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Criteria for periodicity and an application to elliptic functions

Published online by Cambridge University Press:  14 August 2020

Ehud de Shalit*
Affiliation:
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

Abstract

Let P and Q be relatively prime integers greater than 1, and let f be a real valued discretely supported function on a finite dimensional real vector space V. We prove that if $f_{P}(x)=f(Px)-f(x)$ and $f_{Q}(x)=f(Qx)-f(x)$ are both $\Lambda $ -periodic for some lattice $\Lambda \subset V$ , then so is f (up to a modification at $0$ ). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.

Type
Article
Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author was supported by ISF grant 276/17.

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