Canonical Lift Research Articles

Tate's Conjecture for K3 Surfaces of Finite Height

This paper extends the proof [16] of the Tate conjecture for ordinary K3 surfaces over a finite field to the more general case of all K3's of finite height. As in [16], our method is to find a of the K3 to characteristic zero with sufficiently many Hodge cycles. In the ordinary case, the so-called canonical lifting of Deligne and Illusie [7] did the job, and a study of the Galois action on p-adic etale cohomology revealed the Hodge cycles. Here we use more general quasi-canonical liftings, and the action of the crystalline Weil group on

𝜔-linear vector fields on manifolds

The classical study of a flow near a fixed point is generalized by composing, at each point in the manifold, the flow derivative with a parallel translation back along the flow. Circumstances under which these compositions form a one-parameter group are studied. From the point of view of the linear frame bundle, the condition is that the canonical lift commute with its horizontal part (with respect to some metric connection). The connection form applied to the lift coincides with the infinitesimal generator of the one-parameter group. Analysis of this matrix provides dynamical information about the flow. For example, if such flows are equicontinuous, they have uniformly bounded derivatives and therefore the enveloping semigroup is a Lie transformation group. Subclasses of ergodic, minimal, and weakly mixing flows with integral invariants are determined according to the eigenvalues of the matrices. Such examples as Lie algebra flows, infinitesimal affine transformations, and the geodesic flows on manifolds of constant negative curvature are examined.

A canonical lift of Frobenius in Morava $E$-theory

We prove that the $p$th Hecke operator on the Morava $E$-cohomology of a space is congruent to the Frobenius mod $p$. This is a generalization of the fact that the $p$th Adams operation on the complex $K$-theory of a space is congruent to the Frobenius mod $p$. The proof implies that the $p$th Hecke operator may be used to test Rezk's congruence criterion.