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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
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