What is the ideal sheaf of a translated subvariety of an abelian variety?

Question

Let $A$ be an abelian variety (over an algebraically closed field $k$) and $X\subset A$ a nonsingular subvariety with ideal sheaf $\mathscr I_X\subset \mathscr O_A$. Let $\tau_a:A\to A$ be the translation automorphism associated to a point $a\in A$. What is the ideal sheaf of $\tau_a(X)\subset A$? Can it be obtained by pulling back $\mathscr I_X$ via $\tau_a$? Thanks for your kind help!

Answer 1

Now solved.

If $\mathscr I_X$ is the ideal sheaf of $X\subset A$, then the ideal sheaf of $T_a(X)\subset X$ is $T_{-a}^\ast\mathscr I_X$.

Indeed, for any morphism $f:Z\to Y$ and a subvariety $Y^\prime\subset Y$ with ideal $I$, the ideal of $Z\times_YY^\prime\subset Z$ is $f^\ast I\cdot \mathscr O_Z$. Apply this with $f=T_{-a}:A\to A$ and $Y^\prime = X$.