An introduction to higher dimensional local fields and adeles

Morrow, Matthew P.

doi:10.48550/arxiv.1204.0586

Cited by 6 publications

(11 citation statements)

References 57 publications

(80 reference statements)

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“…However, as our main result concerns surfaces only, the reader may easily restrict himself by this from the very beginning. Notice that the definition of adeles on a surface has a much more explicit version, [10], [8], [6].…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Intersections of adelic groups on a surface

Budylin¹,

Gorchinskiy²

2013

Sb. Math.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

Intersections of adelic groups on a surface

Budylin¹,

Gorchinskiy²

2013

Sb. Math.

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“…Let S be a two dimensional, irreducible, Noetherian scheme and let x ∈ y ⊂ S be a complete flag of irreducible closed subschemes. If m is a local equation for x and p is a local equation for y, then let O = O S,x and K x,y = Frac( (O) pO ), see, for example, [1], [24], [12], [11,Part 1] and [21,Sections 6,7]. If x is a smooth point of y, then K x,y is an example of a two-dimensional local field; it is a complete discrete valuation field whose residue field is a one-dimensional local field.…”

Section: Two-dimensional Local Fieldsmentioning

confidence: 99%

“…To such a triple (X, M, T ), one can associate an abelian group A(X, M, T ) of adeles. We will call these groups "geometric adeles" and recommend the following references for details [23], [24], [1], [15], [11] and [21,Section 8]. The adelic group A(X, M, T ) can be interpreted as a restricted product over T of local factors, which are obtained by localising and completing along each flag.…”

Section: Analytic Adelesmentioning

confidence: 99%

Zeta integrals on arithmetic surfaces

Oliver

2016

St. Petersburg Math. J.

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Given a (smooth, projective, geometrically connected) curve over a number field, one expects its Hasse-Weil L-function, a priori defined only on a right half-plane, to admit meromorphic continuation to C and satisfy a simple functional equation. Aside from exceptional circumstances, these analytic properties remain largely conjectural. One may formulate these conjectures in terms of zeta functions of two-dimensional arithmetic schemes, on which one has non-locally compact "analytic" adelic structures admitting a form of "lifted" harmonic analysis first defined by Fesenko for elliptic curves. In this paper we generalize his global results to certain curves of arbitrary genus by invoking a renormalizing factor which may be interpreted as the zeta function of a relative projective line. We are lead to a new interpretation of the "gamma factor" (defined in terms of the Hodge structures at archimedean places) and an (two-dimensional) adelic interpretation of the "mean-periodicity correspondence", which is comparable to the conjectural automorphicity of Hasse-Weil L-functions.

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“…It can be shown that a field K in Ring c k admits at most one structure of n-dimensional local field (see e.g. [Mo,Remark 2.3]). This implies that the forgetful functor LF n k → Ring c k is fully faithful.…”

Section: Topological Local Fieldsmentioning

confidence: 99%

Local Beilinson–Tate operators

Yekutieli¹

2015

Algebra Number Theory

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Abstract. In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher dimensional algebraic varieties by Beilinson in 1980. However Beilinson's paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson-Tate approach to residues in higher dimensions.Our paper presents a variant of Beilinson's operator-theoretic construction. We consider an n-dimensional topological local field K, and define a ring of operators E(K) that acts on K, which we call the ring of local Beilinson-Tate operators. Our definition is of an analytic nature (as opposed to the original geometric definition of Beilinson). We study various properties of the ring E(K). In particular we show that E(K) has an n-dimensional cubical decomposition, and this gives rise to a residue functional in the style of Beilinson-Tate. Presumably this residue functional coincides with the residue functional that we had constructed in 1992; but we leave this as a conjecture.

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An introduction to higher dimensional local fields and adeles

Abstract: These notes are an introduction to higher dimensional local fields and higher dimensional adèles. As well as the foundational theory, we summarise the theory of topologies on higher dimensional local fields and higher dimensional local class field theory.

Cited by 6 publications

References 57 publications

Intersections of adelic groups on a surface

Intersections of adelic groups on a surface

Zeta integrals on arithmetic surfaces

Local Beilinson–Tate operators

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