ON THE FORMAL ARC SPACE OF A REDUCTIVE MONOID

Let X be a scheme of finite type over a finite field k, and let ℒX denote its arc space; in particular, ℒX(k) = X(k[[t]]). Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of ℒX in the neighborhood of non-degenerate arcs, we show that a canonical &quo...

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Bibliographic Details
Published in:American journal of mathematics 2016-02, Vol.138 (1), p.81-108
Main Authors: Bouthier, A., Ngô, B. C., Sakellaridis, Y.
Format: Article
Language:English
Subjects:
Algebra
Mathematical functions
Mathematical models
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Summary:Let X be a scheme of finite type over a finite field k, and let ℒX denote its arc space; in particular, ℒX(k) = X(k[[t]]). Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of ℒX in the neighborhood of non-degenerate arcs, we show that a canonical "basic function" can be defined on the non-degenerate locus of ℒX(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.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2016.0004