View source for Jacobi inversion problem

The problem of inverting the Abelian integrals (cf. [[Abelian integral|Abelian integral]]) of the first kind of an arbitrary algebraic function field (cf. [[Algebraic function|Algebraic function]]). In other words, the problem of inverting the Abelian integrals of the first kind on the compact [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540701.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540702.png" /> that corresponds to a given algebraic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540703.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540704.png" /> be a basis of the Abelian differentials of the first kind on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540705.png" /> (cf. [[Abelian differential|Abelian differential]]). The inversion of a single Abelian integral, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540706.png" />, that is, the representation of all possible rational functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540707.png" />, in particular, the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540708.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407010.png" />, is only meaningful when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407011.png" />. In this case one speaks of the [[Inversion of an elliptic integral|inversion of an elliptic integral]], and this leads to doubly-periodic elliptic functions (cf. [[Elliptic function|Elliptic function]]). For example, the inversion of an integral of the first kind in Legendre normal form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407012.png" /></td> </tr></table>

leads to the Jacobi elliptic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407013.png" /> (cf. [[Jacobi elliptic functions|Jacobi elliptic functions]]).

As C.G.J. Jacobi already observed (1832), the inversion problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407014.png" /> must be considered for the set of all Abelian integrals of the first kind, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407015.png" />, since one has to obtain functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407016.png" /> periods. In the general case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407017.png" />, the rational statement of the Jacobi inversion problem is as follows: Suppose one is given a system of equalities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407019.png" /> are fixed points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407021.png" /> are variable points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407023.png" /> are arbitrary but given complex numbers. It is required to find how and under what conditions the system (1) can be inverted, that is, to obtain a representation of all possible symmetric rational functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407025.png" />, as functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407026.png" />.

Since the Abelian integrals in (1), as functions of the upper limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407027.png" />, depend on the form of the path on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407028.png" /> that joins <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407030.png" />, they are multiple-valued: When the path is varied they can increase by an integer linear combination of periods. Hence it follows that (1) is essentially a system of congruences modulo the periods of the differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407031.png" />. The values of the functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407032.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407033.png" />. Consequently, they are special Abelian functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407034.png" /> independent periods (cf. [[Abelian function|Abelian function]]).

For the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407035.png" />, that is, for an [[Elliptic integral|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407036.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407037.png" /> of the first order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407038.png" /> complex variables with half-integer characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407039.png" /> (cf. [[Theta-function|Theta-function]]).

The period matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407040.png" /> of the basis Abelian differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407041.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407043.png" /></td> </tr></table>

and the Riemann relations (see [[Abelian function|Abelian function]]) between the periods ensure that the series representing the theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407044.png" />, constructed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407045.png" />, converge uniformly on compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407046.png" />. By using the theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407047.png" /> of zero characteristic one can construct the superposition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407048.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407049.png" /></td> </tr></table>

is the vector of Abelian integrals and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407050.png" /> is a system of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407051.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407052.png" /> is called the [[Riemann theta-function|Riemann theta-function]]. For a given set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407053.png" />, either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407054.png" /> has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407055.png" /> a unique system of zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407056.png" /> (the normal case), or it vanishes identically (the exceptional case). The zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407057.png" /> give a solution to the Jacobi inversion problem. The exceptional points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407058.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407059.png" /> form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407060.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407061.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407062.png" />. Consequently, the squares of the quotients are unchanged when periods are added to the argument, and one obtains Abelian functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407063.png" /> periods.

Closely related to the Jacobi inversion problem is the important problem of constructing for a given system of theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407064.png" /> with a common matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407065.png" /> satisfying the convergence conditions the corresponding algebraic function fields and the corresponding Riemann surface. For this construction to be possible, the distinct elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407067.png" /> (there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407068.png" /> of them) must satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407069.png" /> additional relations, the investigation of which when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407070.png" /> is a very difficult problem (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]], [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.G. Chebotarev,   "The theory of algebraic functions" , Moscow-Leningrad  (1948)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Clebsch,   P. Gordan,   "Theorie der Abelschen Funktionen" , Teubner  (1866)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Conforto,   "Abelsche Funktionen und algebraische Geometrie" , Springer  (1956)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Mumford,   "Structure of moduli spaces of curves and Abelian varieties" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 457–465</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.L. Siegel,   "Topics in complex function theory" , '''2''' , Interscience  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Mumford,   "Tata lectures on Theta" , '''1''' , Birkhäuser  (1983–1984)</TD></TR></table>