Let $X=\frac{V}{\Lambda}$ be a complex torus of dimension $2$. Suppose $C$ is a curve in $X$. Consider $C+a=C_1$, the curve which is a translate of $C$ by $a\in X$. Are $C$ and $C_1$ linearly equivalent to each other?
My argument is as follows: If $z$ and $z_1$ are the generic points of $C$ and $C_1$, $z_1$ will be $z+a$. Consider the equation defining $C$ at $z$ say $f$. Then the equation defining $C_1$ at $z_1=z+a$, will be $g$ defined as $g(x)=f(x-a)$. Then consider $f/g$. This will be a meromorphic function with zeros along $C$ and poles along $C_1$. That is $[C]-[C_1]=(f/g)$, therefore $C\sim C_1$. Is this right?