Canonical lifts and delta-structures

Lance Gurney (University of Amsterdam)

Wednesday 27th February 16:00-17:00 Maths 311B

Abstract

The Serre-Tate theorem says that ordinary abelian varieties over finite fields can be "canonically" lifted to the Witt vectors of the base field. However, both ordinariness and Witt vectors make sense over any base S where p is nilpotent. This leads to the question: can a family of ordinary abelian schemes A/S be "canonically" lifted to the Witt vectors W(S)?

In the first half of the talk I'll recast the notion of "canonical" lifts in terms of a universal property of Witt vectors leading to the statement: "A theory of canonical lifts for a moduli problem is a delta-structure on the corresponding moduli space." In the second half I'll give various examples and in doing so answer the question above in the affirmative: every family of ordinary abelian schemes A/S can be canonically lifted to the Witt vectors W(S).

This is joint work with J. Borger.

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