TITLE: isomorphism of abelian varieties
QUESTION [12 upvotes]: Let $A, B, C$ and $D$ be abelian varieties (over $\mathbb{C}$) such that $A \times B \cong C \times D$, and $A \cong C$.  From the irreducibility of abelian varieties, we can say that $B$ and $D$ are isogeneous.  But do we actually have $B \cong D$?

REPLY [20 votes]: This is false even for elliptic curves over $\mathbb{C}$. This was proved by T. Shioda in "Some remarks on abelian varieties"  J. Fac. Sci. Univ. Tokyo Sect. IA Math.  24  (1977), no. 1, 11-21, http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6164/1/jfs240102.pdf.