# Abelian function

A generalization of the concept of an elliptic function of one complex variable to the case of several complex variables. A function $ f(z) $
in the variables $ z _ {1} \dots z _ {p} $,
$ z = (z _ {1} \dots z _ {p} ) $,
which is meromorphic in the complex space $ \mathbf C ^ {p} $,
$ p \geq 1 $,
is called an Abelian function if there exist $ 2p $
row vectors in $ \mathbf C ^ {p} $

$$ w _ \nu = ( \omega _ {1 \nu } \dots \omega _ {p \nu } ) , \ \nu = 1 \dots 2p , $$

which are linearly independent over the field of real numbers and are such that $ f(z + w _ \nu ) = f(z) $ for all $ z \in \mathbf C ^ {p} $, $ \nu = 1 \dots 2p $. The vectors $ w _ \nu $ are called the periods or the system of periods of the Abelian function $ f(z) $. All periods of the Abelian function $ f(z) $ form an Abelian group $ \Gamma $ under addition, which is known as the period group (or the period module). A basis of this group is known as a basis system of periods of the Abelian function, or also as a system of basic (or primitive) periods. An Abelian function $ f (z) $ is said to be degenerate if there exists a linear transformation of the variables $ z _ {1} \dots z _ {p} $ which converts $ f(z) $ into a function of fewer variables; otherwise $ f(z) $ is said to be a non-degenerate Abelian function. Degenerate Abelian functions are distinguished by having infinitely small periods, i.e. for any number $ \epsilon > 0 $ it is possible to find a period $ w _ \nu $ for which

$$ \| w _ {p} \| = \sqrt {\sum _ {k = 1 } ^ { p } | \omega _ {kp} | ^ {2} } < \epsilon . $$

If $ p = 1 $, the non-degenerate Abelian functions are elliptic functions of one complex variable. Each Abelian function with period group $ \Gamma $ is naturally identified with a meromorphic function on the complex torus $ \mathbf C ^ {p} / \Gamma $, i.e. the quotient space $ \mathbf C ^ {p} / \Gamma $( cf. Quasi-Abelian function).

The study of Abelian functions began in the 19th century in connection with the inversion of Abelian integrals (cf. Abelian integral) of the first kind (cf. Jacobi inversion problem, [1], [2]). The Abelian functions obtained in solving this problem are called special Abelian functions; sometimes, in earlier work, only such functions were considered to be Abelian functions. Let $ u _ {1} \dots u _ {p} $ be linearly independent normal Abelian integrals of the first kind, constructed on a Riemann surface $ F $:

$$ u _ {1} = \int\limits _ {c _ {1} } ^ { x } du _ {1} \dots u _ {p} = \int\limits _ {c _ {p} } ^ { x } du _ {p} , $$

and let

$$ z _ {j} = \int\limits _ {c _ {1} } ^ { {x } _ {1} } du _ {j} + \dots + \int\limits _ {c _ {p} } ^ { {x } _ {p} } du _ {j} ,\ j = 1 \dots p , $$

be a given system of sums in which the lower integration limits $ c _ {1} \dots c _ {p} $ are considered as fixed on the surface $ F $. It is then possible to define the special Abelian functions as all rational functions in the $ p $ coordinates of the upper limits $ x _ {1} \dots x _ {p} $, the latter being in turn considered as functions of $ p $ points $ z _ {1} \dots z _ {p} $ on $ F $. In the symbolic notation introduced by K. Weierstrass, any special Abelian function $ Al (z) $ may be represented as

$$ Al (z ) = Al ( z _ {1} \dots z _ {p} ) = $$

$$ = \ R [ x _ {1} ( z _ {1} \dots z _ {p} ) \dots x _ {p} ( z _ {1} \dots z _ {p} ) ]. $$

The complex tori $ \mathbf C ^ {p} / \Gamma $ corresponding to special Abelian functions are the Jacobi varieties of algebraic curves.

The matrix $ W $ whose columns form a period basis of the Abelian function $ f(z) $ has dimension $ p \times 2p $ and is known as the period matrix of the Abelian function $ f(z) $. A necessary and sufficient condition for a given matrix $ W $ of dimension $ p \times 2p $ to be the period matrix of some non-degenerate Abelian function $ f(z) $ is for the matrix to satisfy the following conditions (the Riemann–Frobenius conditions). It must be a Riemann matrix, i.e. there must exist an anti-symmetric non-degenerate square matrix $ M $ with integer elements, of order $ 2p $, and such that 1) $ WM W ^ {T} = 0 $, where $ W ^ {T} $ is the transposed matrix of $ W $; and 2) the matrix $ iWM {\overline{W}\; } ^ {T} $ defines a positive-definite Hermitian form [3]. If the conditions 1) and 2) are expressed as equations and inequalities respectively, a system of $ p(p - 1)/2 $ Riemann equations and $ p(p - 1)/2 $ Riemann inequalities is obtained. The number $ p $ is called the genus of the matrix $ W $ and of the corresponding Abelian function $ f (z) $. The columns $ w _ \nu = \mathop{\rm Re} w _ \nu + i \mathop{\rm Im} w _ \nu $ of $ W $, regarded as vectors in the real Euclidean space $ \mathbf R ^ {2p} $, define the period parallelotope of $ f(z) $.

