In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Jacobian variety ''J''(''C'') of a non-singular
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
''C'' of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
''g'' is the
moduli space of degree 0
line bundles. It is the
connected component of the identity in the
Picard group of ''C'', hence an
abelian variety.
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, 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 coo ...
''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 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 functions.
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 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, 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 – period matrices are a useful technique for computing the Jacobian of a curve
*
Hodge structure – these are generalizations of Jacobians
*
Honda–Tate theorem – classifies abelian varieties over finite fields up to isogeny
*
Intermediate Jacobian
References
Computation techniques
*
* – techniques for constructing Jacobians
Isogeny classes
*
*
Abelian varieties isogenous to no Jacobian
Cryptography
*
Curves, Jacobians, and Cryptography
General
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