The field-of-norms functor and the Hilbert symbol for higher local fields

Abrashkin, Victor; Jenni, Ruth

doi:10.5802/jtnb.786

Cited by 12 publications

(4 citation statements)

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“…This is easily checked directly, and is a special case of [AJ12, Proposition 5.2], which proves a generalization to higher-dimensional local fields; see also [Lau88], where the analogous result is proved for general APF extensions (strictly speaking, the result of [Lau88] does not apply as written in our situation, as the extension is not Galois; but, in fact, the argument still works). In brief, it is enough to check separately the cases that is either unramified or totally ramified; in the former case the result is immediate, while the latter case follows from Dwork’s description of Artin’s reciprocity map for totally ramified abelian extensions [Ser79, XIII §5 Corollary to Theorem 2].◻…”

Section: Resultsmentioning

confidence: 59%

Explicit Serre weights for two-dimensional Galois representations

Calegari¹,

Emerton²,

Gee³

et al. 2017

Compositio Math.

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Section: Resultsmentioning

confidence: 59%

Explicit Serre weights for two-dimensional Galois representations

Calegari¹,

Emerton²,

Gee³

et al. 2017

Compositio Math.

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“…(here exp is the truncated exponential). The series ω(t) p is a kind of approximation of the p-adic period of the formal multiplicative group, which appears usually in explicit formulas for the Hilbert symbol (e.g., [17]). In other words, we obtain another geometric condition characterizing "arithmetic" of the Γ E0 -module H E0 .…”

Section: Now We Can Use the Identificationmentioning

confidence: 99%

Ramification filtration and differential forms

Abrashkin

2023

Izv. Math.

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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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“…The correspondence L → G(L) induces equivalence of the categories of finite Lie Z/p M -algebras and finite p-groups of exponent p M of the same nilpotent class 1 s 0 < p. This equivalence can be extended to the similar categories of profinite Lie algebras and groups. [8,9] Let…”

Section: Categories Of P-groups and Liementioning

confidence: 99%