Higher-level Canonical Subgroups in Abelian Varieties

1.1. Motivation. Let E be an elliptic curve over a p-adic integer ring R, and assume that E has supersingular reduction. Consider the 2-dimensional Fp-vector space of characteristic-0 geometric p-torsion points in the associated 1-parameter formal group Ê over R. It makes sense to ask if, in this vector space, there is a line whose points x are nearer to the origin than are all other points (with nearness measured by |X(x)| for a formal coordinate X of Ê over R; the choice of X does not affect |X(x)|). Such a subgroup may or may not exist, and when it does exist it is unique and is called the canonical subgroup. This notion was studied by Lubin [Lu] in the more general context of 1-parameter commutative formal groups, and its scope was vastly extended by Katz [K] in the relative setting for elliptic curves over p-adic formal schemes and for analytified universal elliptic curves over certain modular curves. Katz’ ideas grew into a powerful tool in the study of p-adic modular forms for GL2/Q. The study of p-adic modular forms for more general algebraic groups and number fields, going beyond the classical case of GL2/Q, leads to the desire to have a theory of canonical subgroups for families of abelian varieties. (See [KL] for an application to Hilbert modular forms.) Ideally, one wants such a theory that avoids restrictions on the nature of formal (or algebraic) integral models for the family of abelian varieties, but it should also be amenable to study via suitable formal models (when available). In this paper we develop such a theory, and our viewpoint and methods are different from those of other authors who have recently worked on the problem (such as [AM], [AG], [GK], and [KL]). The theory in this paper (in conjunction with methods in [KL]) has been recently used by K. Tignor to construct 1-parameter p-adic families of non-ordinary automorphic forms on some 3-variable general unitary groups associated to CM fields. Roughly speaking, if A is an abelian variety of dimension g over an analytic extension field k/Qp (with the normalization |p| = 1/p) then a level-n canonical subgroup Gn ⊆ A[p] is a k-subgroup with geometric fiber (Z/pZ) such that (for k ∧ /k a completed algebraic closure) the points in Gn(k ∧ ) ⊆ A[pn](k) are nearer to the identity than are all other points in A[p](k ∧ ). Here, nearness is defined in terms of absolute values of coordinates in the formal group of the unique formal semi-abelian “model” AR′ for A over the valuation ring R′ of a sufficiently large finite extension k′/k. (See Theorem 2.1.9 for the characterization of AR′ in terms of the analytification A, and see Definitions 2.2.5 and 2.2.7 for the precise meaning of “nearness”.) In particular, Gn[p] is a level-m canonical subgroup for all 1 ≤ m ≤ n. By [C4, Thm. 4.2.5], for g = 1 this notion of higher-level canonical subgroup is (non-tautologically) equivalent to the one defined in [Bu] and [G]. (Although the theory for n > 1 can be recursively built from the case n = 1 when g = 1, which is the viewpoint used in [Bu] and [G], it does not seem that this is possible when g > 1 because it is much harder to work with multi-parameter formal groups than with one-parameter formal groups.) In contrast with the Galois-theoretic approach in [AM], our definition of canonical subgroups is not intrinsic to the torsion subgroups of A but rather uses the full structure of the formal group of a formal semi-abelian model AR′ . Moreover, since our method is geometric rather than Galois-theoretic it can be applied at the level of geometric points (where Galois-theoretic methods are not applicable). If a level-n