arXiv subjects

Weronika Czerniawska

Publications and source records attributed to Weronika Czerniawska.

3 recordsLinked to original sources

arXiv2025

Selective Integration on Higher Adeles and the Euler Characteristic of Surfaces

The space of two-dimensional geometric adeles of a surface is far from being a locally compact space and there is no translation countably additive invariant nontrivial measure on it. At the same time, certain subquotients of the adeles are direct limits of compact subquotients or inverse limits of discrete subquotients, compatible in a special way. Using this fact, the paper defines a translation invariant measure and integration on certain subquotients of the geometric adeles of surfaces. This theory is considerably different from the theory of integration on analytic adeles of surfaces. After revising aspects of one-dimensional theory, the paper includes a full definition of two-dimensional geometric adeles. A number of their new topological properties are established. The new translation invariant measure and integration on selective subquotients of the geometric adeles is used for integrals of certain functions in a two-dimensional method describing the size of adelic cohomology groups of surfaces, without using standard adelic complexes. A formula for the Euler characteristic of the surface and its divisor in terms of integrals over subquotients of geometric adeles is proved. Using the Euler characteristic, a new two-dimensional adelic intersection number is introduced. For geometric surfaces it is a positive multiple of the standard intersection number. Several results in the previous study of geometric adeles are given new proofs.

math.NT↗

arXiv2024

Harmonic analysis approach to the relative Riemann-Roch theorem on global fields

In this paper we generalize and put in a new light part of ``Fouier analysis on Number fields and Hecke's zeta function''[14] by Tate. We express the relative Euler characteristic using purely adelic language. By using certain natural normalization of Haar measure on adeles we obtain the relative Riemann-Roch theorem. In particular we show that using our relative normalization of the Haar measure on adeles we can obtain the relative Riemann-Roch theorem from the adelic Poisson summation formulae. In addition, using our methods we define the relative 'size of cohomology' numbers, i.e. extract the $h^0$ and $h^1$ part of the relative Euler characteristic. Our theory not only covers both absolute and relative cases, but also the case of an arithmetic curve and a nonsingular, projective curve over a finite field.

math.NT↗

arXiv2019

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

We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed adeles is algebraically and topologically self-dual and fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing. We show that the Arakelov intersection pairing can be lifted to an idelic intersection pairing.

math.NT↗