Algebraic functions of complexity one, a Weierstrass theorem, and three arithmetic operations

Abstract

The Weierstrass theorem concerning functions admitting an algebraic addition theorem enables us to give an explicit description of algebraic functions of two variables of analytical complexity one. Their description is divided into three cases: the general case, which is elliptic, and two special ones, a multiplicative and an additive one. All cases have a unified description; they are the orbits of an action of the gauge pseudogroup. The first case is a 1-parameter family of orbits of “elliptic addition,” the second is the orbit of multiplication, and the third of addition. Here the multiplication and addition can be derived from the “elliptic addition” by passages to a limit. On the other hand, the elliptic orbits correspond to complex structures on the torus, the multiplicative orbit corresponds to the complex structure on the cylinder, and the additive one to that on the complex plane.

This work was financially supported by the Russian Foundation for Basic Research under grants nos. 14-00709-a and 13-01-12417-ofi-m2.