分野別セミナー

Let $\mathscr{O}_K$ be a $p$-adic discrete valuation ring
with perfect residue field $k$. We classify $p$-divisible groups
and $p$-power order finite flat group schemes over $\mathscr{O}_K$
in terms of certain Frobenius modules over $\mathfrak{S}:=W(k)[[u]]$.
We also show the compatibility with crystalline Dieudonn\'e theory
and Tate module functor (as Galois representation).
The classification was fully known when $p>2$, and for
connected $p$-divisible groups and finite flat group schemes
for any $p$. (Both cases are due to Kisin.)
So we focus on the case with $p=2$, and will explain the motivation
and application of the classification, as well as the sketch
of the proof (especially when $p=2$).


Independently, Eike~Lau generalized display theory to arbitrary
$p$-divisible groups (allowing $p=2$).
Our approach differs from Lau's and we additionally recover
the Tate module from the classification, while Lau's proved
the classification over more general base without recovering
Tate module from the classification.