Elliptic Crystals and Modular Motives

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: 4E W. These data are required to satisfy various compatibilities: for example, A(k E) is the kernel of idk 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 X k be the reduction modulo p of a smooth log curve Y W over W, let (E, B) be an elliptic crystal on Y W, and let (E, A) be the restriction of (E, B) to X k. 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 Y W or on X k. If f: X$ X is a morphism of smooth log curves over k then a lifting g: Y$ Y of f determines a lifting g*(BEY) ( f *E)Y$ of A( f *E)X$ k . 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 *EX is a bijection. As a consequence we show (reprising some of the results of Mochizuki) in Theorem (2.4) that (when the degree of AEX is positive and p is odd) an indigenous bundle on X k determines a unique lifting Y W, characterized by the fact doi:10.1006 aima.2001.1976, available online at http: www.idealibrary.com on