Jacobians of curves

Jacobians of curves
Name	Jacobian
Type	Abelian variety
Dimension	genus of curve

Jacobians of curves

The Jacobian of a smooth projective algebraic curve is a principally polarized Abelian variety that encodes divisor-class and linear-equivalence information of the curve. Introduced in the work of Jacobi and systematized by Riemann and Weil, Jacobians provide a bridge among the theories of Abel, Picard, Albanese, Poincaré, and modern Mordell–Weil arithmetic. They play central roles in the studies initiated by Moduli theory, the Torelli theorem, and the arithmetic of Fermat-type and Diophantine equation problems.

Definition and basic properties

For a smooth projective curve C of genus g over a field k, the Jacobian J(C) is the g-dimensional principally polarized Abelian variety whose k-points parametrize degree-zero line bundles or divisor classes on C. Historically motivated by the inversion problem studied by Abel and Jacobi, the construction and properties were clarified by Riemann, who introduced theta functions, and by Weil, who axiomatized Abelian varieties. Key structural properties connect J(C) to the Picard and Albanese functors; endomorphisms of J(C) relate to the theory developed by Mumford and Tate, and polarizations link to the Poincaré and Rosati. Over C, the analytic description uses the period lattice via integrals of holomorphic differentials, a perspective with roots in the work of Riemann–Roch and Abel–Jacobi.

Construction (Abelian varieties, Picard and Albanese interpretations)

There are multiple equivalent constructions: as a principally polarized Abelian variety obtained by quotienting the g-dimensional complex vector space of holomorphic differentials by the period lattice (analytic construction traceable to Riemann and Jacobi); as the connected component of the identity of the Picard scheme Pic^0_C (algebraic construction advanced by Grothendieck and Mumford); and as the universal target of maps from C to Abelian varieties, the Albanese variety viewpoint used by Serre and Lang. The Poincaré line bundle on C × J(C) realizes a universal family of degree-zero line bundles; this construction was formalized in the work of Poincaré and later in the representability results due to Grothendieck and Raynaud. The polarization is given by the theta divisor, whose theory was developed by Riemann, Igusa, and later refined by Mumford and Andreotti.

Examples and explicit descriptions (elliptic curves, hyperelliptic curves, genus 2)

For genus g = 1, J(C) is isomorphic to C itself, giving the classical elliptic curve theory studied by Weierstrass, Tate, and Silverman. For hyperelliptic curves, explicit models and Cantor’s algorithm for the group law were developed in the computational tradition of Cantor and applied in works of Hess and Masser. Genus 2 Jacobians admit explicit theta-function descriptions and explicit arithmetic via models studied by Igusa, Faltings (in arithmetic contexts), and Bolza; such Jacobians often decompose or admit explicit endomorphisms connected to complex multiplication studied by Shimura and Taniyama. Examples central to classical algebraic geometry include Jacobians of plane quartics analyzed by Clebsch and Gordan, while modern explicit approaches cite computations by Brill-Noether practitioners and algorithmic treatments in the style of Lange and Birch.

Torelli theorem and moduli interpretations

The Torelli theorem, proved classically by Torelli and given modern proofs by Mumford and Deligne, states that a smooth projective curve is determined up to isomorphism by its principally polarized Jacobian. This result underpins the embedding of the moduli space of curves M_g into the moduli space A_g of principally polarized Abelian varieties, a perspective central to the work of Igusa, Deligne-Mumford, Faltings, and Harris–Morrison. The interplay with compactifications involves contributions from Satake, Mumford et al., and the study of degenerations by Namiki and Deligne. Torelli-type questions have prompted generalizations by Narasimhan and Seshadri in vector-bundle settings and inspired investigations into Prym varieties initiated by Recillas and Beauville.

Arithmetic of Jacobians (rational points, torsion, L-functions)

Arithmetic questions about J(C) link to the Mordell–Faltings theorem on rational points, the BSD conjecture for Abelian varieties, and the study of Galois representations developed by Serre, Deligne, and Fontaine. The structure of rational points J(C)(k) is governed by the Mordell–Weil theorem proved by Mordell and expanded by Weil; torsion subgroups are constrained by results like the Mazur for elliptic curves and generalizations by Merel. L-functions of Jacobians, their analytic continuation, and functional equations are central in the Langlands program as developed by Langlands, Shimura, and Faltings; special-value conjectures extend BSD-style statements to higher dimensions and have been examined by Gross–Zagier and Kolyvagin in related contexts.

Applications in algebraic geometry and number theory (integrable systems, cryptography)

Jacobians feature in the theory of integrable systems where finite-gap solutions of nonlinear equations were connected to algebraic-geometric data by Krichever, Novikov, and Dubrovin using Baker–Akhiezer functions and theta-function techniques originating with Riemann. In arithmetic geometry, Jacobians are used in descent methods and Chabauty–Coleman approaches following Chabauty, Coleman, and McCallum to bound rational points; they also appear in visibility techniques of Mazur and in isogeny-based constructions related to the Tate–Shafarevich group studied by Cassels. In cryptography, Jacobians of hyperelliptic curves provide group law settings for discrete-logarithm based protocols developed by Koblitz and Menezes, while isogeny problems on Abelian varieties influenced post-quantum proposals investigated by Charles, Goren, and Lauter.

Category:Abelian varieties