On the formal arc space of a reductive monoid

TL;DR

This paper constructs a canonical function on the arc space of a scheme over a finite field, linking it to local L-functions, and confirms a conjecture for L-monoids.

Contribution

It defines a basic function on the arc space of reductive monoids and computes it explicitly for affine toric varieties and L-monoids, confirming a conjecture.

Findings

The basic function corresponds to the trace of Frobenius on intersection complex stalks.

For affine toric varieties and L-monoids, the function matches the generating function of local unramified L-functions.

The conjecture for L-monoids is proved.

Abstract

Let $X$ be a scheme of finite type over a finite field $k$, and let $\mathcal L X$ denote its arc space; in particular, $\mathcal L X(k) = X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of $\mathcal L X$ in the neighborhood of non-degenerate arcs, we show that a canonical "basic function" can be defined on the non-degenerate locus of $\mathcal L X(k)$, which corresponds to the trace of Frobenius on the stalks of the intersection complex of any finite-dimensional model. We then proceed to compute this function when $X$ is an affine toric variety or an "$L$-monoid". Our computation confirms the expectation that the basic function is a generating function for a local unramified $L$-function; in particular, in the case of an $L$-monoid we prove a conjecture formulated by the second-named author.