Search ResultsShowing 1-20 of 37
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arXiv:2306.06588 (Published 2023-06-11)
Waring Problem for Matrices over Finite Fields
Comments: 33 pages. Comments are very welcomeLet $k$ be a natural number. The Waring problem for square matrices over finite fields, solved by Kishore and Singh, asked for the existence of a constant $C(k)$ such that for all finite fields $\mathbb F_q$ with $q\ge C(k)$, each matrix $A\in M_n(\mathbb F_q)$ with $n\ge 2$ is a sum $A=B^k+C^k$ of two $k$-th powers. We present a new proof of this result with the explicit bound $C(k)=(k-1)^4+6k$ and we generalize and refine this result in many cases, including those when $B$ and $C$ are required to be (invertible) (split) semisimple or invertible cyclic, when $k$ is assumed to be coprime to $p$ (the improved bound being $k^3-3k^2+3k$), or when $n$ is big enough.
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arXiv:2208.00286 (Published 2022-07-30)
The $δ$-invariant theory of Hecke correspondences on $\mathcal A_g$
Categories: math.NTLet $p$ be a prime, let $N\geq 3$ be an integer prime to $p$, let $R$ be the ring of $p$-typical Witt vectors with coefficients in an algebraic closure of $\mathbb F_p$, and consider the correspondence $\mathcal A'_{g,1,N,R}\rightrightarrows \mathcal A_{g,1,N,R}$ obtained by taking the union of all prime to $p$ Hecke correspondences on Mumford's moduli scheme of principally polarized abelian schemes of relative dimension $g$ endowed with symplectic similitude level-$N$ structure over $R$-schemes. It is well-known that the coequalizer $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ of the above correspondence exists and is trivial in the category of schemes, i.e., is $\text{Spec}(R)$. We construct and study in detail such a coequalizer (categorical quotient) in a more refined geometry (category) referred to as {\it $\delta$-geometry}. This geometry is in essence obtained from the usual algebraic geometry by equipping all $R$-algebras with {\it $p$-derivations}. In particular, we prove that our substitute of $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ in $\delta$-geometry has the same `dimension' as $\mathcal A_{g,1,N,R}$, thus solving a main open problem in the work of Barc\u{a}u--Buium. We also give applications to the study of various Zariski dense loci in $\mathcal A_{g,1,N,R}$ such as of isogeny classes and of points with complex multiplication. To prove our results we develop a Serre--Tate expansion theory for {\it Siegel $\delta$-modular forms} of arbitrary genus which we then combine with old and new results from the geometric invariant theory of multiple quadratic forms and of multiple endomorphisms.
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arXiv:1809.05141 (Published 2018-09-13)
Purity results for Barsotti--Tate groups in dimension at least 3
Comments: 32 pagesLet $p$ be a prime. Let $R$ be a regular local ring of mixed characteristic $(0,p)$ and of dimension d\ge 3. We assume that its completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ as a Cohen ring of the same residue field $k$ as $R$ and with $h\in C(k)[[x_1,\ldots,x_d]]$ 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 show that each Barsotti--Tate group over the field of fractions of $R$ which extends to every local ring of $R$ 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 an integral regular scheme flat over $\mathbb Z_{(p)}$ and such that the completion of each local ring of $Y$ at a closed point of $Y$ of characteristic $p$ is a ring of formal power series over some complete discrete valuation ring of mixed characteristic $(0,p)$ and absolute index of ramification $e\le p-1$, then each Barsotti--Tate group over the field of fractions 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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arXiv:1210.6629 (Published 2012-10-24)
On the Tate and Langlands--Rapoport conjectures for special fibres of integral canonical models of Shimura varieties of abelian type
Comments: 55 pagesWe prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prove that integral canonical models of compact Shimura varieties of Hodge type that are of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type, are closed subschemes of integral canonical models of Siegel modular varieties.
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Dimensions of group schemes of automorphisms of truncated Barsotti--Tate groups
Comments: 52 pages. Final version as close to the galley proofs as possible. To appear in IMRNLet $D$ 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 $n_D\le cd$, $(\gamma_D(i+1)-\gamma_D(i))_{i\in\Bbb N}$ is a decreasing sequence in $\Bbb N$, for $cd>0$ we have $\gamma_D(1)<\gamma_D(2)<...<\gamma_D(n_D)$, and for $m\in\{1,...,n_D-1\}$ 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.
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Stratifications of Newton polygon strata and Traverso's conjectures for p-divisible groups
Comments: 50 pages, to appear in Annals of MathematicsSubjects: 14L05The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D.
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Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristic
Comments: 25 pages. Final version to appear in J. Number TheoryJournal: J. Number Theory 132 (2012), no. 9, 2003-2019Keywords: finite flat group schemes, discrete valuation ring, mixed characteristic, boundedness results, strengthens tates extension theoremTags: journal articleLet $p$ 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 Tate's extension theorem for homomorphisms of $p$-divisible groups.