All Abelian functions corresponding to the same period matrix $ W $ form an Abelian function field $ K _ {W} $. If the field $ K _ {W} $ contains a non-degenerate Abelian function, its degree of transcendence over the field $ \mathbf C $ is $ p $; the torus $ \mathbf C ^ {p} / \Gamma $ is then an Abelian variety, and $ K _ {W} $ turns out to be its field of rational functions. If, on the other hand, all Abelian functions of $ K _ {W} $ are degenerate, then $ K _ {W} $ is isomorphic to the field of rational functions on an Abelian variety of dimension lower than $ p $. See also Quasi-Abelian function.

As in the case of elliptic functions, any Abelian function can be represented as the quotient of two entire transcendental theta-functions (cf. Theta-function), which in turn can be represented as theta-series. A given Riemann matrix $ W $ determines a class of theta-series by means of which all Abelian functions of the field $ K _ {W} $ can be constructed.

For special Abelian functions, the matrix $ W $ can always be brought — by means of a linear transformation of the independent variables $ z _ {1} \dots z _ {p} $— to the form

$$ W = \left \| \begin{array}{cccccc} \pi i &\dots & 0 &a _ {11} &\dots &a _ {1p} \\ \dots &\dots &\dots &\dots &\dots &\dots \\ 0 &\dots &\pi i &a _ {p1} &\dots &a _ {pp} \\ \end{array} \ \right \| . $$

The Riemann relations between the elements of the matrix $ \| a _ {jk} \| $, $ j, k = 1 \dots p $, now ensure the symmetry of the matrix, $ a _ {jk} = a _ {kj} $, and the positive definiteness of the matrix of real parts $ \| \mathop{\rm Re} a _ {jk} \| $, $ j, k = 1 \dots p $. However, for $ p > 3 $ there will be only $ 3 (p - 1 ) $ independent elements among the elements $ a _ {jk} $ of $ \| a _ {jk} \| $, i.e. as many as the number of conformal moduli for the Riemann surface $ F $ on which the inversion problem is solved (cf. Moduli of a Riemann surface). In addition to the Riemann relations there are in such a case $ (p - 2)(p - 3)/2 $ transcendental relations between the $ a _ {jk} $, the explicit form of which for the case $ p = 4 $ was first found by E. Schottky in 1886; for a review of subsequent advances in the field see [5].

Special Abelian functions can be represented as the quotient of two entire theta-functions with half-integral characteristics of a special kind. These representations yield a number of properties of special Abelian functions which generalize many properties of elliptic functions. Thus, derivatives of an Abelian function $ f(z) $ with respect to any argument $ z _ {j} $ are Abelian functions; any $ p + 1 $ Abelian functions are related by some algebraic equation; any Abelian function can be rationally expressed in terms of some $ p + 1 $ Abelian functions, e.g. by means of an arbitrary Abelian function and its $ p $ partial derivatives of the first order; Abelian functions satisfy the addition theorem, i.e. the value of an Abelian function at the point $ a + b \in \mathbf C ^ {p} $ may be rationally expressed in terms of the values of a certain $ ( p + 1 ) $- tuple of Abelian functions at the points $ a, b \in \mathbf C ^ {p} $.

Abelian functions are very important in algebraic geometry as a means of uniformization of algebraic varieties of certain types.

#### References

[1] | G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) MR0092855 Zbl 0078.06602 |

[2] | N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian) |

[3] | C.L. Siegel, "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen (1955) |

[4] | A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958) |

[5] | D. Mumford, "Curves and their Jacobians" , Univ. Michigan Press (1975) MR0419430 Zbl 0316.14010 |

[6] | H. Stahl, "Theorie der Abelschen Funktionen" , Leipzig (1896) Zbl 35.0473.03 |

[7] | A. Clebsch, P. Gordan, "Theorie der Abelschen Funktionen" , Teubner (1866) |

[8] | F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956) MR0079316 Zbl 0074.36601 |

#### Comments

A basis system of periods is also known as a period basis. Instead of [5] one can also consult the more recent [a1]. The Riemann–Frobenius conditions are also known as the Riemann bilinear relations or simply as the Riemann relations. The Schottky problem concerns the description of the space of all Jacobians in the space of matrices satisfying the Riemann conditions. It has recently been solved, cf. Jacobi variety.

#### References

[a1] | E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985) MR0770932 Zbl 0559.14017 |

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Abelian function.

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