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Fourier transform for étale motivic cohomology

Ivan Rosas-Soto

Abstract

In the present article, we study the integral aspects of the Fourier transform of an abelian variety $A$ over a field $k$, using étale motivic cohomology, following the ideas and theory given by Moonen, Polishchuk and later by Beckman and de Gaay Fortman. We prove that there exists a PD-structure over the positive degree part of the étale Chow ring $\text{CH}^{\text{ét}}_{>0}(A)$ with respect to the Pontryagin product.

Fourier transform for étale motivic cohomology

Abstract

In the present article, we study the integral aspects of the Fourier transform of an abelian variety over a field , using étale motivic cohomology, following the ideas and theory given by Moonen, Polishchuk and later by Beckman and de Gaay Fortman. We prove that there exists a PD-structure over the positive degree part of the étale Chow ring with respect to the Pontryagin product.

Paper Structure

This paper contains 4 sections, 17 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Etale Fourier transform
  3. PD-structure
  4. Etale Fourier transform

Key Result

Corollary 1

Let $M$ be a commutative monoid with identity in the category of quasi-projective $k$-schemes, such that the product morphism $\mu : M\times M \to M$ is proper. Let $p_d:\text{Sym}^d(M)\to M$ be the morphism induced by the iterated multiplication map $M^d \to M$. Then the maps $\gamma_d^M: \text{CH}

Theorems & Definitions (32)

  • Corollary : Corollary \ref{['coro1']}
  • Corollary : Corollary \ref{['coro2']}
  • Definition 1.1
  • Theorem : Theorem \ref{['teoEquiv']}
  • Theorem : Corollary \ref{['coro']}
  • Lemma 2.1: MP
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 22 more