On a purely inseparable analogue of the Abhyankar conjecture for affine curves

Otabe, Shusuke

doi:10.1112/s0010437x18007194

Cited by 6 publications

(22 citation statements)

References 35 publications

(98 reference statements)

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“…For example, finite local torsors , which are sometimes called purely inseparable coverings , over X make the group scheme

π^{normalN} false(X false)

larger. In [29], the author formulated a purely inseparable analogue of the Abhyankar conjecture to estimate the difference between these two fundamental groups

π_{1}^{ét} false(X false)

and

π^{normalN} false(X false)

from the viewpoint of the inverse Galois problem. Question (Purely inseparable analogue of the Abhyankar conjecture [29, Question 3.3].)…”

Section: Introductionmentioning

confidence: 99%

“…In [29], the author formulated a purely inseparable analogue of the Abhyankar conjecture to estimate the difference between these two fundamental groups

π_{1}^{ét} false(X false)

and

π^{normalN} false(X false)

from the viewpoint of the inverse Galois problem. Question (Purely inseparable analogue of the Abhyankar conjecture [29, Question 3.3].) Let U be an affine smooth curve over an algebraically closed field k of positive characteristic

p > 0

and let G be a finite local k ‐group scheme.…”

Section: Introductionmentioning

confidence: 99%

“…Here recall that if a finite local k ‐group scheme G appears as a quotient of

π^{normalN} false(U false)

, then the character group

X (G)

must be embedded into the group

{true(Q_{p} / Z_{p} true)}^{\oplus γ + n - 1}

(cf. [29, Proposition 3.1]).…”

Section: Introductionmentioning

confidence: 99%

“…This is a generalization of one of the previous results due to the author (cf. [29, Proposition 3.4]). Corollary (Cf.…”

Section: Introductionmentioning

confidence: 99%

“…In §4, after reviewing a recent work due to Biswas–Borne on tamely ramified torsors [7, §3], we shall give a proof of Theorem 1.8. The idea is that a quite similar argument as in the proof of [29, Proposition 3.4] still works even if one replaces an affine curve U by the pro‐system of root stacks

{true{\sqrt[p^{boldr}]{boldD / X_{0}} true}}_{r}

. The main ingredients of the proof are Lieblich's work on twisted sheaves [24] (cf.…”

Section: Introductionmentioning

confidence: 99%

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An embedding problem for finite local torsors over twisted curves

Otabe

2021

Mathematische Nachrichten

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In his previous paper, the author proposed as a problem a purely inseparable analogue of the Abhyankar conjecture for affine curves in positive characteristic and gave a partial answer to it, which includes a complete answer for finite local nilpotent group schemes. In the present paper, motivated by the Abhyankar conjectures with restricted ramifications due to Harbater and Pop, we study a refined version of the analogous problem, based on a recent work on tamely ramified torsors due to Biswas–Borne, which is formulated in terms of root stacks. We study an embedding problem to conclude that the refined analogue is true in the solvable case.

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“…For example, finite local torsors , which are sometimes called purely inseparable coverings , over X make the group scheme

π^{normalN} false(X false)

larger. In [29], the author formulated a purely inseparable analogue of the Abhyankar conjecture to estimate the difference between these two fundamental groups

π_{1}^{ét} false(X false)

and

π^{normalN} false(X false)

from the viewpoint of the inverse Galois problem. Question (Purely inseparable analogue of the Abhyankar conjecture [29, Question 3.3].)…”

Section: Introductionmentioning

confidence: 99%

“…In [29], the author formulated a purely inseparable analogue of the Abhyankar conjecture to estimate the difference between these two fundamental groups

π_{1}^{ét} false(X false)

and

π^{normalN} false(X false)

p > 0

and let G be a finite local k ‐group scheme.…”

Section: Introductionmentioning

confidence: 99%

“…Here recall that if a finite local k ‐group scheme G appears as a quotient of

π^{normalN} false(U false)

, then the character group

X (G)

must be embedded into the group

{true(Q_{p} / Z_{p} true)}^{\oplus γ + n - 1}

(cf. [29, Proposition 3.1]).…”

Section: Introductionmentioning

confidence: 99%

“…This is a generalization of one of the previous results due to the author (cf. [29, Proposition 3.4]). Corollary (Cf.…”

Section: Introductionmentioning

confidence: 99%

{true{\sqrt[p^{boldr}]{boldD / X_{0}} true}}_{r}

. The main ingredients of the proof are Lieblich's work on twisted sheaves [24] (cf.…”

Section: Introductionmentioning

confidence: 99%

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An embedding problem for finite local torsors over twisted curves

Otabe

2021

Mathematische Nachrichten

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A generalized Abhyankar’s conjecture for simple Lie algebras in characteristic p>5

2021

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The arithmetic local Nori fundamental group

Romagny¹,

Tonini

Zhang

2021

Trans. Amer. Math. Soc.

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In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field k k . We give particular attention to the case of fields: to any field extension K / k K/k we attach a pro-local group scheme over k k . We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of the local fundamental group of an open, of its generic point (as it happens for the étale fundamental group) and even of any smooth neighborhood.

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On a purely inseparable analogue of the Abhyankar conjecture for affine curves

Cited by 6 publications

References 35 publications

An embedding problem for finite local torsors over twisted curves

An embedding problem for finite local torsors over twisted curves

A generalized Abhyankar’s conjecture for simple Lie algebras in characteristic p>5

The arithmetic local Nori fundamental group

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