Jacobi inversion problem

The problem of inverting the Abelian integrals (cf. Abelian integral) of the first kind of an arbitrary algebraic function field (cf. Algebraic function). In other words, the problem of inverting the Abelian integrals of the first kind on the compact Riemann surface $ F $ of genus $ p \geq 1 $ that corresponds to a given algebraic equation $ F ( z, w) = 0 $.

Let $ \phi _ {1} \dots \phi _ {p} $ be a basis of the Abelian differentials of the first kind on $ F $( cf. Abelian differential). The inversion of a single Abelian integral, for example, $ \int _ {c _ {1} } ^ {w _ {1} } \phi _ {1} \equiv u _ {1} ( w _ {1} ) = z _ {1} $, that is, the representation of all possible rational functions of $ w _ {1} $, in particular, the representation of $ w _ {1} $ as a function of $ z _ {1} $, $ w _ {1} = w _ {1} ( z _ {1} ) $, is only meaningful when $ p = 1 $. In this case one speaks of the inversion of an elliptic integral, and this leads to doubly-periodic elliptic functions (cf. Elliptic function). For example, the inversion of an integral of the first kind in Legendre normal form

$$ u _ {1} ( w _ {1} ) = \ \int\limits _ { 0 } ^ { {w _ 1} } \frac{dt}{\sqrt {( 1 - t ^ {2} ) ( 1 - k ^ {2} t ^ {2} ) } } = z _ {1,} $$

leads to the Jacobi elliptic function $ w _ {1} = \mathop{\rm sn} z _ {1} $( cf. Jacobi elliptic functions).

As C.G.J. Jacobi already observed (1832), the inversion problem for $ p> 1 $ must be considered for the set of all Abelian integrals of the first kind, $ \int \phi _ {1} \dots \int \phi _ {p} $, since one has to obtain functions with $ 2p $ periods. In the general case when $ p \geq 1 $, the rational statement of the Jacobi inversion problem is as follows: Suppose one is given a system of equalities

\begin{equation} \label{eq1} \int\limits _ { c _ {1} } ^ { {w _ 1 } } \phi _ {j} + \dots + \int\limits _ { c _ {p} } ^ { {w _ p } } \phi _ {j} = z _ {j} ,\ \ j = 1 \dots p, \end{equation}

where $ c _ {1} \dots c _ {p} $ are fixed points on $ F $, $ w _ {1} \dots w _ {p} $ are variable points on $ F $ and $ z = ( z _ {1} \dots z _ {p} ) $ are arbitrary but given complex numbers. It is required to find how and under what conditions the system \eqref{eq1} can be inverted, that is, to obtain a representation of all possible symmetric rational functions of $ w _ {k} $, $ k = 1 \dots p $, as functions of $ z = ( z _ {1} \dots z _ {p} ) $.

Since the Abelian integrals in \eqref{eq1}, as functions of the upper limit $ w _ {k} $, depend on the form of the path on $ F $ that joins $ c _ {k} $ and $ w _ {k} $, they are multiple-valued: When the path is varied they can increase by an integer linear combination of periods. Hence it follows that \eqref{eq1} is essentially a system of congruences modulo the periods of the differentials $ \phi _ {1} \dots \phi _ {p} $. The values of the functions of $ z= ( z _ {1} \dots z _ {p} ) $ obtained by solving the Jacobi inversion problem must remain unchanged when the argument is increased by any integer combination of the periods of the differentials $ \phi _ {1} \dots \phi _ {p} $. Consequently, they are special Abelian functions with $ 2p $ independent periods (cf. Abelian function).

For the case $ p = 1 $, that is, for an elliptic integral, the elliptic functions solving the inversion problem can be constructed by using the comparatively-simple Jacobi theta-functions in a single complex variable $ z $, and meromorphic elliptic functions can be constructed as quotients of entire theta-functions. It is also possible to solve the general Jacobi inversion problem by using the theta-functions $ \theta _ {H} ( z) = \theta _ {H} ( z _ {1} \dots z _ {p} ) $ of the first order in $ p $ complex variables with half-integer characteristics $ H $( cf. Theta-function).

