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Let $p$ be a prime. Let $V$ be a discrete valuation ring of mixed characteristic $(0,p)$ and index of ramification $e$. Let $f: G rightarrow H$ be a homomorphism of finite flat commutative group schemes of $p$ power order over $V$ whose generic fiber is an isomorphism. We provide a new proof of a result of Bondarko and Liu that bounds the kernel and the cokernel of the special fiber of $f$ in terms of $e$. For $e < p-1$ this reproves a result of Raynaud. Our bounds are sharper that the ones of Liu, are almost as sharp as the ones of Bondarko, and involve a very simple and short method. As an application we obtain a new proof of an extension theorem for homomorphisms of truncated Barsotti--Tate groups which strengthens Tates extension theorem for homomorphisms of $p$-divisible groups.
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Let $cO_K$ be a complete discrete valuation ring of residue characteristic $p>0$, and $G$ be a finite flat group scheme over $cO_K$ of order a power of $p$. We prove in this paper that the Abbes-Saito filtration of $G$ is bounded by a simple linear function of the degree of $G$. Assume $cO_K$ has generic characteristic 0 and the residue field of $cO_K$ is perfect. Fargues constructed the higher level canonical subgroups for a Barsotti-Tate group $cG$ over $cO_K$ which is not too supersingular. As an application of our bound, we prove that the canonical subgroup of $cG$ of level $ngeq 2$ constructed by Fargues appears in the Abbes-Saito filtration of the $p^n$-torsion subgroup of $cG$.
Let $(R, mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/mathfrak{m} = mathbb{F}_{q}$. Given a monic polynomial $P(t) in R[t]$ whose reduction modulo $mathfrak{m}$ gives an irreducible polynomial $bar{P}(t) in mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $mathrm{coker}(P(A))$, where $A in mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $mathrm{Mat}_{n}(R)$ of $n times n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the $bar{P}$-part of a random matrix $bar{A} in mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime $p$, any finite abelian $p$-group (i.e., $mathbb{Z}_{p}$-module) $H$ occurs as the $p$-part of the class group of a random imaginary quadratic field extension of $mathbb{Q}$ with a probability inversely proportional to $|mathrm{Aut}_{mathbb{Z}}(H)|$. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems. For proofs, we use some concrete combinatorial connections between $mathrm{Mat}_{n}(R)$ and $mathrm{Mat}_{n}(mathbb{F}_{q})$ to translate our problems about a Haar-random matrix in $mathrm{Mat}_{n}(R)$ into problems about a random matrix in $mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution.
We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when $k$ is replaced by an appropriate $K$-regular, not necessarily commutative $k$-algebra.
Let $p$ be a prime. Let $(R,ideal{m})$ be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over $Spec Rsetminus{ideal{m}}$ extends to an abelian scheme over $Spec R$. We show that such extensions always exist if $ele p-1$, exist in most cases if $ple ele 2p-3$, and do not exist in general if $ege 2p-2$. The case $ele p-1$ implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring $O$ of mixed characteristic $(0,p)$ and index of ramification at most $p-1$. This leads to large classes of examples of Neron models over $O$. If $p>2$ and index $p-1$, the examples are new.
Let $Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $lambda$ of the field of algebraic numbers which is prime to p, consider the $lambda$-adic pro-semisimple completion of $Pi$ as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on $lambda$. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its crystalline analog (proved by T. Abe) using a reconstruction theorem in the spirit of Kazhdan-Larsen-Varshavsky. We also formulate two related conjectures, each of which is a reciprocity law involving a sum over all $l$-adic cohomology theories (including the crystalline theory for $l=p$).
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