Stable reduction of modular curves

Details Stable reduction of modular curves Stable reduction of modular curves

2002-10-23

arxiv.org/abs/math/0210363v1

Mathematics

Algebraic Geometry

18 pages, 2 figures

Scientific paper

We determine the stable reduction at $p$ of all three point covers of the projective line with Galois group ${\rm SL}_2(p)$. As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves $X_0(p)$ and $X_1(p)$. Our method does not use the fact that modular curves are moduli spaces. Instead, we rely on results of Raynaud and the authors which describe the stable reduction of three point covers whose Galois group is strictly divisible by $p$.

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