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تسجيل مستخدم جديدEquivariant formal group laws and complex-oriented spectra over primary cyclic groups: Elliptic curves, Barsotti-Tate groups, and other examples
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We explicitly construct and investigate a number of examples of $mathbb{Z}/p^r$-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and $p$-divisible groups, as well as some other related examples.
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Let $S$ be the spectrum of a complete discrete valuation ring with fraction field of characteristic 0 and perfect residue field of characteristic $pgeq 3$. Let $G$ be a truncated Barsotti-Tate group of level 1 over $S$. If ``$G$ is not too supersingu
lar, a condition that will be explicitly expressed in terms of the valuation of a certain determinant, we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fibre to a subgroup scheme of $G$, finite and flat over $S$. We call it the canonical subgroup of $G$.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,din mathbb{N}$ be such that $h=c+d>0$. Let $H$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. For $minmathbb{N}^ast$ let $H[p^m]=ker([p^m]:Hrightarrow H
)$. It is a finite commutative group scheme over $k$ of $p$ power order, called a Barsotti-Tate group of level $m$. We study a particular type of $p$-divisible groups $H_pi$, where $pi$ is a permutation on the set ${1,2,dots,h}$. Let $(M,varphi_pi)$ be the Dieudonne module of $H_pi$. Each $H_pi$ is uniquely determined by $H_pi[p]$ and by the fact that there exists a maximal torus $T$ of $GL_M$ whose Lie algebra is normalized by $varphi_pi$ in a natural way. Moreover, if $H$ is a $p$-divisible group of codimension $c$ and dimension $d$ over $k$, then $H[p]cong H_pi[p]$ for some permutation $pi$. We call these $H_pi$ canonical lifts of Barsotti-Tate groups of level $1$. We obtain new formulas of combinatorial nature for the dimension of $boldsymbol{Aut}(H_pi[p^m])$ and for the number of connected components of $boldsymbol{End}(H_pi[p^m])$.
Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same
codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $gamma_D(i):=dim(pmb{Aut}(D[p^i]))$. We show that $n_Dle cd$, $(gamma_D(i+1)-gamma_D(i))_{iinBbb N}$ is a decreasing sequence in $Bbb N$, for $cd>0$ we have $gamma_D(1)<gamma_D(2)<...<gamma_D(n_D)$, and for $min{1,...,n_D-1}$ there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$. Different generalizations to $p$-divisible groups with a smooth integral group scheme in the crystalline context are also proved.
Let $G$ be a finite cyclic group of order $n ge 2$. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdot ... cdot (n_lg)$ where $gin G$ and $n_1,..., n_l in [1,ord(g)]$, and the index $ind (S)$ of $S$ is defined as the minimum of $(n_
1+ ... + n_l)/ord (g)$ over all $g in G$ with $ord (g) = n$. In this paper we prove that a sequence $S$ over $G$ of length $|S| = n$ having an element with multiplicity at least $frac{n}{2}$ has a subsequence $T$ with $ind (T) = 1$, and if the group order $n$ is a prime, then the assumption on the multiplicity can be relaxed to $frac{n-2}{10}$. On the other hand, if $n=4k+2$ with $k ge 5$, we provide an example of a sequence $S$ having length $|S| > n$ and an element with multiplicity $frac{n}{2}-1$ which has no subsequence $T$ with $ind (T) = 1$. This disproves a conjecture given twenty years ago by Lemke and Kleitman.
Let $p$ be a prime. Let $R$ be a regular local ring of dimension $dge 2$ whose completion is isomorphic to $C(k)[[x_1,ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $hin C(k)[[x_1,ldots,x_d]]$ such that its
reduction modulo $p$ does not belong to the ideal $(x_1^p,ldots,x_d^p)+(x_1,ldots,x_d)^{2p-2}$ of $k[[x_1,ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $text{Frac}(R)$ which extends to every local ring of $text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $ele p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.
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