Is it true that if an isogeny between two principally polarized abelian varieties respects the polarization, then it is in fact an isomorphism?



Morphism between polarized abelian varieties
That should be true, yes.


A polarization of $A$ is given by a bilinear form on $H\_1(A, Z)$; this is equivalent to a map $H\_1(A,Z) \to H\_1(A,Z)^\vee$, which is an isomorphism if the polarization is principal.


A map between two abelian varieties is given by a corresponding linear map $H\_1(A\_1, Z) \to H\_1(A\_2, Z)$. The map between the varieties is an isomorphism if the map on $H\_1$ is.


The map induced on the bilinear form then is the composition
$$
H\_1(A\_1,Z)\to H\_1(A\_2,Z) \to H\_1(A\_2,Z)^\vee \to H\_1(A\_1,Z)^\vee
$$
If the form is respected by this map, then this is an isomorphism. Consequently, the left-hand map must be as well, as claimed.


