Twisted divided powers and applications

Abstract

In order to give a formal treatment of differential equations in positive characteristic p, it is necessary to use divided powers. One runs into an analog problem in the theory of q-difference equations when q is a pth root of unity. We introduce here a notion of twisted divided powers (relative to q) and show that one can recover the twisted Weyl algebra and obtain a twisted p-curvature map that describes the center of the twisted Weyl algebra. We also build a divided p-Frobenius that will give, by duality, a formal Azumaya splitting of the twisted Weyl algebra as well as a twisted Simpson correspondence.

Twisted differential operators of negative level and prismatic crystals

“…We recast here some results from [GLQ19] (see also section 2 of [GSQ20]) and extend them to negative level.…”

“…as shown in lemma 1.2 of [GLQ19]. We will need to understand how blowing-up and frobenius act on twisted powers and we can already notice the following:…”

“…Let us now prove our assertion. Existence follows from propositions 2.2 and 2.1 in [GLQ19]. For uniqueness, we may first replace A with some polynomial ring (in an infinite number of variables) over Zrqs, then replace Zrqs with its fraction field Qpqq and finally rely on proposition 2.1 of [GLQ19] again.…”

“…commutative. This requires some work but this is done in section 7 of [GLQ19] where we showed that rF s is explicitly given by…”

Twisted calculus on affinoid algebras

“…Actually, there exists also a formal confluence theorem (theorem 9.13 of [LQ18a]) in this situation but it is a lot more technical. This is however very interesting because, as the first author showed with Michel Gros in [GL14] (see also the more recent [GLQ17]), there exists a quantum Simpson correspondence when q is a root of unity. One can hope that an ultrametric version of these theorems could provide an ultrametric Simpson correspondence.…”