In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus 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 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 of all global holomorphic differentials on C and L is the lattice of all elements of V of the form
[ γ ] : ω ↦ ∫ γ ω {\displaystyle [\gamma ]:\ \omega \mapsto \int _{\gamma }\omega }
where γ is a closed path in C. In other words,
J ( C ) = H 0 ( Ω C 1 ) ∗ / H 1 ( C ) , {\displaystyle J(C)=H^{0}(\Omega _{C}^{1})^{*}/H_{1}(C),}
with H 1 ( C ) {\displaystyle H_{1}(C)} embedded in H 0 ( Ω C 1 ) ∗ {\displaystyle H^{0}(\Omega _{C}^{1})^{*}} 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 Weil (1948) 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 Schindler, Bernhard (1993). "Period Matrices of hyperelliptic curves". Manuscripta Mathematica. 78 (4): 369–380. doi:10.1007/BF02599319. S2CID 122944746. Anderson, Greg W. (2002). "Abeliants and their application to an elementary construction of Jacobians". Advances in Mathematics. 172 (2): 169–205. arXiv:math/0112321. doi:10.1016/S0001-8708(02)00024-5. S2CID 2458575. – techniques for constructing Jacobians
Isogeny classes Howe, Everett W. (2005). "Infinite Families of Pairs of Curves over Q with Isomorphic Jacobians". Journal of the London Mathematical Society. 72 (2): 327–350. arXiv:math/0304471. doi:10.1112/S0024610705006812. S2CID 5742703. Chai, Ching-Li; Oort, Frans (2012). "Abelian varieties isogenous to a Jacobian". Annals of Mathematics. 176: 589–635. doi:10.4007/annals.2012.176.1.11. S2CID 3153696. Abelian varieties isogenous to no Jacobian
Cryptography Curves, Jacobians, and Cryptography
General P. Griffiths; J. Harris (1994), Principles of Algebraic Geometry, Wiley Classics Library, Wiley Interscience, pp. 333–363, ISBN 0-471-05059-8 Jacobi, C.G.J. (1832). "Considerationes generales de transcendentibus Abelianis". Journal für die reine und angewandte Mathematik (Crelle's Journal). 1832 (9): 394–403. doi:10.1515/crll.1832.9.394. S2CID 120125760. Jacobi, C.G.J. (1835), "De functionibus duarum variabilium quadrupliciter periodicis, quibus theoria transcendentium abelianarum innititur", J. Reine Angew. Math., 13: 55–78 J.S. Milne (1986), "Jacobian Varieties", Arithmetic Geometry, New York: Springer-Verlag, pp. 167–212, ISBN 0-387-96311-1 Mumford, David (1975), Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., MR 0419430 Shokurov, V.V. (2001) [1994], "Jacobi variety", Encyclopedia of Mathematics, EMS Press Weil, André (1948), Variétés abéliennes et courbes algébriques, Paris: Hermann, MR 0029522, OCLC 826112 Hartshorne, Robin (19 December 1977), Algebraic Geometry, New York: Springer, ISBN 0-387-90244-9
