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Traverso’s isogeny conjecture for pdivisible groups
, 2008
"... Let k be an algebraically closed field of characteristic p> 0. Let c, d ∈ N. Let bc,d ≥ 1 be the smallest integer such that for any two pdivisible groups H and H ′ over k of codimension c and dimension d the following assertion holds: If H[pbc,d ′ bc,d ′] and H [p] are isomorphic, then H and H ..."
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Let k be an algebraically closed field of characteristic p> 0. Let c, d ∈ N. Let bc,d ≥ 1 be the smallest integer such that for any two pdivisible groups H and H ′ over k of codimension c and dimension d the following assertion holds: If H[pbc,d ′ bc,d ′] and H [p] are isomorphic, then H and H are isogenous. We show that bc,d = ⌈ cd c+d⌉. This proves Traverso’s isogeny conjecture for pdivisible groups over k.
Stratifications of Newton polygon strata and Traverso’s conjectures for pdivisible groups
, 2012
"... The isomorphism number (resp. isogeny cutoff) of a pdivisible group D over an algebraically closed field of characteristic p is the least positive integer m such that D[pm] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of ..."
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The isomorphism number (resp. isogeny cutoff) of a pdivisible group D over an algebraically closed field of characteristic p is the least positive integer m such that D[pm] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of pdivisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of pdivisible 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[pm] to D.
De JongOort purity for pdivisible groups, in
 Progress in Mathematics 270, Birkhäuser, 2010. Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115
"... De JongOort purity states that for a family of pdivisible groups X → S over a noetherian scheme S the geometric fibres have all the same Newton polygon if this is true outside a set of codimension bigger than 2. A more general result was first proved in [JO] and an alternative proof is given in [V ..."
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De JongOort purity states that for a family of pdivisible groups X → S over a noetherian scheme S the geometric fibres have all the same Newton polygon if this is true outside a set of codimension bigger than 2. A more general result was first proved in [JO] and an alternative proof is given in [V1]. We present
Subtle invariants for pdivisible groups and Traverso’s conjectures
"... We report on the joint works [1] and [2] aimed towards the classification of pdivisible groups over an algebraically closed field k of positive characteristic p. Let W (k) be the ptypical Witt ring of k. Let σ be the Frobenius automorphism of W (k). Let D be a pdivisible group over k of positive ..."
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We report on the joint works [1] and [2] aimed towards the classification of pdivisible groups over an algebraically closed field k of positive characteristic p. Let W (k) be the ptypical Witt ring of k. Let σ be the Frobenius automorphism of W (k). Let D be a pdivisible group over k of positive height r and let (M, φ, ϑ) be its (contravariant) Dieudonné module. Thus M is a free W (k)module of rank r, φ: M → M is a σlinear endomorphism, and ϑ: M → M is a σ −1linear endomorphism such that we have φ ◦ ϑ = ϑ ◦ φ = p1M. Let d = dimk(M/φ(M)) and c = dimk(M/ϑ(M)) be the dimension and the codimension (respectively) of D. We have c + d = r. Let t: = min{c, d}. 2. What one would like to achieve? • Classify all D’s. This means: (a) list all isomorphism classes with c and d fixed; (b) decide when another pdivisible group E over k specializes to D in the sense below; (c) understand the abstract groups Hom(E[p m], D[p m]) for all m ∈ N ∗ ; and (d) identify good invariants of D (which go up or down under specializations). • Refine the Newton polygon stratifications associated to pdivisible groups over Fpschemes using good invariants. • Generalize to quadruples of the form (M, φ, ϑ, G), where G is an integral, closed subgroup subscheme of GLM subject to the axioms of [1], Definition 2. [Here we will concentrate on the case G = GLM which corresponds simply to D.] Definition 1. We say that E specializes to D if there exists a pdivisible group D over k[[x]] such that Dk = D and D k((x)) is isomorphic to E k((x)) 3. Classical invariants It is well known that we have a direct sum decomposition of Fisocrystals