Abstract
Modular curves are quotients of the upper half-plane by congruence subgroups of SL2(ℤ). The most prominent examples are X 0 (N),X1 (N) and X(N), for N ≥ 1. Modular curves are also moduli spaces for elliptic curves endowed with a level structure. Therefore, they are defined over small number fields and have a rich arithmetic structure. Deligne and Rapoport [4] determine the reduction behavior of X 0 (p) and X 1 (p) at the prime p. Using this result, they prove a conjecture of Shimura saying that the quotient of the Jacobian of X 1 (p) by the Jacobian of X 0 (p) acquires good reduction over ℚ(ςp). Both the reduction result and its corollary have been generalized by Katz and Mazur [10] to arbitrary level N and various level structures.
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Bouw, I.I., Wewers, S. (2004). Stable Reduction of Modular Curves. In: Cremona, J.E., Lario, JC., Quer, J., Ribet, K.A. (eds) Modular Curves and Abelian Varieties. Progress in Mathematics, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7919-4_1
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