A Tauberian Theorem for $\ell$-adic Sheaves on $\mathbb A^1$

Details A Tauberian Theorem for $\ell$-adic Sheaves on $\mathbb A^1$ A Tauberian Theorem for $\ell$-adic Sheaves on $\mathbb A^1$

2010-06-04

arxiv.org/abs/1006.0789v1

Mathematics

Algebraic Geometry

To appear in Science in China, an issue dedicated to Wang Yuan on the occation of his 80th birthday

Scientific paper

Let $K\in L^1(\mathbb R)$ and let $f\in L^\infty(\mathbb R)$ be two functions on $\mathbb R$. The convolution $$(K\ast f)(x)=\int_{\mathbb R}K(x-y)f(y)dy$$ can be considered as an average of $f$ with weight defined by $K$. Wiener's Tauberian theorem says that under suitable conditions, if $$\lim_{x\to \infty}(K\ast f)(x)=\lim_{x\to \infty} (K\ast A)(x)$$ for some constant $A$, then $$\lim_{x\to \infty}f(x)=A.$$ We prove the following $\ell$-adic analogue of this theorem: Suppose $K,F, G$ are perverse $\ell$-adic sheaves on the affine line $\mathbb A$ over an algebraically closed field of characteristic $p$ ($p\not=\ell$). Under suitable conditions, if $$(K\ast F)|_{\eta_\infty}\cong (K\ast G)|_{\eta_\infty},$$ then $$F|_{\eta_\infty}\cong G|_{\eta_\infty},$$ where $\eta_\infty$ is the spectrum of the local field of $\mathbb A$ at $\infty$.

Affiliated with

Also associated with

No associations

Rating

A Tauberian Theorem for $\ell$-adic Sheaves on $\mathbb A^1$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Tauberian Theorem for $\ell$-adic Sheaves on $\mathbb A^1$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Tauberian Theorem for $\ell$-adic Sheaves on $\mathbb A^1$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-722547