Torsion-stabilized modular curves of level $p$

This is the first paper of a project on new integral models $\mathcal{X}(N)$ of the modular curve $X(N)$.

The final results for a general level $N$ will be obtained in the second paper, while this paper is devoted to giving all necessary background and definitions applicable to any $N$ and then working out the case of $\mathcal{X}(p)$ with all possible details.

We define $\mathcal{X}(N)$ as the closure of $Y(N)$ in the space $\overline{\mathcal{M}}_{1,N^2}=\overline{\mathcal{M}}_{1,\Gamma}$, where $\Gamma=(\mathbb{Z}/N\mathbb{Z})^2$, and show that for $N=p$ it is the blowup of the Katz-Mazur model $\widetilde{\mathcal{X}}(p)$ at all supersingular points, and hence $(\mathcal{X}(p),Y(p))$ is the minimal toroidal resolution of $(\widetilde{\mathcal{X}}(p),Y(p))$.

In fact, it is even log smooth over $(\mathbb{Z},\mathbb{Z}[1/p])$, but this is special for the case when $p=N$.

One can tautologically view $\mathcal{X}(p)$ as the moduli space of $\Gamma$-stabilized genus-1 curves $(E,\Gamma)$ which can be smoothed to an elliptic curve labelled by its $N$-torsion, but our main results provide explicit criteria of the smoothability: $\mathcal{X}(p)$ parameterizes $\Gamma$-equivariant stable genus-1 curves $(E,\Gamma)$ such that the action satisfies two explicit conditions formulated in the paper.