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arXiv · 2208.10424

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

Weronika Czerniawska
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Abstract

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.

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Weronika Czerniawska. 2022-08-22. Harmonic analysis approach to the relative Riemann-Roch theorem on global fields. https://arxiv.org/abs/2208.10424

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