Period matrix In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
Ehresmann's theorem
Let be a holomorphic submersive morphism. For a point ''b'' of ''B'', we denote ...
– period matrices are a useful technique for computing the Jacobian of a curve
*
Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
– these are generalizations of Jacobians
*
Honda–Tate theorem – classifies abelian varieties over finite fields up to isogeny
*
Intermediate Jacobian
References
Computation techniques
Period Matrices of Hyperelliptic CurvesAbeliants and their application to an elementary construction of Jacobians– techniques for constructing Jacobians
Isogeny classes
*
Infinite families of pairs of curves over Q with isomorphic Jacobians
Abelian varieties isogenous to a JacobianAbelian varieties isogenous to no Jacobian
Cryptography
*
Curves, Jacobians, and Cryptography
General
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