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Register a new userCanonical subgroups of Barsotti-Tate groups
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Authors
Yichao Tian
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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 supersingular, 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 $G$.
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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 $p$ 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 $Y$.
Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fite, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimuras theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in Gap, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.
We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).
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