Given a prime number $p>2$, I'm looking for a smooth projective hyperelliptic curve $C$ defined over $\mathbb{Q}$ whose Jacobian $J(C)$ has a subgroup isomorphic to $\mu_p^2$ as a $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$-module (or, if you prefer, as a group scheme over $\mathbb{Q}$). Equivalently, there exists a connected Galois cover of $C$ (defined over $\mathbb{Q}$) with group $(\mathbb{Z}/p\mathbb{Z})^2$.

I am looking for such an example in order to illustrate a technique that allows to specialize covers of curves to class groups of number fields.

Remark 1: if $p=3$ or $p=5$, then such examples are known, using the following trick: one looks for a hyperelliptic curve $C$ whose Jacobian is isomorphic to the product of two elliptic curves, each of them containing a $\mu_p$ subgroup.

Remark 2: if I ask the same question but replacing $\mu_p^2$ by $(\mathbb{Z}/p\mathbb{Z})^2$, then the answer is positive, as shows the following example: let $C_p$ be the hyperelliptic curve defined by the affine equation $y^2=x^p+1$. It is clear from the identity $x^p=(y-1)(y+1)$ that $J(C_p)(\mathbb{Q})$ contains an element of order $p$. Let $C_{2p}$ be the hyperelliptic curve with equation $y^2=x^{2p}+1$. Then one checks that there are two independent maps $C_{2p}\to C_p$ of degree $2$. Using the existence of these covers, one obtains an isogeny $J(C_p)\times J(C_p)\to J(C_{2p})$ with degree a power of $2$. Therefore, $J(C_{2p})(\mathbb{Q})$ contains a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^2$. Equivalently, there exists a connected $\mu_p^2$-torsor over $C_{2p}$.


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