Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Lance Gurney
Affiliation:

University of Amsterdam
Date:

Wed, 2019-04-24 14:30 - 15:30 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.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3444

[3] https://www.mpim-bonn.mpg.de/node/246