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Let $\mathcal{A}_{g,D}$ be the moduli space of abelian varieties of dimension $g$ and polarization $D$ of type $(d_1, \ldots, d_g)$.

Let $\mathcal{M}$ be the moduli space parametrizing pairs $(A, \mathcal{L})$, where $A \in \mathcal{A}_{g,D}$ and $\mathcal{L}$ is a non-trivial $2$-torsion line bundle on $A$, i.e. a non-zero element of $\textrm{Pic}^0(A)$ such that $\mathcal{L}^{\otimes 2}=\mathcal{O}_A$.

Then there is a covering

$\pi \colon \mathcal{M} \to longrightarrow \mathcal{A}_{g,D}$

of degree $2^{2g}-1$, given by $[A, \mathcal{L}] \to A$.

Question Is the monodromy group of $\pi$ transitive? Or, equivalently, is $\mathcal{M}$ connected?

The answer is yes when $g=1$, i.e. for elliptic curves. In fact in this case $\mathcal{M}$ is a particular case of a more general construction called the moduli space of spin curves, which was studied by several authors (Cornalba, Verra, Farkas, etc).

What about the case $g \geq 2$? Is there any reference? I'm particularly interested to the case where $g=2$ and $D=(1,2)$.

EDIT Let me explain better the case I'm interested in, hoping that this can be helpful. Let $(A, D)$ be an abelian surface with polarization of type $(1,2)$, which I assume to be not of product type. The linear system $|D|$ is a pencil, that is $h^0(D)=2$, its general element is irreducible and up to a translation we can take $\mathcal{O}_A(D)$ symmetric, i.e. $(-1)_A^* \mathcal{O}_A(D)= \mathcal{O}_A(D)$. Therefore the base locus of $|D|$ is given by the zero element $o$ of $A$ and by three $2$-division points $e_1$, $e_2$, $e_3$ such that $e_1+e_2=e_3$.

There are exactly three $2$-torsion line bundles $\mathcal{L}_1$, $\mathcal{L}_2$, $\mathcal{L}_3$ on $A$ such that there exists an element in the "translated pencil" $|D \otimes \mathcal{L}_i|$ having a double point at $o$ (which is easily proven to be a node). The set

$\{\mathcal{O}_A, \mathcal{L}_1, \mathcal{L}_2, \mathcal{L}_3\}$

form a subgroup of $\textrm{Pic}^0(A)$ isomorphic to $\mathbb{Z}/(2) \times \mathbb{Z}/(2)$.mathbb{Z}/(2)$, which is exactly the image of the map

$\phi \colon A[2] \longrightarrow \textrm{Pic}^0(A)[2]$,

see Laurent Moret-Bailly's answer.
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Since there is no way to distinguish between non-trivial $2$-torsion line bundles on $A$, my guess is that the

The answer should be yes.

This is true yes when $g=1$, i.e. for elliptic curves. In fact in this case $\mathcal{M}$ is a particular case of a more general construction called the moduli space of spin curves, which was studied by several authors (Cornalba, Verra, Farkas, etc).

EDIT Let me explain better the case I'm interested in, hoping that this can be helpful. Let $(A, D)$ be an abelian surface with polarization of type $(1,2)$, which I assume to be not of product type. The linear system $|D|$ is a pencil, that is $h^0(D)=2$, its general element is irreducible and up to a translation we can take $\mathcal{O}_A(D)$ symmetric, i.e. $(-1)_A^* \mathcal{O}_A(D)= \mathcal{O}_A(D)$.Therefore the base locus of $|D|$ is given by the zero element $o$ of $A$ and by three $2$-division points $e_1$, $e_2$, $e_3$ such that $e_1+e_2=e_3$.

There are exactly three $2$-torsion line bundles $\mathcal{L}_1$, $\mathcal{L}_2$, $\mathcal{L}_3$ on $A$ such that there exists an element in the "translated pencil" $|D \otimes \mathcal{L}_i|$ having a double point at $o$ (which is easily proven to be a node). The set

$\{\mathcal{O}_A, \mathcal{L}_1, \mathcal{L}_2, \mathcal{L}_3\}$

form a subgroup of $\textrm{Pic}^0(A)$ isomorphic to $\mathbb{Z}/(2) \times \mathbb{Z}/(2)$.
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$2$-torsion line bundles on abelian varieties

Let $\mathcal{A}_{g,D}$ be the moduli space of abelian varieties of dimension $g$ and polarization $D$ of type $(d_1, \ldots, d_g)$.

Let $\mathcal{M}$ be the moduli space parametrizing pairs $(A, \mathcal{L})$, where $A \in \mathcal{A}_{g,D}$ and $\mathcal{L}$ is a non-trivial $2$-torsion line bundle on $A$, i.e. a non-zero element of $\textrm{Pic}^0(A)$ such that $\mathcal{L}^{\otimes 2}=\mathcal{O}_A$.

Then there is a covering

$\pi \colon \mathcal{M} \to \mathcal{A}_{g,D}$

of degree $2^{2g}-1$, given by $[A, \mathcal{L}] \to A$.

Question Is the monodromy group of $\pi$ transitive? Or, equivalently, is $\mathcal{M}$ connected?

Since there is no way to distinguish between non-trivial $2$-torsion line bundles on $A$, my guess is that the answer should be yes.

This is true when $g=1$, i.e. for elliptic curves. In fact in this case $\mathcal{M}$ is a particular case of a more general construction called the moduli space of spin curves, which was studied by several authors (Cornalba, Verra, Farkas, etc).

What about the case $g \geq 2$? Is there any reference? I'm particularly interested to the case where $g=2$ and $D=(1,2)$.