I'm new to abelian varieties. Here's the question:
Let $f: A\to B$ be a homomorphism of $n$-dimensional abelian varieties with real multiplication by $\mathcal{O}_L$, where $[L:\mathbb{Q}] = n$. If $f$ is $\mathcal{O}_L$-linear and non-zero, then $f$ is an isogeny. Finiteness of the kernel is easy if assuming surjectivity, but I don't see why $\mathcal{O}_L$-linearity gives the surjectivity.