Criteria for periodicity and an application to elliptic functions

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 fP(x) = f (Px) - f (x) and fQ(x) = f (Qx) - f (x) are both Λ-periodic for some lattice Λ ⊂ V, then so is f (up to a modification at ý). 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 theoremand explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.