Elliptic Crystals and Modular Motives

Abstract

The goal of this paper is to investigate the crystalline properties of families of elliptic curves or, more precisely, to study the families of crystals (elliptic crystals) attached to families of elliptic curves.Elliptic crystals over a point can be defined and classified very simply in terms of semilinear algebra, and this is how we begin (1.1). Our definition is cast in the logarithmic setting, so that it applies also to semistable (degenerate) elliptic curves. Roughly speaking, an elliptic crystal over a (logarithmic) perfect field k of characteristic p is a free W-module E of rank two equipped with a Frobenius-linear endomorphism 8, a nilpotent linear endomorphism N, a two-step Hodge filtration A on k E, and an isomorphism tr: 4 2 E Ä W. These data are required to satisfy various compatibilities: for example, A 1 (k E) is the kernel of id k 8. When N is not zero, such a crystal is quite rigid, and in particular its canonical coordinates are very precisely determined (1.3). Liftings of elliptic crystals from k to its Witt ring are determined by specifying a lifting B of the Hodge filtration A of k E, and the classification is made explicit in (1.5).Our first main result (2.2) concerns the relationship between deformation theory and elliptic crystals on a curve. Let XÂk be the reduction modulo p of a smooth log curve YÂW over W, let (E, B) be an elliptic crystal on YÂW, and let (E, A) be the restriction of (E, B) to XÂk. Following Mochizuki [8] and Gunning [6], we shall say that E is indigenous if its Kodaira-Spencer mapping is an isomorphism, a condition which can be checked either on YÂW or on XÂk. If f: X$ Ä X is a morphism of smooth log curves over k then a lifting g: Theorem (2.2) shows that if p is odd and E is indigenous, the resulting map from the set of liftings of f to the set of liftings of the mod p Hodge filtration of f *E X is a bijection. As a consequence we show (reprising some of the results of Mochizuki) in Theorem (2.4) that (when the degree of A 1 E X is positive and p is odd) an indigenous bundle on XÂk determines a unique lifting YÂW, characterized by the fact

Integral canonical models of Shimura varieties of preabelian type

“…see also [Fa1, and the reference to it in 2.2.1 c)) reduces the situation to the case when D is the 2-divisible group of an ordinary elliptic curve. For this last case we refer to [Og,3.14] (from loc. cit.…”

“…c) For any supersingular elliptic curve E s over k we get a good additive coordinate t of its moduli formal scheme M 1 of deformations. This reobtains [Og,3.15]. Warning: in loc.…”

“…So x i dz i = d(ln(z i + 1)). Such a connection allows us to treat the situation as if n = 1: as the product of any two elements of Lie( Ñ ), viewed as endomorphisms of M , is 0, based on [Og,3.14], we get immediately that the number of Frobenius lifts of R n which makes g R n to be the identity element is at least 2 n . All such lifts are obtained (cf.…”

Frobenius and the Hodge Spectral Sequence

“…All these results apply to cohomology with coefficients in an F-crystal, and in a subsequent article [6] we shall analyze (as our main motivating example) the F-crystals introduced by Scholl in his study of p-adic properties of modular forms of higher weight. It was in fact Scholl who suggested that I apply the machinery of [7] to his crystals, and the difficulties I encountered in attempting to carry out this calculation led eventually to the ideas developed here.…”

“…Thus our result generalizes the original result of Mazur. As we shall see in a subsequent article [6], even when this is no longer the case, it is sometimes possible to use additional information provided by a closer examination of the spectral sequences andÂor duality to determine the Hodge numbers of 8 exactly.…”