Computing isomorphism numbers of F -crystals using the level torsions

Xiao, Xiao

doi:10.1016/j.jnt.2012.05.035

Cited by 2 publications

(10 citation statements)

References 13 publications

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“…For each

j \in J

, there exists a pair

{scriptT}_{j} = false(B_{j}, η_{j} false)

such that

(() M_{j}, φ_{π_{j}})

is isomorphic to

{scriptM}_{j} false(T_{j} false)

. If

{scriptM}_{j} false(T_{j} false)

is ordinary, then

n_{{scriptM}_{j} false(T_{j} false)} \leq 1

; see [, Proposition 2.9 and Corollary 2.11]. If

{scriptM}_{j} false(T_{j} false)

is non‐ordinary, we have

ℓ_{{scriptM}_{j} false(T_{j} false)} \leq ℓ_{scriptM}

by Lemma .…”

Section: Minimal Bold-italicf‐crystalsmentioning

confidence: 98%

“…Compared to the upper bound provided in [, Theorem 1.2] which works for all isoclinic F ‐crystals, the upper bound provided by Theorem applies only to isosimple F ‐crystals. Although somewhat limited in generality, Theorem does provide a sharper upper bound than the one in [, Theorem 1.2] in some cases and it can also provides optimal upper bounds as well; see Examples and . On the other hand, the upper bound provided in [, Theorem 1.2] can be slightly better than the one in Theorem in some other cases; see Example .…”

Section: Introductionmentioning

confidence: 93%

“…We compare our new upper bound with some other upper bounds in , and in the following examples. Remark The upper bound of Theorem can be restated using λ as

n_{scriptM} \leq 2 \max \{⌊ (r - false⌈ \frac{s}{e} false⌉) λ ⌋, ⌊ ⌊ \frac{s}{e} ⌋ (e - λ) ⌋\} + 1 .

…”

Section: Estimatesmentioning

confidence: 99%

“…For example, Lau, Nicole and Vasiu proved that if

scriptM

is a Dieudonné module with dimension d and codimension c , then

n_{scriptM} \leq ⌊ 2 c d / (c + d) ⌋

and the inequality is optimal in the sense that there exists an isoclinic Dieudonné module

scriptM

such that

n_{scriptM} = ⌊ 2 c d / (c + d) ⌋

. In the case of isoclinic F ‐crystals, upper bounds that are optimal in many cases but not in general have also been found in .…”

Section: Introductionmentioning

confidence: 99%

“…Although somewhat limited in generality, Theorem does provide a sharper upper bound than the one in [, Theorem 1.2] in some cases and it can also provides optimal upper bounds as well; see Examples and . On the other hand, the upper bound provided in [, Theorem 1.2] can be slightly better than the one in Theorem in some other cases; see Example .…”

Section: Introductionmentioning

confidence: 99%

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Minimal F-crystals and isomorphism numbers of isosimple F-crystals

Xiao

2016

Math. Nachr.

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Abstract. In this paper we generalize minimal p-divisible groups defined by Oort to minimal F -crystals over algebraically closed fields of positive characteristic. We prove a structural theorem of minimal F -crystals and give an explicit formula of the Frobenius endomorphism of the basic minimal F -crystals that are the building blocks of the general minimal F -crystals. We then use minimal F -crystals to generalize minimal heights of p-divisible groups and give an upper bound of the isomorphism numbers of F -crystals, whose isogeny type are determined by simple F -isocrystals, in terms of their ranks, Hodge slopes and Newton slopes.

show abstract

“…For each

j \in J

, there exists a pair

{scriptT}_{j} = false(B_{j}, η_{j} false)

such that

(() M_{j}, φ_{π_{j}})

is isomorphic to

{scriptM}_{j} false(T_{j} false)

. If

{scriptM}_{j} false(T_{j} false)

is ordinary, then

n_{{scriptM}_{j} false(T_{j} false)} \leq 1

; see [, Proposition 2.9 and Corollary 2.11]. If

{scriptM}_{j} false(T_{j} false)

is non‐ordinary, we have

ℓ_{{scriptM}_{j} false(T_{j} false)} \leq ℓ_{scriptM}

by Lemma .…”

Section: Minimal Bold-italicf‐crystalsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 93%

“…We compare our new upper bound with some other upper bounds in , and in the following examples. Remark The upper bound of Theorem can be restated using λ as