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Does the product $Y_1(Np) \times Y_1(Np)$ admit a semistable model over $\mathbf{Z}_p[\zeta_p]$ with a natural moduli-space interpretation?

Less telegraphically: let $p$ be a prime, and $N \ge 4$ coprime to $p$. Then Katz and Mazur define in their book Arithmetic moduli of elliptic curves a semistable model of $Y_1(Np)$ over $R = \mathbf{Z}_p[\zeta_p]$, which has an interpretation as a moduli space for elliptic curves with level $N$ structure plus a "balanced level $\Gamma_1(p)$" Drinfeld level structure of determinant $\zeta_p$.

In general the product of two semistable curves is not semistable; one has to make some blow-ups. If I'm not mistaken, applying the methods of this paper gives a scheme $\mathscr{Y} \twoheadrightarrow Y_1(Np)^2$ over $R$ which is an isomorphism away from points $(E_1, E_2)$ with $E_i$ both supersingular, which get blown up to a $\mathbf{P}^1$. [Edit: As Will Sawin points out, there are two ways of doing this, and we choose the one where flipping the two factors corresponds to $z \mapsto z^{-1}$ on $\mathbf{P}^1$.]

This seems to match up with the observation in Katz--Mazur (remark 1.10.4) that the group scheme $\alpha_p \times \alpha_p$ has a whole $\mathbf{P}^1$ of $\alpha_p$ subgroups inside it, and the zero-section is a Drinfeld generator of all of them at once. I'm wondering if this is "the same $\mathbf{P}^1$" as appears in $\mathscr{Y}$.

Can one interpret my $\mathscr{Y}$ as a moduli space for pairs $(E_1, E_2)$ of elliptic curves with $\Gamma_1(p)^{\mathrm{bal}}$ structures $(P_i, Q_i)$ on $E_i$, together with the additional data of a degree $p$ subgroup $C \subset E_1 \times E_2$ such that $(P_1, P_2)$ and $(Q_1, Q_2)$ are Drinfeld generators of $C$ and $C^\vee$? [EDIT: Now answered -- see below.]

Second question, motivated by the fact that scaling both $P_1$ and $P_2$ by the same unit in $(\mathbf{Z}/p)^\times$ won't change the subgroup $C$:

Does there exist a semistable $R$-model of $(Y_1(Np) \times Y_1(Np)) / \Delta$ with a natural moduli interpretation, where $\Delta = (\mathbf{Z} / p)^\times$ acting via the diagonal action of diamond operators $(\langle d\rangle, \langle d \rangle)$?

EDIT: Using the answers to my question Homomorphisms between Oort–Tate group schemes, I realised that the answer to the first question is "yes" -- this follows by applying the Tate-Oort classification of order $p$ group schemes, and using the description of $X_1(p)$ in terms of Tate-Oort theory given in Deligne-Rapoport. I still don't know the answer to the second question so I will leave this open for now.

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    $\begingroup$ The supersingular points will be nodes, with local equations of the form $xy = p^v$. So the product of two supersingular points will be given by an equation of the form $xy = zw = p^v$. So the resolution should be the same as the resolution for a space with equation $xy=zw$, where there's two different ways to blow it up to a $\mathbb P^1$. I think at most one of them can be this moduli space, but possibly one of them is. $\endgroup$
    – Will Sawin
    Commented Jan 12, 2022 at 16:17
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    $\begingroup$ Blowing up along the component with ideal $(x,z)$ introduces a $\mathbb P^1$ with coordinate $x/z$. Similarly the other four introduce $\mathbb P^1$s with coordinates $x/w$, $y/z$, $y/w$. We have $x/z =w/y$ and $x/w=z/y$ so there are two pairs that give isomorphic blowups, but I don't think the blowups are isomorphic between the pairs. Specifically, I think you want to blowup either the component where both subgroups are $\mathbb Z/p$ or the component where both subgroups are $\mu_p$ so that the coordinate of your new $\mathbb P^1$ will get inverted as you swap the two elliptic curves. $\endgroup$
    – Will Sawin
    Commented Jan 12, 2022 at 16:41
  • $\begingroup$ Yes, that certainly seems like the "natural" choice for the moduli space interpretation. $\endgroup$
    – David Loeffler
    Commented Jan 12, 2022 at 16:48

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