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We develop a motivic integration version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums. We also study division algebras over the function field, and obtain relations among the motivic Fourier transforms of a test function at different completions. We use these to prove, in a special case, a motivic version of a theorem of Deligne-Kazhdan-Vigneras.
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We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers-Comte-Loeser about $p$-adic local density. We prove some results about geometry of definable sets in Henselian valued fields of characteristic zero, both in semi-algebraic and subanalytic languages, and study Lipschitz continuous maps between such sets. We prove existence of regular stratifications satisfying analogous of Verdier condition $(w_f)$. Using Cluckers-Loeser theory of motivic integration, we define a notion of motivic local density with values in the Grothendieck ring of the theory of the residue sorts. We then prove the existence of a distinguished tangent cone and that one can compute the local density on this cone endowed with appropriate motivic multiplicities. As an application we prove a uniformity theorem for $p$-adic local density.
We generalize Dahmen-Micchelli deconvolution formula for Box splines with parameters. Our proof is based on identities for Poisson summation of rational functions with poles on hyperplanes.
We show that the analogue of the Peterson conjecture on the action of Steenrod squares does not hold in motivic cohomology.
We define a notion of colimit for diagrams in a motivic category indexed by a presheaf of spaces (e.g. an etale classifying space), and we study basic properties of this construction. As a case study, we construct the motivic analogs of the classical extended and generalized powers, which refine the categoric
The first author and Bump defined Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of $mathrm{GL}_3$ over $mathbb{Q}$ they proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit their conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ng^o, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. We prove the Poisson summation formula for certain schemes closely related to Schubert varieties and use it to refine and establish the conjecture of the first author and Bump in many cases.
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