Abelian integral

algebraic integral

An integral of an algebraic function, i.e. an integral of the form

$$ \tag{1 } \int\limits _ {z _ {0} } ^ { {z } _ {1} } R ( z, w ) dz , $$

where $ R (z, w ) $ is some rational function in variables $ z, w $ that are related by an algebraic equation

$$ \tag{2 } F ( z, w ) \equiv a _ {0} ( z ) w ^ {n} + a _ {1} ( z ) w ^ {n-1 } + \dots + a _ {n} ( z ) = 0 $$

with coefficients $ a _ {j} (z) $ that are polynomials in $ z $, $ j = 0, \dots, n $. To equation (2) there corresponds a compact Riemann surface $ F $ which is an $ n $-sheeted covering of the Riemann sphere. On this Riemann surface $ z, w $, and consequently also $ R (z, w) $, can be considered as single-valued functions of the points on $ F $.

The integral (1) is then given as the integral $ \int _ {L} \omega $ of the Abelian differential $ \omega = R (z, w) dz $ on $ F $ taken along some rectifiable path $ L $. In general, specifying only the initial and end points $ z _ {0} $ and $ z _ {1} $ of the path $ L $ does not completely determine the value of the Abelian integral (1), or, which is the same, the integral (1) turns out to be a multi-valued function of the initial and end points of the path $ L $.

The behaviour of an Abelian integral on $ F $ depends, first of all, on the topological structure of $ F $, in particular on the topological invariant called the genus $ g $ of the surface $ F $ (cf. Genus of a surface). The genus $ g $ is connected with the number of sheets $ n $ and with the number of branch points $ \nu $( counted with multiplicities) by the relation $ g = ( \nu / 2 ) - n + 1 $. For $ g = 0 $ the variables $ z $ and $ w $ can be rationally expressed in some parameter $ t $, and the Abelian integral becomes the integral of a rational function in $ t $. This will happen, for example, in the elementary cases $ w ^ {2} = a _ {0} z + a _ {1} $ and $ w ^ {2} = a _ {0} z ^ {2} + a _ {1} z + a _ {2} $.

If $ g \geq 1 $, any Abelian integral can be expressed in the form of a linear combination of elementary functions and canonical Abelian integrals of the three kinds. The integral $ \int _ {L} \omega $ is called an Abelian integral of the first kind if $ \omega $ is an Abelian differential of the first kind. In other words, Abelian integrals of the first kind are characterized by the fact that for a fixed initial point $ z _ {0} $ of the path $ L $ they are a function of the upper bound $ z _ {1} $ that is an everywhere finite, usually multi-valued, function on $ F $. Such a characterization may be used, for example, to construct analogues of Abelian integrals of the first kind on non-compact Riemann surfaces. Any Abelian integral of the first kind can be represented in the form of a linear combination of $ g $ linearly-independent normal Abelian integrals of the first kind

$$ u _ {1} = \int\limits _ { L } \phi _ {1}, \dots, u _ {g} = \int\limits _ { L } \phi _ {g} $$

of differentials $ \phi _ {1}, \dots, \phi _ {g} $ that constitute a canonical basis for the Abelian differentials of the first kind. If the surface $ F $ is cut along the cycles $ a _ {1} b _ {1} \dots a _ {g} b _ {g} $ of a canonical basis for the homology, a simply-connected domain $ F ^ {*} $ is obtained. The integrals $ \int _ {L ^ {*} } \phi _ {i} $ are single-valued functions of the upper bound $ z _ {1} $ for all paths $ L ^ {*} \subset F ^ {*} $ with fixed initial point $ z _ {0} $ and fixed end point $ z _ {1} $. The multi-valuedness of the integrals $ u _ {i} = \int _ {L} \phi _ {i} $ along an arbitrary path $ L \subset F $ joining $ z _ {0} $ with $ z _ {1} $ is now completely described by the fact that it differs from the integrals $ \int _ {L ^ {*} } \phi _ {i} $ only by an integral linear combination of the $ A $-periods $ A _ {ij} $ and the $ B $-periods $ B _ {ij} $ of a basis of the differentials of the first kind. These constitute the period matrix of dimension $ g \times 2g $, which satisfies the bilinear Riemann relations (cf. Abelian differential).

An integral $ \int _ {L} \omega $ where $ \omega $ is an Abelian differential of the second kind is said to be an Abelian integral of the second kind. Considered as a function of the upper bound, it has no singularities anywhere on $ F $ except for poles. An Abelian integral of a normalized Abelian differential of the second kind is known as a normal Abelian integral of the second kind.

