monodromy on kernel of polarization for abelian surfaces Let $(A,{\mathcal L})$ be a $(1,n)$-polarized complex abelian surface. By general theory, the polarization corresponds to a map $A \to A^\vee$, and let $K({\mathcal L})$ be its kernel. Then $K({\mathcal L}) \simeq {\mathbb Z}/n \times {\mathbb Z}/n$.
Now let $M$ be the moduli space of $(1,n)$-polarized complex abelian surfaces and $a \in M$ corresponding to $(A,{\mathcal L})$. It looks like there should be a monodromy map
$$ \psi:\pi_1(M,a) \longrightarrow \mathrm{Aut}_{gp}( K({\mathcal L}) ). $$
Question. What is the image of $\psi$?
I presume that all the automorphisms in $Im(\psi)$ preserve the skew symmetric pairing (Mumford, Abelian Varieties, p. 227)
$$ K({\mathcal L}) \times K({\mathcal L}) \to {\mathbb C}^\times. $$
Is the converse true?
A reference would be greatly appreciated, though even a confirmation that this is correct is useful. Also, this clearly connects somewhat to the issue of monodromy on cohomology, and a reference for that would also be appreciated.
 A: I am going to discuss the case that $n=p$ is a prime number, the squarefree case can be reduced to this case.
Let $d$ and $N$ be coprime integers (where $d$ is either $p$ or $1$). We will adopt the (fairly standard) notation that $\mathcal{A}_{2,d,N}$ is the moduli (stack) over $\mathbb{C}$ of abelian surfaces $A$ with a polarisation $\lambda$ of degree $d$ and a symplectic level $N$-structure, that is, an isomorphism $ \alpha: A[N] \to (\mathbb{Z}/N \mathbb{Z})^{\oplus 4}$, preserving the symplectic form coming from the Weil-pairing.
The moduli stack of $(1,p)$-polarised abelian surfaces is just $\mathcal{A}_{2,p,1}$.
For $N \ge 3$ the stack $\mathcal{A}_{2,d,N}$ is actually a scheme, and it sometimes useful to work with it instead, the natural map $\mathcal{A}_{2,d,N} \to \mathcal{A}_{2,d,1}$ is a $\operatorname{Sp}_{4}(\mathbb{Z}/N\mathbb{Z})$-torsor. Moreover, $\mathcal{A}_{2,d,N}$ is (geometrically) connected for all choices of $d$ and $N$ (again $d$ is either $1$ or $p$, just to be safe).
Let $K(\lambda)$ be the kernel of $\lambda$ over $\mathcal{A}_{2,p,1}$, this is an étale group scheme over $\mathcal{A}_{2,p,1}$ equipped with a symplectic form. There is a $\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$-torsor $\mathcal{E}$ over $\mathcal{A}_{2,1,p}$ parametrising trivialisations of $K(\lambda)$, but it is clear that there is an $\operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$ torsor $\mathcal{E}' \subset \mathcal{E}$, consisting of trivialisations of $K(\lambda)$ that preserve the symplectic form. Thus for a point $x=(A, \lambda) \in \mathcal{A}_{2,1,p}(\mathbb{C})$ we get a monodromy representation $$\psi:\pi_1^{\text{ét}}(\mathcal{A}_{2,1,p},x) \to \operatorname{SL}_{2}(\mathbb{Z}/p\mathbb{Z}).$$
Claim: The map $\psi$ is surjective.
Let $V$ be a one-dimensional subspace of $(\mathbb{Z}/p \mathbb{Z})^{\oplus 4}$, then there is a morphism
$$f_V:\mathcal{A}_{2,1,p} \to \mathcal{A}_{2,p,1} $$
which takes a principally polarised abelian surface $(B, \mu)$ together with an isomorphism $B[p] \simeq (\mathbb{Z}/p \mathbb{Z})^{\oplus 4}$ to the abelian surface $A=B/V$, together with the polarisation $\lambda$ of degree $p$ induced from $p \cdot \mu$. This polarisation descends because $V \subset \ker p \cdot \mu$ is an isotropic subspace. The kernel of the polarisation $\lambda$ is canonically identified with $V^{\perp}/V$, and so the map $f_V$ factors through $\mathcal{E}' \to \mathcal{A}_{2,p,1}$. It now suffices to show that the map $\mathcal{A}_{2,1,p} \to \mathcal{E}'$ is surjective.
