On the zeroes of certain periodic functions over valued fields of positive characteristic

Let C be an algebraically closed field of positive characteristic p   and complete with respect to a non-archimedean absolute value | . || . | and Λ⊂CΛ⊂C a discrete FpFp-submodule. Suppose there exists an FpFp-basis {λ0,λ1,…}{λ0,λ1,…} of Λ   such that 0<|λ0|<|λ1|<⋯0<|λ0|<|λ1|<⋯ and |λi|→∞|λi|→∞. For k∈Nk∈N define the meromorphic functionCk,Λ(z)=∑λ∈Λ1(z−λ)k on C. We show that all the zeroes x   of Ck,ΛCk,Λ satisfyequation(⁎)|x|=|λi||x|=|λi| for some i. Furthermore, the number (counted with multiplicities) of zeroes for which (⁎) holds depends only on i and the p-adic expansion coefficients of k, but not on Λ.