# A Tauberian Theorem for $ell$-adic Sheaves on $mathbb A^1$ - Mathematics > Algebraic Geometry

A Tauberian Theorem for $ell$-adic Sheaves on $mathbb A^1$ - Mathematics > Algebraic Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: Let $K\in L^1\mathbb R$ and let $f\in L^\infty\mathbb R$ be two functionson $\mathbb R$. The convolution $$K\ast fx=\int {\mathbb R}Kx-yfydy$$can be considered as an average of $f$ with weight defined by $K$. Wiener-sTauberian theorem says that under suitable conditions, if $$\lim {x\to\infty}K\ast fx=\lim {x\to \infty} K\ast Ax$$ for some constant $A$,then $$\lim {x\to \infty}fx=A.$$ We prove the following $\ell$-adic analogueof this theorem: Suppose $K,F, G$ are perverse $\ell$-adic sheaves on theaffine line $\mathbb A$ over an algebraically closed field of characteristic$p$ $p ot=\ell$. Under suitable conditions, if $$K\astF| {\eta \infty}\cong K\ast G| {\eta \infty},$$ then $$F| {\eta \infty}\congG| {\eta \infty},$$ where $\eta \infty$ is the spectrum of the local field of$\mathbb A$ at $\infty$.

Author: Lei Fu

Source: https://arxiv.org/