I'm currently studying the book 'Complex Abelian Varieties' by Birkenhake and Lange. On page 74, after lemma 1.5, the authors make the following statement:
'Another observation, which will prove useful on multiple occasions, is as follows: by definition, $t_x^*D = D - x$, which implies that $y \in t_x^*D$ if and only if $x \in t_y^*D$.
However, I'm having trouble understanding the equation $t_x^*D = D - x$. For instance, if we take $D = \sum n_p(p)$, it seems to me that the pullback of a translation by $x$ of the divisor should be $D = \sum n_p(p+x)$. Can someone clarify this for me?"