## Semistable models for modular curves of arbitrary level.(English)Zbl 1357.14034

Stable or semistable models for the modular curves $$X_0(Np^m)$$ have been known for $$m \leq 3$$ due to the works of many authors – P. Deligne and M. Rapoport [Lect. Notes Math. 349, 143–316 (1973; Zbl 0281.14010)] for $$m = 1$$, B. Edixhoven [Ann. Inst. Fourier 40, No. 1, 31–67 (1990; Zbl 0679.14009)] for $$m = 2$$, K. McMurdy and R. Coleman [Algebra Number Theory 4, No. 4, 357–431 (2010; Zbl 1215.11060)] for $$m = 3$$.
In this paper, for any $$m \geq 1$$, the author produces an integral model for the curve $$X(Np^m)$$ over the ring of integers of a sufficiently ramified extension of $${\mathbb Q}_p$$ whose special fiber is a semistable curve in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of $$X(Np^m)$$, which is a union of copies of a Lubin-Tate curve. The author points out that inasmuch a Lubin-Tate curve (over $${\mathbb Q}_p$$) appears as the rigid space attached to the $$p$$-adic completion of a modular curve at one of its mod $$p$$ supersingular points, the problem of finding a semistable model for a modular curve is essentially the same as finding one for the corresponding Lubin-Tate curve. For technical convenience it is essential here to work with the Lubin-Tate curve not at level $$p^m$$ but rather at infinite level. The author shows that the infinite level Lubin-Tate space (in arbirary dimension, over an arbitrary non-archimedean local field) has the structure of a perfectoid space, which is simpler – one could say “as usual” – than the Lubin-Tate spaces of finite level.
Let us go into some more technical detail. Fix a non-archimedean local field $$K$$ with uniformizer $$\pi$$ and residue field $$k \cong {\mathbb F}_q$$. The Lubin-Tate tower is a projective system of formal schemes $$\mathcal{M}_m$$ which parametrize deformations with level $$\pi^m$$ structure of a one-dimensional formal $$\mathcal{O}_K$$-module of height $$n$$ over $$\bar{\mathbb F}_q$$. After extending scalars to a separable closure of $$K$$, the Lubin-Tate tower admits an action of $$\mathrm{GL}_n(K) \times D^{\times} \times W_K$$, where $$D/K$$ is the central division algebra of invariant $$1/n$$, and $$W_K$$ is the Weil group of $$K$$. Note that the $$\ell$$-adic étale cohomology of the Lubin-Tate tower realizes both the Jacquet-Langlands correspondence (between $$\mathrm{GL}_n(K)$$ and $$D^{\times}$$) and the local Langlands correspondence (between $$\mathrm{GL}_n(K)$$ and $$W_K$$).
Focusing on the case that $$q$$ is odd and $$n = 2$$ (the first step of the so-called “non-abelian Lubin-Tate theory”), the author constructs a compatible family of semi-stable models $$\widehat{\mathcal M}_m$$ for each $${\mathcal M}_m$$ over the ring of integers of a sufficiently ramified extension of $$K$$. This means that the rigid generic fiber of $$\widehat{\mathcal M}_m$$ is the same as that of $${\mathcal M}_m$$, but that the special fiber of $$\widehat{\mathcal M}_m$$ is a locally finitely presented scheme of dimension 1 with only ordinary double points as singularities. Note that it is not possible to arrange for the semi-stable models $$\widehat{\mathcal M}_m$$ to be compatible, i.e., there is no tower $$\dots\to \widehat{\mathcal M}_2 \to \widehat{\mathcal M}_1$$ with finite transition maps because, roughly speaking, the singularities of the $$\widehat{\mathcal M}_m$$, when $$m \to\infty$$, accumulate around the so-called CM points. This problem can be remedied by entirely removing the CM points (in a precise technical way).
The main result of the paper then reads: “Assume that $$q$$ is odd. For each $$m \geq 1$$, there is a finite extension $$L_m/\widehat{K}_{nr}$$ for which $${\mathcal M}_m$$ admits a semistable model $$\widehat{\mathcal M}_m$$; every connected component of the special fiber of $$\widehat{\mathcal M}_m$$ admits a purely inseparable morphism to one of the following smooth projective curves over $$\bar{\mathbb F}_q$$:
1.
The projective line $$\mathbb P^1$$
2.
The curve with affine model $$xy^q-x^qy=1$$
3.
The curve with affine model $$y^q+y=x^{q+1}$$
4.
The curve with affine model $$y^q-y=x^2$$.”
Note that the mere existence of a semistable model of $${\mathcal M}_m$$ (after a suitable finite extension of scalars) follows from the corresponding result about proper algebraic curves. Besides, the Drinfeld-Carayol theorem allows to get hold of the field $$L_m$$ over which a semistable model appears. So the genuine content of the main theorem is the list of curves it produces.

### MSC:

 14G35 Modular and Shimura varieties 11G05 Elliptic curves over global fields

### Keywords:

semistable models; modular curves; Lubin-Tate tower

### Citations:

Zbl 0281.14010; Zbl 0679.14009; Zbl 1215.11060
Full Text:

### References:

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