Abelian Integral - Modern View

Modern View

In Riemann surface theory, an abelian integral is a function related to the indefinite integral of a differential of the first kind. Suppose we are given a Riemann surface and on it a differential 1-form that is everywhere holomorphic on, and fix a point on, from which to integrate. We can regard

as a multi-valued function, or (better) an honest function of the chosen path drawn on from to . Since will in general be multiply connected, one should specify, but the value will in fact only depend on the homology class of .

In the case of a compact Riemann surface of genus 1, i.e. an elliptic curve, such functions are the elliptic integrals. Logically speaking, therefore, an abelian integral should be a function such as .

Such functions were first introduced to study hyperelliptic integrals, i.e. for the case where is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic functions, where is a polynomial of degree . The first major insights of the theory were given by Niels Abel; it was later formulated in terms of the Jacobian variety . Choice of gives rise to a standard holomorphic mapping

of complex manifolds. It has the defining property that the holomorphic 1-forms on, of which there are g independent ones if g is the genus of S, pull back to a basis for the differentials of the first kind on S.