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Purity results for $p$-divisible groups and abelian schemes over regular bases of mixed characteristic
Comments: 28 pages. Final version identical (modulo style) to the galley proofs. To appear in Doc. MathJournal: Doc. Math. 15 (2010), 571--599Tags: journal articleLet $p$ 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 $\Spec R\setminus\{\ideal{m}\}$ extends to an abelian scheme over $\Spec R$. We show that such extensions always exist if $e\le p-1$, exist in most cases if $p\le e\le 2p-3$, and do not exist in general if $e\ge 2p-2$. The case $e\le p-1$ 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 N\'eron models over $O$. If $p>2$ and index $p-1$, the examples are new.
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Three methods to prove the existence of integral canonical models of Shimura varieties of Hodge type
Comments: 15 pages. Survey that grew out from seminar talksThis is a survey of the three main methods developed in the last 15 years to prove the existence of integral canonical models of Shimura varieties of Hodge type. The only new part is formed by corrections to results of Kisin.
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Breuil's classification of $p$-divisible groups over regular local rings of arbitrary dimension
Comments: 20 pages. Final version to appear in Advanced Studies in Pure Mathematics, Proceeding of Algebraic and Arithmetic Structures of Moduli Spaces, Hokkaido University, Sapporo, Japan, September 2007Journal: Advanced Studies in Pure Mathematics 58 (2010), 461-479Keywords: regular local rings, divisible groups, arbitrary dimension, breuils classification, perfect fieldTags: journal articleLet $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.
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Purity of level m stratifications
Comments: Final version 38 pages. To appear in Ann. Sci. \'Ec. Norm. SupLet $k$ be a field of characteristic $p>0$. Let $D_m$ be a $\BT_m$ over $k$ (i.e., an $m$-truncated Barsotti--Tate group over $k$). Let $S$ be a\break $k$-scheme and let $X$ be a $\BT_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}(X)$ is pure in $S$ i.e., the immersion $S_{D_m}(X) \hookrightarrow S$ is affine. For $p\in\{2,3\}$, we prove purity if $D_m$ satisfies a certain property depending only on its $p$-torsion $D_m[p]$. For $p\ge 5$, we apply the developed techniques to show that all level $m$ stratifications associated to Shimura varieties of Hodge type are pure.
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arXiv:0712.1840 (Published 2007-12-11)
Geometry of Shimura varieties of Hodge type over finite fields
Comments: 47 pages. Enlarged version of the three lectures we gave in July 2007 during the summer school "Higher dimensional geometry over finite fields", June 25 - July 06, 2007, Mathematisches Institut, Georg-August-Universit\"at G\"ottingenJournal: Proceedings of the NATO Advanced Study Institute on Higher dimensional geometry over finite fields, G\"ottingen, Germany, June 25 - July 06 2007, 197--243, IOS Press.Tags: lecture notes, journal articleWe present a general and comprehensive overview of recent developments in the theory of integral models of Shimura varieties of Hodge type. The paper covers the following topics: construction of integral models, their possible moduli interpretations, their uniqueness, their smoothness, their properness, and basic stratifications of their special fibres.
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Good Reductions of Shimura Varieties of Hodge Type in Arbitrary Unramified Mixed Characteristic, Part II
Comments: 29 pagesWe prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.
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Good Reductions of Shimura Varieties of Hodge Type in Arbitrary Unramified Mixed Characteristic, Part I
Comments: 56 pages. Up-dated version based on connection made with local models and parahoric subgroupsWe prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.
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Level m stratifications of versal deformations of p-divisible groups
Comments: 35 pages. Accepted (in final form) for publication in J. Alg. GeomJournal: J. Alg. Geom. 17 (2008), no. 4, 599-641Tags: journal articleLet $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d,m$ be positive integers. Let $D$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. Let $\scrD$ be a versal deformation of $D$ over a smooth $k$-scheme $\scrA$ which is equidimensional of dimension $cd$. We show that there exists a reduced, locally closed subscheme $\grs_D(m)$ of $\scrA$ that has the following property: a point $y\in\scrA(k)$ belongs to $\grs_D(m)(k)$ if and only if $y^*(\scrD)[p^m]$ is isomorphic to $D[p^m]$. We prove that $\grs_D(m)$ is {\it regular and equidimensional} of {\it dimension} $cd-\dim(\pmb{\text{Aut}}(D[p^m]))$. We give a proof of {\it Traverso's formula} which for $m>>0$ computes the codimension of $\grs_D(m)$ in $\scrA$ (i.e., $\dim(\pmb{\text{Aut}}(D[p^m]))$) in terms of the Newton polygon of $D$. We also provide a criterion of when $\grs_D(m)$ satisfies the {\it purity property} (i.e., it is an affine $\scrA$-scheme). Similar results are proved for {\it quasi Shimura $p$-varieties of Hodge type} that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic $(0,p)$.
