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21 Multidimensional Theta FunctionsApplications

§21.8 Abelian Functions

An Abelian function is a 2g-fold periodic, meromorphic function of g complex variables. In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. For every Abelian function, there is a positive integer n, such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. For further information see Igusa (1972, pp. 132–135) and Markushevich (1992).