Pure and Applied Mathematics Quarterly Volume 2, Number 1 (Special Issue: In honor of John H. Coates, Part 1 of 2 ) 29|110, 2006 Modular Curves and Rigid-Analytic Spaces Brian Conrad Contents 1. Introduction 30 1.1. Motivation 30 1.2. Overview of results 30 1.3. Notation and terminology 32 2. Analytic generalized elliptic curves 33 2.1. Basic de¯nitions over a rigid-analytic space 33 2.2. Global structure of generalized elliptic curves over rigid spaces 41 3. Universal structures 52 3.1. Relative rigid Tate curves and a variant on Berthelot's functor 52 3.2. Degenerate ¯bers and modular curves 58 4. Canonical subgroups 69 4.1. The p-torsion case 69 4.2. Higher torsion-level canonical subgroups 75 4.3. Frobenius lifts 81 Appendix A. Some properties of morphisms 86 Received July 23, 2005. 1991 Mathematics Subject Classi¯cation. Primary 14G22; Secondary 14H52. This research was partially supported by NSF grant DMS-0093542, the Alfred P. Sloan Foun- dation, and the Clay Mathematics Institute. I would like to thank L. Lipshitz and M. Temkin for helpful discussions, and K. Buzzard for sustained encouragement while this circle of ideas evolved. 30 Brian Conrad A.1. Properties of ¯nite morphisms 86 A.2. Change of base ¯eld 90 Appendix B. Algebraic theory of generalized elliptic curves 101 B.1. Deligne{Rapoport semistable curves of genus 1 101 B.2. Generalized elliptic curves over schemes 103 B.3. Reduction-type and formal canonical subgroups 106 References 109 1. Introduction 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 de¯ne a relative theory of a \canonical subgroup" in p-adic fam- ilies of elliptic curves whose reduction types are good but not too supersingular. The theory initiated by Katz has been re¯ned 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 tech- niques in rigid geometry, the de¯nitions and results in the theory can be made applicable to families over rather general rigid-analytic spaces over arbitrary ana- lytic extensions k=Qp (including base ¯elds 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 ¯bers), 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 de¯nitions and theorems, and to avoid unnecessary reliance on the ¯ne structure of integral models of modular curves. The 1-dimensional case exhibits special features (such as moduli-theoretic com- pacti¯cation 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 possi- ble 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. 1.2. Overview of results. The starting point for our work is [C3], where we develop a rigid-analytic theory of relative ampleness for line bundles and a rigid- analytic theory of fpqc descent for both morphisms and proper geometric objects Modular Curves and Rigid-Analytic Spaces 31 equipped with a relatively ample line bundle. (For coherent sheaves, the theory of fpqc descent was worked out by Bosch and GÄortz[BG].) In x2.1{x2.2 we use [C3] to construct a global theory of generalized elliptic curves over arbitrary rigid- analytic spaces. Without a doubt, the central di±culty is to establish ¯niteness properties for torsion, such as ¯niteness of the morphism [d]: Esm ! Esm when E ! S is a rigid-analytic generalized elliptic curve (with relative smooth locus Esm) such that the non-smooth ¯bers are geometrically d-gons. Such ¯niteness properties are used as input in the proof of \algebraicity" locally on the base for any rigid-analytic generalized elliptic curve E ! S, as well as algebraicity for the action map on E by the S-group Esm that is generally not quasi-compact over S. The main ingredient in these arguments is the compatibility of representing objects for algebraic and rigid-analytic Hilbert and Hom functors. To illustrate the general theory, in x3.1 we discuss relative Tate curves from a viewpoint that applies to higher-dimensional moduli spaces for which reasonable proper integral models are available. In x3.2 we prove rigid-analytic universality of analyti¯ed algebraic modular curves (the point being to include the cusps on an equal footing). Our arguments are applied in the higher-dimensional setting in [C4], and so we use