Über die absolute Galoisgruppe dyadischer Zahlkörper.

doi:10.1515/crll.1984.350.152

Cited by 4 publications

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“…In a sense this would mean that this should correspond to a surface of genus N 2 + 1. For the full proof we refer to [11] and [4], it is based on a theory of H. Koch [14] which axiomizes the fact that for every finite, tamely ramified extension L/K the group G L (p) is a Demuskin group.…”

Section: Explicit Outer Automorphismmentioning

confidence: 99%

Surfaces and p-adic fields I: Dehn twists

Gropper¹

2023

Preprint

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Under the philosophy of arithmetic topology, one would like to have an analogy between surfaces S, and p-adic fields K.In the following we describe a point of view which helps look at surfaces and p-adic fields in a "uniform way", and show that results on mapping class groups can be extended to this point of view, and thus be applied to GK , the absolute Galois groups of the p-adic field K.By moving both groups to the world of pro-p groups (for GK we take its maximal pro-p quotient, and for π1(S) we take its pro-p completion), we see they both are pro-p Poincare duality groups of dimension 2, also known as Demuskin groups. Such groups have a very nice classification in terms of generators and relations.Next, in order to move geometric ideas to a purely group theoretic setting, we use the language of graphs of groups and Bass-Serre trees.By restating results for surfaces purely in such a manner, we can (after proving a few technical results) get similar results for p-adic fields.We start by examining discrete groups with Demuskin type relations, for which we show a few pro-p rigidity type results.Using this we show that all splittings of a Demuskin group come from a discrete splitting, which in turn helps us show that Dehn twists make sense in such a context. This gives us a family of infinite order Outer automorphisms of GK (p), the maximal pro-p quotient of GK , which are "arithmetic Dehn twists". On the other hand, when specializing this to Demuskin groups coming from surface groups, one gets back the usual definition of Dehn twists on surfaces.We also get a curve complex for p-adic fields, namely the complex whose edges are graph of groups with GK (p) as their fundamental group, and 2 vertices have an edge between them if the graphs corresponding to the vertices are compatible. By the relations between discrete and pro-p Demuskin groups, we get a nice way to understand said curve complex.As a finally corollary, we show that there in an infinite family of nonisomorphic discrete groups, having isomorphic pro-l completions for all primes l (which are free pro-l for l = p and Demuskin for l = p). his DPhil and before it. In particular he would like to thank his viva examiners, Kobi Kremnitzer and Tomer Schlank, for helpful comments and corrections. He would also like to thank Jonathan Fruchter, Ido Grayevsky and Alex Lubotzky, for helpful discussions on materials related to the current paper, and to Dani Wise for suggesting the use of Whitehead's algorithm in the proof of the final corollary.The author would also like to thank Uri Bader and Tsachik Gelander, and their Midrasha on groups, where he learned most of his geometric group theory background.

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Section: Explicit Outer Automorphismmentioning

confidence: 99%

Surfaces and p-adic fields I: Dehn twists

Gropper¹

2023

Preprint

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The Galois‐equivariant Structure of the Norm Residue Symbol in Tamely Ramified‐extensions of 2‐adic Number Fields

Simons

1994

Mathematische Nachrichten

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Let L be a p-adic number field of degree d over Q,,, which contains the p' roots of unity pps (s 2 1). Then V, = L x / ( L x ) p s is a free Z/p"Z-module of rank d + 2, and the @-norm residue symbol on L" induces on K, a hi-multiplicative form It is well known (see, e.g., [6]) that (-, -), is nondegeneratc and antisymmetric; if p = 2 and L doesn't contain a fourth root of unity, this form is non-alternating: there is an element a E L such that (a, a), = (a, -1), = -1. When L is a Galois extension of the subfield k with Galois group G, then (-, -), is Galois-equivalent: for a, h E 5, and g E G , (aY, bq), = (a, b): .Of special interest to us is the case where k is fixed and the Galois extension Llk is a tamely ramified of split type (defined in the next section). Here the Galois-module structure of VL has been explicitly determined, and precise knowledge of the symplectic structure of the norm residue symbol on this space when ps + 2 has led to a purely group-theoretical characterization of the absolute Galois group of k (KOCH [9]) and even to explicit presentations of this group (JAKOVLEV [3] and JANNSEN and WINCRERG [4] for p = I 2; ZELVENSKII [ 121 and DIEKERT [I] for p = 2 and s 2 2). This paper will address the remaining case, when k ( W ) / k is a ramified extension of the 2-adic number field k. In [8], H. KOCH obtained a structure Theorem for VL when L/k is unramified of 2-power order, and in [11] the author extended KOCH'S result to the case where L / k is tamely ramified abelian. The main result of this paper gives a characterization of the symplectic structure of V, for L/k tamely ramified of split type.

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Hagen¹

1995

Informationsqualität Von Nachrichten

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Über die absolute Galoisgruppe dyadischer Zahlkörper.

Cited by 4 publications

References 6 publications

Surfaces and p-adic fields I: Dehn twists

Surfaces and p-adic fields I: Dehn twists

The Galois‐equivariant Structure of the Norm Residue Symbol in Tamely Ramified‐extensions of 2‐adic Number Fields

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