Ramification filtration via deformations

Abrashkin, Viktor Aleksandrovich

doi:10.1070/sm9322

“…The form is related to the behaviour of ramification subgroups under the map π H : I −→ I(H) via explicit description of the ramification filtration from [1], cf. also [5].…”

Section: Consider H ∈ Mγ Liementioning

confidence: 99%

“…Describe briefly the construction of ω(H). The group morphism I −→ I(H) admits an explicit description via the nilpotent version of Artin-Schreier theory from [1,2,5]. This allows us to specify a choice of a basis m = (m 1 , .…”

Section: Consider H ∈ Mγ Liementioning

confidence: 99%

Ramification filtration and differential forms

Abrashkin¹

2021

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Let K = k 0 ((t 0 )) where [k 0 : F p ] < ∞. Denote by K tr the maximal tamely ramified extension of K in K sep and letsatisfying the following condition. Suppose F = r∈Q t −r 0 l r , where all l r ∈ L(H) ⊗ Fp . For v > 0, consider the minimal ideal L(H) (v) in L(H) such that L(H) (v) ⊗ Fp contains all l r wirh r v. Then the image of the ramification subgroup in the upper numbering ΓThe form ω(H) is defined in terms of the matrices of Frobenius and connection on the φ-module associated with H. In the end of paper we sketch how the same result can be obtained for arbitrary Z p [Γ K ]-modules.

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“…In this case l H determines the group homomorphism from Γ K to the opposite group G 0 (L(H)) (this group is isomorphic to G(L(H)) via the map g → g −1 ). The results from the papers [8]- [10] were obtained in terms of the contravariant version, but the results from [11]- [13], [19] used the covariant version. We can easily switch from one theory to another via the automorphism − id L(H) .…”

Section: Now We Can Use the Identificationmentioning

confidence: 99%

Ramification filtration and differential forms

Abrashkin

2023

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Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$.

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Ramification filtration and differential forms

Abrashkin

¹

2023

Известия Российской академии наук. Серия математическая

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Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$. Bibliography: 21 titles.

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Ramification filtration via deformations

Abstract: Let be a field of formal Laurent series with coefficients in a finite field of characteristic , Show more

Cited by 4 publications

References 15 publications

Ramification filtration and differential forms

Ramification filtration and differential forms

Ramification filtration and differential forms

Ramification filtration and differential forms

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