Minimal slope conjecture of F-isocrystals

Tsuzuki, Nobuo

doi:10.1007/s00222-022-01146-5

Cited by 2 publications

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References 35 publications

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“…Since , we have . By the recent work [Tsu23] and its improvement done in [D'A23, Theorem 4.1.3], one has so that . Hence, the natural composite map is injective and it induces an isomorphism .…”

Section: On the Injectivity Of The éTale Abel–jacobi Mapmentioning

confidence: 83%

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Perfect points of abelian varieties

Ambrosi

2023

Compositio Math.

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Let $p$ be a prime number, $k$ a finite field of characteristic $p>0$ and $K/k$ a finitely generated extension of fields. Let $A$ be a $K$ -abelian variety such that all the isogeny factors are neither isotrivial nor of $p$ -rank zero. We give a necessary and sufficient condition for the finite generation of $A(K^{\mathrm {perf}})$ in terms of the action of $\mathrm {End}(A)\otimes \mathbb {Q}_p$ on the $p$ -divisible group $A[p^{\infty }]$ of $A$ . In particular, we prove that if $\mathrm {End}(A)\otimes \mathbb {Q}_p$ is a division algebra, then $A(K^{\mathrm {perf}})$ is finitely generated. This implies the ‘full’ Mordell–Lang conjecture for these abelian varieties. In addition, we prove that all the infinitely $p$ -divisible elements in $A(K^{\mathrm {perf}})$ are torsion. These reprove and extend previous results to the non-ordinary case.

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Section: On the Injectivity Of The éTale Abel–jacobi Mapmentioning

confidence: 83%

“…Since is semisimple in , we can apply recent advances in -adic cohomology (see [Tsu23] and its improvement done in [D'A23]) to construct, from the splitting of , an idempotent in with the desired properties, which, since is finitely generated over a finite field, lifts to , by the -adic Tate conjecture for abelian varieties.…”

Section: Introductionmentioning

confidence: 99%

Perfect points of abelian varieties

Ambrosi

2023

Compositio Math.

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Etale and crystalline companions, I

Kedlaya¹

2022

Épijournal De Géométrie Algébrique

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Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell \neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$. Using the Langlands correspondence for global function fields in both the \'etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general $X$; building on work of Deligne, Drinfeld showed that any \'etale coefficient object has \'etale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has \'etale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of \'etale coefficient objects; this subject will be pursued in a subsequent paper.

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Minimal slope conjecture of F-isocrystals

Cited by 2 publications

References 35 publications

Perfect points of abelian varieties

Perfect points of abelian varieties

Etale and crystalline companions, I

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