A Tauberian theorem for ℓ-adic sheaves on $\mathbb{A}$ ¹

Abstract

Let K ∈ L ¹(ℝ) and let f ∈ L ^∞(ℝ) be two functions on ℝ. The convolution

$$ \left( {K*F} \right)\left( x \right) = \int_\mathbb{R} {K\left( {x - y} \right)f\left( y \right)dy} $$

can be considered as an average of f with weight defined by K. Wiener’s Tauberian theorem says that under suitable conditions, if

$$ \mathop {\lim }\limits_{x \to \infty } \left( {K*F} \right)\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \int_\mathbb{R} {\left( {K*A} \right)\left( x \right)} $$

for some constant A, then

$$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = A $$

We prove the following ℓ-adic analogue of this theorem: Suppose K, F, G are perverse ℓ-adic sheaves on the affine line $\mathbb{A}$ over an algebraically closed field of characteristic p (p ≠ ℓ). Under suitable conditions, if $\left( {K*F} \right)|_{\eta _\infty } \cong \left( {K*G} \right)|_{\eta _\infty }$, then $F|_{\eta _\infty } \cong G|_{\eta _\infty }$, where η _∞ is the spectrum of the local field of $\mathbb{A}$ at ∞.