Abelian integrals

Abelian integrals
Name	Abelian integrals
Discipline	Mathematics
Subdiscipline	Algebraic geometry, Complex analysis
Introduced	19th century
Notable people	Niels Henrik Abel, Carl Gustav Jacob Jacobi, Bernhard Riemann

Abelian integrals Abelian integrals are complex integrals of algebraic differential forms on algebraic curves studied in the 19th century and central to modern Algebraic Geometry and Complex Analysis. They generalize elliptic integrals encountered by Niels Henrik Abel, Carl Gustav Jacob Jacobi, and Ferdinand Georg Frobenius, and they underpin the theory developed by Bernhard Riemann, Hermann Amandus Schwarz, and Karl Weierstrass. Their study connects with the work of Henri Poincaré, David Hilbert, and A. Grothendieck in topology, singularity theory, and scheme theory.

Definition and basic examples

An Abelian integral is obtained by integrating an algebraic differential on a compact algebraic curve such as a projective curve considered by Bernhard Riemann and studied in the context of the Riemann–Roch theorem associated with Alexander Grothendieck's later reforms. Classical examples include elliptic integrals arising from cubic curves like those in the investigations of Niels Henrik Abel and Carl Gustav Jacob Jacobi and hyperelliptic integrals studied by Adolf Hurwitz and Henri Poincaré. For a curve defined by a polynomial studied in the tradition of David Hilbert and Emmy Noether, integrating a meromorphic differential yields multivalued functions whose branch behaviour was analyzed by Augustin-Louis Cauchy and Karl Weierstrass.

Historical background and development

The subject traces through contributions by Niels Henrik Abel and Carl Gustav Jacob Jacobi on inversion problems, which were reframed by Bernhard Riemann in terms of surfaces and periods and by Ferdinand Georg Frobenius in matrix terms. Later formalism involved the Riemann–Roch theorem and the study of Jacobian varieties by André Weil, André Hurwitz, and Oscar Zariski, while algebraic foundations were shaped by Emmy Noether and Alexander Grothendieck. Analytical and topological aspects drew input from Henri Poincaré and George David Birkhoff, and the interaction with integrable systems was influenced by Sofia Kovalevskaya and Jacques Hadamard.

Connection with algebraic curves and abelian differentials

Abelian integrals are integrals of abelian differentials on algebraic curves such as those appearing in the work of Bernhard Riemann and formalized via Jacobian varieties introduced by Carl Ludwig Siegel and André Weil. The decomposition of spaces of differentials relates to the Riemann–Hurwitz formula and moduli problems later addressed by David Mumford and Pierre Deligne. The classification of holomorphic differentials on curves interacts with Igor Shafarevich's arithmetic geometry and with the Torelli theorem associated with Andreotti and Mumford's circle.

Periods, monodromy, and the period lattice

Periods of Abelian integrals define lattices in complex vector spaces leading to Jacobian varieties studied by Bernhard Riemann, Carl Ludwig Siegel, and André Weil, and they connect with monodromy representations analyzed by Henri Poincaré and Ralph Fox. The period lattice underlies the inversion problem addressed by Niels Henrik Abel and the uniformization ideas of Felix Klein and Henri Poincaré, and it plays a role in the work of Alexander Grothendieck on motives and the period conjectures influenced by Maxim Kontsevich and Yuri Manin.

Applications in dynamical systems and Hilbert's 16th problem

Abelian integrals appear in perturbation theory of planar polynomial vector fields studied by David Hilbert in his 16th problem and later by Jean Écalle, Yu. Ilyashenko, and A. Glutsyuk. The number of zeros of Abelian integrals is central to bifurcation analysis investigated by Nikolai Bautin and Jean-Pierre Françoise, and to limit cycle counts pursued by Igor Ilyashenko and Anatoly Pyartli. Connections to Poincaré maps recall Henri Poincaré's foundational work in dynamical systems and to perturbation techniques advanced by Alexander Shnol and Vladimir Arnold.

Computation and special functions

Computational approaches to Abelian integrals link to classical special functions such as elliptic functions of Niels Henrik Abel and Carl Gustav Jacob Jacobi, and to hyperelliptic functions studied by Adolf Hurwitz and Henri Poincaré. Algorithmic work on period matrices and abelian integrals has been advanced by researchers associated with David Mumford's school, computational algebraists influenced by Emmy Noether and Ernst Kummer, and numerical analysts following John von Neumann and Alston Householder. Symbolic integration techniques draw on ideas from Joseph Liouville and modern computer algebra systems developed in the tradition of Stephen Wolfram and Richard Fateman.

Generalizations and related concepts

Generalizations include multidimensional integrals on higher-dimensional varieties in the spirit of Alexander Grothendieck's cohomology theories, period maps studied by Phillip Griffiths and Wilfried Schmid, and mixed Hodge structures developed by Pierre Deligne. Relations to motives, regulators, and special values connect to conjectures of Alexander Beilinson, Pierre Deligne, and Gerd Faltings, while links to integrable systems involve names such as Sergei Novikov and Boris Dubrovin. Contemporary research spans algebraic geometry, arithmetic geometry, and mathematical physics with contributions from Edward Witten and Maxim Kontsevich.

Category:Algebraic geometry