Has anyone come across anything along the following lines?

Let $X_1(p)$ be the compactification of the quotient of upper half plane by $\Gamma_1(p)$ for some unspecified large prime. Let $X_1(p) \to \mathbb{P}^n$ be some embedding into some projective space. Has anyone explicitly constructed a flat family of dimension one subschemes of $\mathbb{P}^n$ such that generic fiber is isomorphic to $X$ but a special fiber is something degenerate, yet explicit. For example, has anyone calculated explicitly Grobner basis or some Okounkov-Newton style binomial degeneration?

My interest is primarily in $\Gamma_1(p)$ and embedding by weight one Eisenstein series, but other embeddings and other modular groups (like $\Gamma(p)$ and $\Gamma_0(p)$) are interesting as well.


1 Answer 1


The algebraic stack $m_{\Gamma_0(p)}$ defines in Deligne-Rapoport has a coarse moduli space $M_{\Gamma_0(p)}$, which is a smooth scheme defines over $\mathbb{Z}[1/p]$ (even he is define over $\mathbb{Z}$ by taking the normalization along the $j$-line) and such that $M_{\Gamma_0(p)}(\mathbb{C})$ is the modular curve for $\Gamma_0(p)$. Deligne-Rapoport proved that the special fibre of $M_{\Gamma_0(p)}$ is two projective lines with an ordinary singularities at the supersingular elliptic curves. The two projective lines classify the generalized elliptic curves with etale (resp. multiplicative) subgroup of order $p$.


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