Mathematics – Number Theory
Scientific paper
2010-10-20
Mathematics
Number Theory
32 pages, 3 figures
Scientific paper
We produce an integral model for the modular curve X(Np^n) over the ring of integers of a sufficiently ramified extension of the p-adic field 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^n) in a manner compatible with the transition maps. By nonabelian Lubin-Tate theory, the cohomology of the tower X(Np^n)realizes the local Langlands and Jacquet-Langlands correspondences for GL(2) over a p-adic field; we tie together nonabelian Lubin-Tate theory to the representation-theoretic point of view afforded by Bushnell-Kutzko types. Our work also applies to the Lubin-Tate tower of curves for a local field of positive characteristic, so that one obtains stable models for Drinfeld modular curves as well.
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