An Abelian integral of the third kind is an arbitrary Abelian integral. It usually has logarithmic singularities on $ F $; however, such singularities can only occur in pairs. An Abelian integral of a normal Abelian differential of the third kind is called a normal Abelian integral of the third kind. Any Abelian integral can be represented as a linear combination of normal Abelian integrals of the first, second and third kinds. Unlike Abelian integrals of the first and second kinds, Abelian integrals of the third kind usually also have the so-called polar periods, beside the $ A $- and $ B $-periods (which are called cyclic periods). Polar periods are taken along cycles which are homologous to zero, but encircle the logarithmic singularities of the Abelian integral. They are caused by the poles of the Abelian differential $ \omega $ with non-zero residues.

A number of relations depending on the topological and conformal structure of $ F $ exist for arbitrary Abelian integrals on the same Riemann surface $ F $. Thus, if $ \omega _ {P _ {i} P _ {j} } $ is a normal Abelian differential of the third kind with simple poles in $ P _ {i} $ and $ P _ {j} $, then the following theorem on the permutation of the parameters and the bounds of an Abelian integral of the third kind holds for all points $ P _ {1} , P _ {2} , P _ {3} , P _ {4} $:

$$ \int\limits _ {P _ {1} } ^ { {P } _ {2} } \omega _ {P _ {3} P _ {4} } = \int\limits _ {P _ {3} } ^ { {P } _ {4} } \omega _ {P _ {1} P _ {2} } . $$

Relations which connect Abelian integrals with rational functions on $ F $ are known as Abelian theorems. In terms of divisors, for example, the Abelian theorem for Abelian integrals of the first kind has the following form: A divisor $ \mathfrak a $ on $ F $ is the divisor of a meromorphic function if and only if there exists a chain $ L $ with $ \partial L = \mathfrak a $ and $ \int _ {L} \omega = 0 $ for all Abelian differentials of the first kind on $ F $. There also exist variants of Abelian theorems for Abelian integrals of the second and third kinds [4]. Abelian integrals and, in particular, Abelian theorems, are the basis of the transcendental construction of the Jacobi variety of a Riemann surface. The question of the inversion of an Abelian integral as a function of its upper bound also leads to the concepts of an Abelian function; an elliptic function; and theta-functions (cf. Theta-function; Jacobi inversion problem).

Historically, the theory of Abelian integrals followed from the consideration of a surface of genus $ g = 1 $. If one writes the corresponding equation in the form

$$ F ( z, w ) \equiv w ^ {2} - f ( z ) = 0, $$

where $ f(z) $ is a polynomial in $ z $ of the third or fourth degree, then one obtains elliptic integrals (cf. Elliptic integral) as the respective Abelian integrals. They first appeared at the end of the 17th century and the beginning of the 18th century as the result of the rectification of curves of the second order in the studies of Jacob and Johann Bernoulli and of G. Fagnano. L. Euler tackled the addition theorem of elliptic integrals, which is a special case of a theorem of N.H. Abel (1752). Abel and C.G.J. Jacobi (1827) stated the problem of inversion of elliptic integrals and obtained the solution. The beginnings of the theory of elliptic functions were thus established. However, some facts concerning this theory had been established by C.F. Gauss early in the 18th century. Abel and Jacobi dealt with the much more difficult case of inversion of Abelian integrals in the case $ g > 1 $. During the very first stages of development stress was laid on hyper-elliptic integrals, where $ F (z, w) = w ^ {2} - f(z) $ with $ f(z) $ a polynomial of the fifth or sixth degree without multiple roots. Here $ g = 2 $ and the difficulty of the inversion problem can already be noticed. The principal advances in the theory of inversion of Abelian integrals are due to B. Riemann (1851), who introduced the concept of Riemann surfaces and formulated and gave proofs of a large number of important results.

Multi-dimensional generalizations of the theory of Abelian integrals form the subject matter of algebraic geometry and the theory of complex manifolds.

Comments

An interesting and useful additional reference is [a1].

References

[1]	N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt.8;9 (In Russian)
[2]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602
[3]	R. Nevanlinna, "Uniformisierung" , Springer (1953) pp. Chapt.5 MR0057335 Zbl 0053.05003
[4]	G.A. Bliss, "Algebraic functions" , Amer. Math. Soc. (1933) MR0203007 MR1502680 Zbl 0008.21004 Zbl 59.0384.03
[5]	H. Stahl, "Theorie der Abelschen Funktionen" , Leipzig (1896) Zbl 35.0473.03
[6]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[a1]	S. Lang, "Introduction to algebraic and Abelian functions" , Addison-Wesley (1972) MR0327780 Zbl 0255.14001