Jacobian variety

Jacobian variety
Name	Jacobian variety
Field	Algebraic geometry
Introduced by	Niels Abel; Carl Gustav Jacob Jacobi

Jacobian variety

The Jacobian variety is a principally polarized abelian variety attached to a compact Riemann surface or a smooth projective algebraic curve, playing a central role in the theories of Niels Abel, Carl Gustav Jacob Jacobi, André Weil, Alexander Grothendieck, and Bernhard Riemann. It connects the study of divisors, line bundles, and period integrals with moduli problems studied by David Mumford, Igor Shafarevich, Jean-Pierre Serre, and Gerd Faltings. The construction uses analytic tools from Riemann surface theory and algebraic tools from the theory of Abelian varietys and Picard schemes, and it is foundational for results like the Torelli theorem and the Abel–Jacobi theorem.

Definition and basic properties

The Jacobian variety of a genus g smooth projective curve C over a field k is an abelian variety J(C) of dimension g representing the degree zero component of the Picard scheme of C, closely tied to the divisor class group used by Évariste Galois-era class field techniques and later generalizations by Emil Artin. It carries a canonical principal polarization coming from the theta divisor studied in the work of Bernhard Riemann, Carl Ludwig Siegel, and André Weil, and it parametrizes equivalence classes of degree zero line bundles as in the formulations of Oscar Zariski and Alexander Grothendieck. For complex curves, the Jacobian is isomorphic to a complex torus C^g/Λ determined by period matrices computed via holomorphic differentials, an approach developed by Hermann Weyl and formalized by Henri Poincaré. Functoriality relates morphisms of curves studied by Mikhail Gromov and Joseph Oesterlé to homomorphisms of Jacobians, used in proofs by Gerd Faltings and in explorations by Jean-Pierre Serre.

Construction (Abel–Jacobi map and complex torus)

Analytically, choose a basis of holomorphic one-forms on a compact Riemann surface S (methods of Riemann and H. F. Baker) to compute periods with respect to a symplectic basis of homology cycles arising in the work of Henri Poincaré and L. E. J. Brouwer. The period lattice Λ ⊂ C^g yields the complex torus C^g/Λ, and the Abel–Jacobi map sending degree-zero divisors to integrals of differentials is the classical map of Niels Abel and Jacobi that embeds S into its Jacobian for non-hyperelliptic curves in the manner studied by Oscar Zariski and Andreotti. Algebraically, representability of the Picard functor by an abelian variety uses techniques from Alexander Grothendieck's theory of schemes and line bundles, as refined by David Mumford and Michael Artin; the Albanese variety construction provides a dual viewpoint used by Grothendieck and Igor Shafarevich.

Algebraic and arithmetic aspects

Over number fields and finite fields, Jacobians are central to questions treated by Gerd Faltings in the proof of the Mordell conjecture and by John Tate in the formulation of the Tate conjecture and Tate module theory. The study of rational points on curves reduces to understanding J(C)(K) via descent and Chabauty methods developed by Claude Chabauty, Manjul Bhargava, and Bjorn Poonen, and explicit computations use algorithms from Henri Cohen and J. H. Silverman. Over finite fields, the Weil conjectures proved by Pierre Deligne constrain the zeta function of J(C), and Honda–Tate theory relates isogeny classes of abelian varieties to eigenvalues of Frobenius considered by John Tate and Tate–Honda-style classifications. Modularity phenomena involving Jacobians appear in the work of Andrew Wiles, Richard Taylor, and Ken Ribet through modular curves and their Jacobians.

Polarization and principal polarization

A polarization of an abelian variety, formalized by André Weil and David Mumford, is an ample line bundle or an isogeny to its dual; for Jacobians the canonical theta divisor provides a principal polarization central to the Schottky problem studied by F. Schottky and later by Igusa and Tatsuya Shiota. Principal polarizations characterize Jacobians among principally polarized abelian varieties via the Torelli theorem and other criteria explored by Nicholas Shepherd-Barron and Samuel Grushevsky. The interaction between polarizations and endomorphism algebras of Jacobians has been examined by Albert, Shimura, and Goro Shimura in the context of complex multiplication and Shimura varietys.

Examples and special cases

For genus 1, the Jacobian of an elliptic curve is the curve itself, a fact central to the theories of Andrew Wiles and Gerhard Frey and used in the arithmetic of Elliptic curves. For hyperelliptic curves studied by Arthur Cayley and George Salmon, explicit models of Jacobians admit description via Mumford coordinates used in computational work by David Cantor and John Cremona. Jacobians of modular curves such as X_0(N) feature in results by Hecke, Atkin–Lehner, and Brian Conrey and decompose up to isogeny into abelian subvarieties associated to newforms analyzed by Goro Shimura and Jean-Pierre Serre. Prym varieties associated to coverings studied by Friedrich Prym provide related examples treated by David Mumford and Igor Dolgachev.

Moduli and Torelli theorem

The moduli space A_g of principally polarized abelian varieties, developed by Bernard Siegel, David Mumford, and Igusa, contains the Jacobian locus of curves and interacts with the Deligne–Mumford moduli space M_g studied by Pierre Deligne and David Mumford; the Torelli theorem of Riemann and rigorous proofs by Andreotti and C. P. Ramanujam assert that the period map from M_g to A_g is injective on isomorphism classes. Schottky problem solutions by Friedrich Schottky, Igusa, and more recently Grushevsky and Samuel Grushevsky characterize which points of A_g arise from Jacobians, with connections to theta functions explored by Bernard Riemann and F. Klein.

Applications in geometry and number theory

Jacobian varieties underpin the proof of the Abel–Jacobi theorem central to Niels Abel and the inversion problem tackled by Carl Gustav Jacob Jacobi, and they feature in modern proofs of the Mordell conjecture by Gerd Faltings. They serve as targets for cycle maps and regulators in the work of Spencer Bloch and Kazuya Kato on higher K-theory and motivic cohomology, and they appear in the study of rational points via Chabauty–Coleman methods linked to Robert Coleman and Claude Chabauty. In complex geometry, Jacobians provide examples of special Kähler manifolds related to Hodge theory developed by Wilhelm Hodge and Phillip Griffiths and to period domains studied by Pierre Deligne and Philip Griffiths. In cryptography, Jacobians of hyperelliptic curves are used in public-key systems studied by Neal Koblitz and Victor Miller.

Category:Abelian varieties