Diophantine approximations for periods of exponential and elliptic functions

Abstract

This. is to announce the results, of the paper [10] which will appear with complete proofs.. Let be a Weierstrass elliptic function with alge- braic inv,ariants g, g,, associated with a period lattice 9 of C. Let (C) be the endomorphism ring of , that is, the ring of complex numbers p such that the lattice pt9 is contained in tO.We know that ) is either the ring Z of rational integers, or a subring of finite index of the ring of integers of a complex quadratic field k.If (:/:Z, we say that has complex multiplication over k.Let w:, .e2be two periods of , which are linearly independent over the field of real numbers R, and e tO be a non-zero period of o.(a) Historical survey.We begin with some history on the transcen- dence measures concerning with these periods.C.L. Siegel [19] observed in 193.2 that the period lattice contains a transcendental number.The transcendence of and u/ follows from a theorem proved by Th.Schneider in 1937 (see for example [18]).If has complex multiplication, the number

Locations

• Proceedings of the Japan Academy Series A Mathematical Sciences - View - PDF