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 \(\bar{\mathbb F}_q\):
The projective line \(\mathbb P^1\)
The curve with affine model \(xy^q-x^qy=1\)
The curve with affine model \(y^q+y=x^{q+1}\)
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.


14G35 Modular and Shimura varieties
11G05 Elliptic curves over global fields
Full Text: DOI arXiv


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