Algebraic addition theorems

✍ Scribed by Stephen A Andrea

• Publisher: Elsevier Science
• Year: 1974
• Tongue: English
• Weight: 585 KB
• Volume: 13
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(74)90063-2

✦ Synopsis

It is a classical result in elliptic function theory that for every elliptic function there is an algebraic addition theorem. In this note we wish to show that such results really depend only on the fact that one is dealing with analytic functions which are finite-to-one in a certain strong sense.

More precisely, let f be an analytic complex-valued function defined on an open subset of a Riemann surface M. We say that f is ofjnite type if (1) the domain off includes all but finitely many points of M and (2) there is a positive integer n such that for all but finitely many complex numbers z, the equation f(u) = z has exactly n solutions u E domain(f) C M. (Repeated roots are counted according to their multiplicity.)

Then one has the following result.

THEOREM.

Let f, g, and h be analytic complex-valued functions deJ'ined on open subsets of M, N, and M x N, respectively, where M and N are Riemann surfaces. Suppose that f and g are of jkite type, and that all but finitely many of the functions u H h(u, v) and v t+ h(u, v) are analytic functions ofJinite type dejked on M and on N, respectivety. Then there is a nontrivial polynomial R(X, z, w) with complex coeficients such that

for all u, v where the three functions are dejined.

As an immediate corollary, one has addition theorems for elliptic functions. For, an elliptic function F defined on the complex plane C is of finite type when its domain is considered to be the torus CjQ, where X2 is the discrete two-dimensional module of periods of F. Then the f, g, h of the theorem are taken to be F(u), F(v), and F(u + v), while the Riemann surfaces M and N are the torus C/L?.