Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

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Ramification Groups of Abelian Local Field Extensions

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-027-9 ◽

1971 ◽

Vol 23 (2) ◽

pp. 271-281 ◽

Cited By ~ 10

Author(s):

Murray A. Marshall

Keyword(s):

Prime Ideal ◽

Local Field ◽

Galois Extension ◽

Residue Class ◽

Abelian Extensions ◽

Ring Of Integers ◽

Field Extensions ◽

Valued Field ◽

Residue Class Field ◽

Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

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Riesz Type Kernels over the Ring of Integers of a Local Field

Journal of Mathematical Analysis and Applications ◽

10.1006/jmaa.1997.5352 ◽

1997 ◽

Vol 208 (2) ◽

pp. 528-552 ◽

Cited By ~ 6

Author(s):

Shijun Zheng

Keyword(s):

Local Field ◽

Ring Of Integers

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Norm subgroups of formal groups over a local field

Ukrainian Mathematical Journal ◽

10.1007/bf01101662 ◽

1977 ◽

Vol 28 (4) ◽

pp. 409-412

Author(s):

G. T. Konovalov

Keyword(s):

Local Field ◽

Formal Groups

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Compatible systems and ramification

Compositio Mathematica ◽

10.1112/s0010437x19007619 ◽

2019 ◽

Vol 155 (12) ◽

pp. 2334-2353 ◽

Cited By ~ 1

Author(s):

Qing Lu ◽

Weizhe Zheng

Keyword(s):

Finite Field ◽

Finite Type ◽

Local Field ◽

Ring Of Integers

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

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