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Criteria for periodicity and an application to elliptic functions

Part of: Difference and functional equations Difference equations Curves Differential and difference algebra

Published online by Cambridge University Press:  14 August 2020

Ehud de Shalit
Ehud de Shalit*
Affiliation:
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
*
e-mail: ehud.deshalit@mail.huji.ac.il
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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.

Keywords

Difference equationsperiodic functionselliptic functions

MSC classification

Primary: 39A10: Difference equations, additive 12H10: Difference algebra
Secondary: 14H52: Elliptic curves

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 530 - 540
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author was supported by ISF grant 276/17.

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