The period matrix $ W $ of the basis Abelian differentials $ \phi _ {j} $ has the form

\begin{equation} \label{eq2} W = \left \| \begin{array}{ccccccc} \pi i & 0 & 0 & 0 &a _ {11} &\dots &a _ {1p} \\ 0 &\dots &\cdot &\cdot &\cdot &\dots &\cdot \\ \cdot &\dots &\pi i & 0 &\cdot &\dots &\cdot \\ 0 & 0 & 0 &\pi i &a _ {p1} &\dots &a _ {pp} \\ \end{array}

\ 

\right \| , \end{equation}

$$ a _ {jk} = a _ {kj} , $$

and the Riemann relations (see Abelian function) between the periods ensure that the series representing the theta-functions $ \theta _ {H} ( z) $, constructed in terms of $ W $, converge uniformly on compact sets in $ \mathbf C ^ {p} $. By using the theta-function $ \theta ( z) = \theta _ {0} ( z) $ of zero characteristic one can construct the superposition

$$ \Phi ( w) = \theta ( u ( w) - z), $$

where

$$ u ( w) = \ \left \{ u _ {1} ( w _ {1} ) = \ \int\limits _ { c _ {1} } ^ { {w _ 1 } } \phi _ {1} \dots u _ {p} ( w _ {p} ) = \int\limits _ { c _ {p} } ^ { {w _ p } } \phi _ {p} \right \} $$

is the vector of Abelian integrals and $ w = ( w _ {1} \dots w _ {p} ) $ is a system of points of $ F $; $ \Phi ( w) $ is called the Riemann theta-function. For a given set of numbers $ z \in \mathbf C ^ {p} $, either $ \Phi ( w) $ has on $ F $ a unique system of zeros $ \eta _ {1} \dots \eta _ {p} $( the normal case), or it vanishes identically (the exceptional case). The zeros $ \eta _ {1} \dots \eta _ {p} $ give a solution to the Jacobi inversion problem. The exceptional points $ z $ for which $ \Phi ( w) \equiv 0 $ form in $ \mathbf C ^ {p} $ a set of lower dimension.

Explicit expressions for special Abelian functions that completely solve the Jacobi inversion problem can be constructed by using quotients of theta-functions of the form $ \theta _ {H} ( z)/ \theta ( z) $, where the common denominator is the theta-function of zero characteristic. When periods are added to the argument, the theta-functions are multiplied by specific multipliers. For quotients of theta-functions, after cancellation the only non-trivial multiplier is $ - 1 $. Consequently, the squares of the quotients are unchanged when periods are added to the argument, and one obtains Abelian functions with $ 2p $ periods.

Closely related to the Jacobi inversion problem is the important problem of constructing for a given system of theta-functions $ \theta _ {H} ( z) $ with a common matrix $ W $ satisfying the convergence conditions the corresponding algebraic function fields and the corresponding Riemann surface. For this construction to be possible, the distinct elements $ a _ {jk} $ of $ W $( there are $ p ( p + 1)/2 $ of them) must satisfy $ ( p - 2) ( p - 3)/2 $ additional relations, the investigation of which when $ p > 3 $ is a very difficult problem (see [1], [3], [4], [5]).

References

[1]	N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)
[2]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
[3]	A. Clebsch, P. Gordan, "Theorie der Abelschen Funktionen" , Teubner (1866)
[4]	F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956)
[5]	D. Mumford, "Structure of moduli spaces of curves and Abelian varieties" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 457–465
[a1]	C.L. Siegel, "Topics in complex function theory" , 2 , Interscience (1971)
[a2]	D. Mumford, "Tata lectures on Theta" , 1 , Birkhäuser (1983–1984)