Modular Curves and Rigid-analytic Spaces

1.1. Motivation. In the original work of Katz on p-adic modular forms [Kz], a key insight is the use of Lubin’s work on canonical subgroups in 1-parameter formal groups to define a relative theory of a “canonical subgroup” in p-adic families of elliptic curves whose reduction types are good but not too supersingular. The theory initiated by Katz has been refined in various directions (as in [AG], [AM], [Bu], [GK], [G], [KL]). The philosophy emphasized in this paper and its sequel [C4] in the higher-dimensional case is that by making fuller use of techniques in rigid geometry, the definitions and results in the theory can be made applicable to families over rather general rigid-analytic spaces over arbitrary analytic extensions k/Qp (including base fields such as Cp, for which Galois-theoretic techniques as in [AM] are not applicable). The aim of this paper is to give a purely rigid-analytic development of the theory of canonical subgroups with arbitrary torsion-level in generalized elliptic curves over rigid spaces over k (using Lubin’s p-torsion theory only over valuation rings, which is to say on fibers), with an eye toward the development of a general theory of canonical subgroups in p-adic analytic families of abelian varieties that we shall discuss in [C4]. An essential feature is to work with rigid spaces and not with formal models in the fundamental definitions and theorems, and to avoid unnecessary reliance on the fine structure of integral models of modular curves. The 1-dimensional case exhibits special features (such as moduli-theoretic compactification and formal groups in one rather than several parameters) and it is technically simpler, so the results in this case are more precise than seems possible in the higher-dimensional case. It is therefore worthwhile to give a separate treatement in the case of relative dimension 1 as we do in the present paper.