Introduction to Mono-anabelian Geometry

Hoshi, Yuichiro

doi:10.5802/pmb.42

Cited by 5 publications

(22 citation statements)

References 6 publications

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“…The amphora of K (more precisely of the topological group G K ) is the collection of all quantities, properties, algebraic structures associated with K which depend only on the anabelomorphism class of K. For example if K is a p-adic local field, then the residue characteristic p of K is amphoric, unramifiedness of K/Q p is an amphoric property, and K * is an amphoric topological group. Reader should consult [Hos17] for proofs of all of these assertions as well as a longer list of amphoric properties (Hoshi has also found a number of new amphoric and unamphoric quantities-see [Hos13], [Hos14], [Hos18]). In this paper I also discuss several quantities associated to K which are not amphoric and which do not appear in the existing literature on anabelian geometry.…”

Section: What Is Anabelomorphy?mentioning

confidence: 99%

“…let α : K L be an abelomorphism of p-adic local fields. Then one has an isomorphism α : K * → L * of topological groups (see [Hos17]). This is the definition of [Moc97].…”

Section: Anabelomorphy Amphoras and Amphoric Quantitiesmentioning

confidence: 99%

“…For example the prime number p, the degree of K/Q p , the ramification index of K/Q p , the inertia subgroup I K and the wild inertia subgroup P K ⊂ I K of G K , are amphoric (see [Moc97], [Hos17]) but the ramification filtration is unamphoric.…”

Section: Anabelomorphy Amphoras and Amphoric Quantitiesmentioning

confidence: 99%

“…(5) The p-adic cyclotomic character χ p : G K → Z * p is amphoric. For modern proof of the first three assertions see [Hos17] for the last assertion see [Moc97]. Hoshi's paper also provides a longer list of amphoric quantities, properties and alg.…”

Section: And Only If There Is a Topological Isomorphism Of Their Galo...mentioning

confidence: 99%

“…Let K * = K − {0} be the multiplicative monoid of non-zero elements of K and K ⊛ = K * ∪ {0} as a multiplicative monoid (with the obvious structure). As one learns from [Hos17], if K is a p-adic field then all of these monoids are amphoric.…”

Section: Monoradicality Is Amphoricmentioning

confidence: 99%

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On Mochizuki's idea of Anabelomorphy and its applications

Joshi

2020

Preprint

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Ordinarity of a local Galois representation is amphoric 17 Φ Sen is unamphoric 19 Artin and Swan Conductor of a local Galois representation are not amphoric 20 Peu and Tres ramifiedness are not amphoric properties 21 Discriminant and Different of a p-adic field are unamphoric 22 Frobenius elements are amphoric 24 Monoradicality is amphoric 24 Anabelomorphy for Tate curves and Abelian varieties with multiplicative reduction 25 Anabelomorphy of group-schemes of type (p, p, • • • , p) over p-adic fields 27 Anabelomorphy for Projective Spaces and Affine spaces 27 The L-invariant is unamphoric 29 Anabelomorphic Connectivity Theorem for Number Fields 30 The Ordinary Synchronization Theorem 37 Automorphic Synchronization Theorems: Anabelomorphy and the local Langlands correspondence 40 Anabelomorphic Density Theorem 44 Anabelomorphic Connectivity Theorem for Elliptic Curves 45 Anabelomorphy and Fundamental Groups 47 Anabelomorphy of Elliptic Curves II: Kodaira Symbol, Exponent of Conductor and Tamagawa Number are all unamphoric 49 Artin Conductors, Swan Conductors and Discriminants II 24 The theory of Tate curves at archimedean primes and Θ-functions at archimedean primes 54 25 Anabelomorphic Density Theorem II 58 26 Perfectoid algebraic geometry as an example of anabelomorphy 61 27 The proof of the Fontaine-Colmez Theorem as an example of anabelomorphy on the Hodge side 62 28 Anabelomorphy for p-adic differential equations 64

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Section: What Is Anabelomorphy?mentioning

confidence: 99%

“…let α : K L be an abelomorphism of p-adic local fields. Then one has an isomorphism α : K * → L * of topological groups (see [Hos17]). This is the definition of [Moc97].…”

Section: Anabelomorphy Amphoras and Amphoric Quantitiesmentioning

confidence: 99%

Section: Anabelomorphy Amphoras and Amphoric Quantitiesmentioning

confidence: 99%

Section: And Only If There Is a Topological Isomorphism Of Their Galo...mentioning

confidence: 99%

Section: Monoradicality Is Amphoricmentioning

confidence: 99%

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On Mochizuki's idea of Anabelomorphy and its applications

Joshi

2020

Preprint

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On the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields

Hoshi

Nishio

2022

Res. number theory

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In the present paper, we study the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of anabelian geometry. Let us recall that it is well-known that the natural homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the absolute Galois group of the given mixed-characteristic local field is injective. One main result of the present paper is that if the mixed-characteristic local field satisfies certain conditions, then the set of conjugates of the image of this injective homomorphism in the outer automorphism group is infinite, which thus implies that the image of this injective homomorphism is not normal in the outer automorphism group. In particular, one may conclude that it is impossible to establish a functorial group-theoretic reconstruction, from the absolute Galois group, of the "field-theoretic" subgroup, i.e., the image of this injective homomorphism, of the outer automorphism group.2010 Mathematics Subject Classification. 11S20. Key words and phrases. mixed-characteristic local field, absolute Galois group, anabelian geometry, monoanabelian geometry, group of MLF-type.

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A study on anabelian geometry of higher local fields

Murotani¹

2023

Int. J. Number Theory

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Anabelian geometry has been developed over a much wider class of fields than Grothendieck, who is the originator of anabelian geometry, conjectured. So, it is natural to ask the following question: What kinds of fields are suitable for the base fields of anabelian geometry?In the present paper, we consider this problem for higher local fields. First, to consider "anabelianness" of higher local fields themselves, we give mono-anabelian reconstruction algorithms of various invariants of higher local fields from their absolute Galois groups. As a result, the isomorphism classes of certain types of higher local fields are completely determined by their absolute Galois groups. Next, we prove that mixed-characteristic higher local fields are Kummer-faithful. This result affirms the above question for these higher local fields to a certain extent.

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Introduction to Mono-anabelian Geometry

Abstract: Les Publications mathématiques de Besançon sont membres du Centre Mersenne pour l'édition scientifique ouverte http://www.centre-mersenne.org/Publications mathématiques de Besançon -2021, 5-44

Cited by 5 publications

References 6 publications

On Mochizuki's idea of Anabelomorphy and its applications

On Mochizuki's idea of Anabelomorphy and its applications

On the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields

A study on anabelian geometry of higher local fields

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