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Deformation subspaces of p-divisible groups as formal Lie groups associated to p-divisible groups
Comments: 36 pages. To appear in J. Alg. GeomJournal: J. Alg. Geom., Vol. 20 (2011), no. 1, 1-45Keywords: p-divisible groups, formal lie groups, deformation subspaces, formal deformation space, construct formal subschemesTags: journal articleLet $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$ which is not isoclinic. Let $\scrD$ (resp. $\scrD_k$) be the formal deformation space of $D$ over $\Spf(W(k))$ (resp. over $\Spf(k)$). We use axioms to construct formal subschemes $\scrG_k$ of $\scrD_k$ that: (i) have canonical structures of formal Lie groups over $\Spf(k)$ associated to $p$-divisible groups over $k$, and (ii) give birth, via all geometric points $\Spf(K)\to\scrG_k$, to $p$-divisible groups over $K$ that are isomorphic to $D_K$. We also identify when there exist formal subschemes $\scrG$ of $\scrD$ which lift $\scrG_k$ and which have natural structures of formal Lie groups over $\Spf(W(k))$ associated to $p$-divisible groups over $W(k)$. Applications to Traverso (ultimate) stratifications are included as well.
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Reconstructing $p$-divisible groups from their truncations of small level
Comments: 32 pages. Final version identical with the galley proofs (modulo style). Paper dedicated to the memory of Angela Vasiu. To appear in Comment. Math. HelvJournal: Comment. Math. Helv. 85 (2010), no. 1, 165--202DOI: 10.4171/CMH/192Keywords: divisible group, small level, traverso truncation conjecture, isomorphic, smallest non-negative integerTags: journal articleLet $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$. Let $n_D$ be the smallest non-negative integer for which the following statement holds: if $C$ is a $p$-divisible group over $k$ of the same codimension and dimension as $D$ and such that $C[p^{n_D}]$ is isomorphic to $D[p^{n_D}]$, then $C$ is isomorphic to $D$. To the Dieudonn\'e module of $D$ we associate a non-negative integer $\ell_D$ which is a computable upper bound of $n_D$. If $D$ is a product $\prod_{i\in I} D_i$ of isoclinic $p$-divisible groups, we show that $n_D=\ell_D$; if the set $I$ has at least two elements we also show that $n_D\le\max\{1,n_{D_i},n_{D_i}+n_{D_j}-1|i,j\in I, j\neq i\}$. We show that we have $n_D\Le 1$ if and only if $\ell_D\Le 1$; this recovers the classification of minimal $p$-divisible groups obtained by Oort. If $D$ is quasi-special, we prove the Traverso truncation conjecture for $D$. If $D$ is $F$-cyclic, we compute explicitly $n_D$. Many results are proved in the general context of latticed $F$-isocrystals with a (certain) group over $k$.
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Traverso's isogeny conjecture for p-divisible groups
Comments: 8 pages, laTex; to appear in Rend. Sem. Mat. Univ. PadovaJournal: Rend. Semin. Mat. Univ. Padova 118 (2007), 73--83Keywords: traversos isogeny conjecture, p-divisible groups, smallest integer, assertion holds, characteristicTags: journal articleLet $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d\in\dbN$. Let $b_{c,d}\ge 1$ be the smallest integer such that for any two $p$-divisible groups $H$ and $H^\prime$ over $k$ of codimension $c$ and dimension $d$ the following assertion holds: If $H[p^{b_{c,d}}]$ and $H^\prime[p^{b_{c,d}}]$ are isomorphic, then $H$ and $H^\prime$ are isogenous. We show that $b_{c,d}=\lceil{cd\over {c+d}}\rceil$. This proves Traverso's isogeny conjecture for $p$-divisible groups over $k$.
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Minimal truncations of supersingular p-divisible groups
Comments: 9 pages, LaTex; to appear in Indiana Univ. Math. JJournal: Indiana Univ. Math. J. {\bf 56} (2007), no. 6, pp. 2887-2897Keywords: supersingular p-divisible group, minimal truncations, traversos truncation conjecture, height 2d, isomorphismTags: journal articleLet k be an algebraically closed field of characteristic p>0. Let H be a supersingular p-divisible group over k of height 2d. We show that H is uniquely determined up to isomorphism by its truncation of level d (i.e., by H[p^d]). This proves Traverso's truncation conjecture for supersingular p-divisible groups. If H has a principal quasi-polarization \lambda, we show that (H,\lambda) is also uniquely determined up to isomorphism by its principally quasi-polarized truncated Barsotti--Tate group of level d (i.e., by (H[p^d],\lambda[p^d])).
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Integral models in unramified mixed characteristic (0,2) of hermitian orthogonal Shimura varieties of PEL type, Part II
Comments: 24 pages, final version to appear in Math. Nachr. Part I is available as math.NT/0307205We construct relative PEL type embeddings in mixed characteristic (0,2) between hermitian orthogonal Shimura varieties of PEL type. We use this to prove the existence of integral canonical models in unramified mixed characteristic (0,2) of hermitian orthogonal Shimura varieties of PEL type.