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Canonical Lifts in Families and delta-structures

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Speaker: 
Lance Gurney
Affiliation: 
University of Amsterdam
Date: 
Wed, 2019-04-24 14:30 - 15:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Serre-Tate proved in the 60's that an ordinary abelian variety E over a finite field k can be canonically lifted to an ordinary abelian variety over the Witt vectors of k. Expressed in terms of the moduli stack M_ord of ordinary abelian schemes, this is precisely a universal lifting property with respect to the Witt vectors (of finite fields).

I will explain how this universal lifting property of M_ord actually holds for arbitrary morphisms, or in other words, every ordinary abelian scheme over a base scheme S on which p is locally nilpotent can be canonically lifted to the Witt vectors W(S) of S.
 
This fact intimately related with the notion of a delta-structure (Buium, Joyal), which encodes (in an intelligent way) a lift of the Frobenius. The Witt vectors are the universal such example of a scheme with a delta-structure and so it follows formally that a delta-structure on a scheme is equivalent to a universal lifting property with respect to the Witt vectors. Thus, the above result can instead be interpreted as saying that the stack M_ord has a delta-structure.
 
Joint work with J. Borger.
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