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On the formal arc space of a reductive monoid
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 138, Number 1, February 2016
- pp. 81-108
- 10.1353/ajm.2016.0004
- Article
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Let $X$ be a scheme of finite type over a finite field $k$, and let ${\cal
L} X$ denote its arc space; in particular, ${\cal L} X(k)=X(k[[t]])$.
Using the theory of Grinberg, Kazhdan, and Drinfeld on the
finite-dimensionality of singularities of ${\cal L} X$ in the neighborhood
of non-degenerate arcs, we show that a canonical ``basic function'' can be
defined on the non-degenerate locus of ${\cal L} X(k)$, 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 author.