theta divisors
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| theta divisors | |
|---|---|
| Name | Theta divisor |
| Type | Divisor on an abelian variety |
| Introduced | 19th century |
| Field | Algebraic geometry; complex analysis; number theory |
theta divisors
A theta divisor is a special effective divisor associated to a theta function or to a theta line bundle on an abelian variety, central to the study of Jacobians, principally polarized abelian varieties, and moduli problems. They connect classical work of Riemann and Jacobi with modern developments involving Torelli-type theorems, Schottky problems, and Hodge-theoretic approaches to singularities. Their geometry and singularities link to questions treated by users of tools such as moduli stacks, deformation theory, and arithmetic geometry.
Definition and basic properties
A theta divisor is defined on an abelian variety carrying a polarization given by an ample line bundle; classically it arises from the zero locus of a theta function introduced by Carl Gustav Jacob Jacobi, Bernhard Riemann, and later studied by Adolf Hurwitz and Ferdinand von Lindemann. For a principally polarized abelian variety linked to a curve via the Jacobian variety construction associated to a compact Riemann surface, the theta divisor encodes the Abel–Jacobi image of effective divisors of degree g−1, a viewpoint developed by Riemann, Appell, and refined by Andre Weil and Alexander Grothendieck. Basic properties include ampleness of the corresponding line bundle studied in the work of David Mumford and symmetry under the inversion map of the underlying abelian variety considered by Igor Dolgachev and Francesco Viviani.
Theta functions and theta line bundles
Theta functions originate in the analytic theory developed by Carl Gustav Jacob Jacobi and Bernhard Riemann and were systematized in the language of line bundles by Andre Weil and David Mumford. The theta line bundle on a complex torus corresponds to a Hermitian form determined by a Riemann form, a construction appearing in the work of Henri Poincaré and later refined by John Tate in arithmetic contexts. Transformations of theta functions under the action of the symplectic group such as Sp(2g,Z) and in relation to modular forms feature in research by Igor Shafarevich, Jean-Pierre Serre, and Pierre Deligne, while applications to automorphic forms and representation theory connect to Harish-Chandra and Robert Langlands.
Theta divisors on Jacobians and principally polarized abelian varieties
On a Jacobian of a curve studied by Bernhard Riemann and later by Enrico Arbarello and Philip Griffiths, the theta divisor is the image of the (g−1)-fold symmetric product of the curve under the Abel–Jacobi map investigated by Gunnar Magnus and formalized in the Torelli theorem of René Torelli and proofs by David Mumford and Igor Krichever. In the context of principally polarized abelian varieties arising in the theory of Shimura varieties and Siegel modular varieties, theta divisors play a role in the Schottky problem tackled by Friedrich Schottky, Igor Krichever, Edwin E. Freitag, and Christophe Birkenhake. The interplay with moduli spaces studied by Joe Harris, David Eisenbud, and Carel Faber relates theta divisors to questions about degenerations treated by Gerd Faltings and Christiane Birkenhake.
Singularities, multiplicities, and geometric criteria
The singularities of theta divisors were investigated by Bernhard Riemann and later by Andreotti and Mayer, whose criteria relate singular loci to the geometry of the underlying curve; the Andreotti–Mayer theorems were advanced by Carlo Ciliberto and Enrico Arbarello. Multiplicity constraints and log-canonical thresholds for theta divisors connect to vanishing theorems by Kunihiko Kodaira and extensions by Jean-Pierre Demailly and Claire Voisin. Geometric criteria detecting whether a principally polarized abelian variety is a Jacobian use singularity loci examined by Mumford, Griffiths, and Riccardo Salvati Manni, while degeneration techniques appear in work of Nicholas Shepherd-Barron and Carel Faber.
Applications in algebraic geometry and number theory
Theta divisors serve in proofs of the Torelli theorem attributed to René Torelli and later refinements by David Mumford and James Harris; they appear in descriptions of the moduli space of curves investigated by Joe Harris and Ian Morrison. In number theory, theta functions and their divisors underpin the construction of arithmetic theta lifts used by Stephen S. Kudla and John Millson, and they are relevant to the study of special values of L-functions considered by Don Zagier and Pierre Deligne. Connections to cryptography through Jacobians of curves used in protocols by Neal Koblitz and Victor Miller exploit explicit knowledge of theta divisors, while relationships with period mappings appear in the work of Phillip Griffiths and Wilfried Schmid.
Examples and explicit computations
Classical examples include theta divisors on elliptic curves linked to Karl Weierstrass ℘-functions and on Jacobians of hyperelliptic curves studied by Adolf Hurwitz and Francesco G. Morosov. Explicit theta constants and theta-null computations appear in the work of F. Klein and F. Fricke, and contemporary explicit formulae are used by David Mumford, Igor Krichever, and Giovanni Faltings in computational approaches to the Schottky problem. Applications to explicit projective embeddings use theta functions as in constructions by Igor Dolgachev and Christian Birkenhake, while algorithmic computations on Jacobians appear in implementations influenced by Andrew Wiles-era arithmetic techniques and computational algebra systems used by Richard Brent and John Cremona.