Jacobian variety

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.

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

:$$

[\gamma]:\ \omega \mapsto \int_\gamma \omega

where γ is a closed path in C. In other words,

:$$

J(C) = H^0(\Omega_C^1)^* / H_1(C),

with $H\_1(C)$ embedded in $H^0(\backslash Omega\_C^1)^*$ via the above map. This can be done explicitly with the use of theta functions.{{Cite book |last=Mumford |first=David |author-link=David Mumford |title=Tata lectures on Theta I |date=2007 |publisher=Birkhäuser |isbn=978-0-8176-4572-4}}

The Jacobian of a curve over an arbitrary field was constructed by {{harvtxt|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.

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.

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.

• 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

{{reflist}}

• {{cite journal |url=https://eudml.org/doc/155814 |title=Period Matrices of hyperelliptic curves |journal=Manuscripta Mathematica |year=1993 |volume=78 |issue=4 |pages=369–380 |last1=Schindler |first1=Bernhard |doi=10.1007/BF02599319 |s2cid=122944746}}
• {{cite journal |doi=10.1016/S0001-8708(02)00024-5 |doi-access=free |title=Abeliants and their application to an elementary construction of Jacobians |year=2002 |last1=Anderson |first1=Greg W. |journal=Advances in Mathematics |volume=172 |issue=2 |pages=169–205 |arxiv=math/0112321 |s2cid=2458575 }} – techniques for constructing Jacobians

• {{cite journal |doi=10.1112/S0024610705006812 |title=Infinite Families of Pairs of Curves over Q with Isomorphic Jacobians |year=2005 |last1=Howe |first1=Everett W. |journal=Journal of the London Mathematical Society |volume=72 |issue=2 |pages=327–350 |arxiv=math/0304471 |s2cid=5742703 }}
• {{cite journal |doi=10.4007/annals.2012.176.1.11 |title=Abelian varieties isogenous to a Jacobian |year=2012 |last1=Chai |first1=Ching-Li |last2=Oort |first2=Frans Oort |journal=Annals of Mathematics |volume=176 |pages=589–635 |s2cid=3153696 |doi-access=free }}
• [https://annals.math.princeton.edu/2020/191-2/p07 Abelian varieties isogenous to no Jacobian]

• Curves, Jacobians, and Cryptography

• {{Citation | author=P. Griffiths | author-link=Phillip Griffiths |author2=J. Harris |author-link2=Joe Harris (mathematician) | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | pages=333–363}}
• {{cite journal |first= C.G.J. |last=Jacobi|s2cid=120125760 |doi=10.1515/crll.1832.9.394 |title=Considerationes generales de transcendentibus Abelianis |journal=Journal für die reine und angewandte Mathematik (Crelle's Journal) |year=1832 |volume=1832 |issue=9 |pages=394–403 }}
• {{citation|first= C.G.J. |last=Jacobi|title=De functionibus duarum variabilium quadrupliciter periodicis, quibus theoria transcendentium abelianarum innititur|journal=J. Reine Angew. Math.|volume= 13 |year=1835|pages= 55–78|url=http://eudml.org/doc/146922 }}
• {{Citation | author=J.S. Milne | chapter=Jacobian Varieties | title=Arithmetic Geometry |publisher=Springer-Verlag|location=New York| year=1986 | pages=167–212|isbn=0-387-96311-1}}
• {{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Curves and their Jacobians | publisher=The University of Michigan Press, Ann Arbor, Mich. |mr=0419430 | year=1975}}
• {{eom|id=J/j054140|first=V.V. |last=Shokurov|title=Jacobi variety}}
• {{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Variétés abéliennes et courbes algébriques | publisher=Hermann | location=Paris | oclc=826112 |mr=0029522 | year=1948}}
• {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | date=19 December 1977 | publisher=Springer | location=New York | isbn=0-387-90244-9}}

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