Translation of meromorphic function which is periodic w.r.t. to a lattice

Question

Let $n\in \mathbb{N}_{>0}$, $\underline{u} \in \mathbb{C}^n\setminus \{ 0\}$, $\underline{v},\underline{w}\in \mathbb{C}^n$ and $L\subseteq \mathbb{C}^n$ a lattice of rank $2n$. Let furthermore $f: \mathbb{C}^n \rightarrow \mathbb{C}$ be a meromorphic function which is periodic with respect to the lattice $L$, i.e. for all $\underline{z}$ and all $\underline{y}\in L$ holds

$$ f(\underline{z}+ \underline{y})= f(\underline{z}).$$

Assume that in addition $f$ satisfies

$$ f(\underline{v} + x\underline{u})= f(\underline{w} + x\underline{u}) $$

for all $x\in \mathbb{R}$ (where we fixed $\underline{u}, \underline{v}$ and $\underline{w}$ in the first line). Does this imply that $\underline{v}$ and $\underline{w}$ are equivalent modulo $L$?

Answer 1

Let

$$F(z_1,z_2) = f_1(z_1)f_2(z_2), \qquad f_1(z) = \wp(z)\wp(2z),\qquad f_2(z) = \wp(z)$$

Then $F$ is $(1,0),(i,0),(0,1),(0,i)$ periodic and since $\wp(a)= \wp(-a)=0$ for some $a$ in the unit square we have $$\forall z \in \mathbb{C}, \qquad F(-a/2,z)= F(a/2,z)=0$$ but $(a,0) \not \in \Lambda$.