Geometric and analytic structures on the higher adèles

Abstract

The adèles of a scheme have local components-these are topological higher local fields. The topology plays a large role since Yekutieli showed in 1992 that there can be an abundance of inequivalent topologies on a higher local field and no canonical way to pick one. Using the datum of a topology, one can isolate a special class of continuous endomorphisms. Quite differently, one can bypass topology entirely and single out special endomorphisms (global Beilinson-Tate operators) from the geometry of the scheme. Yekutieli's "Conjecture 0.12" proposes that these two notions agree. We prove this.

On the local residue symbol in the style of Tate and Beilinson

“…A word of warning: In (2), while there always exists an isomorphism K j ≃ k j ((t 1 )) • • • ((t n )) such that the above claims hold, it is by no means true that any isomorphism between these fields has these properties. This very subtle behaviour is discussed extensively in [Yek15] and [BGW16a].…”

Local abelian Kato-Parshin reciprocity law: A survey

“…In general, the inversion K × ι − → K × on K × is not sequentially continuous with respect to the induced topology of T K on K × . Look at [5,6] for an overview of sequential algebraic structures; -The map K → K defined by multiplication with a fixed non-zero…”

Adelic geometry on arithmetic surfaces II: Completed adeles and idelic Arakelov intersection theory

“…Since ξ is a standard character of A −1 /A 0 , any other character in A −1 which is trivial on A 0 is of the form ξ 1 (g ·) for g ∈ A −1 . Consider any ψ ∈ A and let i = c ψ the minimum integer i ∈ Z such that ψ(A i ) = 1, note that this integer always exists thanks to (5) and the fact that T has no small subgroups. Then:…”

“…Proof. See [18, 2.4], [18,5] and [18,3] for (1), ( 4) and the non-archimedean part of (3) respectively. For the archimedean case of (3) see [24, Corollary of theorem 3].…”