methods that may appear to be more abstract than necessary for the case of relative dimension 1. In x4.1 these results are applied over general analytic extension ¯elds k=Qp to develop a relative rigid-analytic theory of the canonical subgroup of Katz and Lubin. The ¯ne structure of modular curves does not play as dominant a role as in other treatments of the theory, and the key to our method is a ¯bral criterion (Theorem A.1.2) for a quasi-compact, at, and separated map of rigid spaces to be ¯nite. Canonical subgroups with higher torsion level are treated in x4.2, where we show that our non-inductive de¯nition for canonical subgroups is equivalent to the inductive one used in [Bu]; in particular, an ad hoc hypothesis on Hasse invariants in the inductive de¯nition in [Bu] is proved to be necessary and su±cient for existence under the non-inductive de¯nition. We conclude in x4.3 by relating level-n canonical subgroups to characteristic-0 lifts of the kernel of the n-fold relative Frobenius map modulo p1¡" for arbitrarily small " > 0. In Appendix A we review some background results in rigid geometry for which we do not know references. In Appendix B we review results of Deligne{Rapoport and Lubin that are used in our work with generalized elliptic curves in the rigid- analytic case. We suggest that the reader look at the de¯nitions in x2.1 and then immediately skip ahead to x4, only going back to the foundational x2{x3 as the need arises (in proofs, examples, and so on). 32 Brian Conrad 1.3. Notation and terminology. If S is a scheme, we sometimes write X=S 0 to denote an S-scheme and X=S0 to denote the base change X £S S over an S- scheme S0. If X ! S is a at morphism locally of ¯nite presentation, we denote by Xsm the (Zariski-open) relative smooth locus on X. We use the same notation in the rigid-analytic category. A non-archimedean ¯eld is a ¯eld k equipped with a non-trivial non-archimedean absolute value with respect to which k is complete. An analytic extension ¯eld k0=k is an extension k0 of k endowed with a structure of non-archimedean ¯eld such that its absolute value extends the one on k. All rigid spaces will tacitly be assumed to be overp a ¯xed non-archimedean ground ¯eld k unless we say otherwise. We write jk£j ⊆ (0; 1) to denote the divisible subgroup of (0; 1) \generated" by the value group of the absolute value on k£. When we say GAGA over ¯elds we mean that the analyti¯cation functor from proper k-schemes to proper rigid-analytic spaces over k is fully faithful, together with the compatibility of analyti¯cation with ¯ber products and the fact that var- ious properties of a map between ¯nite type k-schemes (such as being ¯nite, at, smooth, an isomorphism, or a closed immersion) hold on the algebraic side if and only if they hold on the analytic side. As a special case, if A is a ¯nite k-algebra then a proper A-scheme (resp. proper rigid space over Sp(A)) can be viewed as a proper k-scheme (resp. proper rigid space over k), so when we are restricting attention to the subcategories of objects and morphisms over such an A, we shall say GAGA over artin rings. The two other aspects of the GAGA principle that we shall frequently invoke are (i) the elementary fact that Xan(Sp(A)) = X(Spec A) for any ¯nite k-algebra A and any locally ¯nite type k-scheme X, and (ii) the less trivial fact that 1-dimensional proper rigid-analytic spaces over k are algebraic (this applies to a proper at rigid-analytic curve over Sp(A) for a ¯nite k-algebra A), the proof of which goes exactly as in the complex-analytic case. The analyti¯cation of a map between locally ¯nite type k-schemes is quasi- separated if and only if the scheme map is separated, so to avoid unnecessary separatedness restrictions on the algebraic side when we consider change of the base ¯eld on the rigid-analytic side it is convenient to introduce a variant on quasi- separatedness, as follows. A map of rigid spaces f : X ! Y is pseudo-separated if the diagonal ¢f : X ! X £Y X factors as X ! Z ! X £Y X where the ¯rst step is a Zariski-open immersion and the second is a closed immersion.