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Preprint Number 557

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557. Antoine Chambert-Loir and François Loeser
Motivic height zeta functions

Submission date: 8 February 2013.


Let \bar C be a projective smooth connected curve over an algebraically closed field of characteristic zero, let F be its field of functions, let C be a dense open subset of \bar C. Let X be a projective flat morphism to C whose generic fiber X_F is a smooth equivariant compactification of G such that D = X_F \setminus G_F is a divisor with strict normal crossings, let U be a surjective and flat model of G over C. We consider a motivic height zeta function, a formal power series with coefficients in a suitable Grothendieck ring of varieties, which takes into account the spaces of sections s of X --> \bar C of given degree with respect to (a model of) the log-anticanonical divisor -K_{X_F}(D) such that s(C) is contained in U. We prove that this power series is rational, that its “largest pole” is at L^{-1}, the inverse of the class of the affine line in the Grothendieck ring, and compute the “order” of this pole as a sum of dimensions of various Clemens complexes at places of \bar C \setminus C. This is a geometric analogue of a result over number fields by the first author and Yuri Tschinkel (Duke Math. J., 2012). The proof relies on the Poisson summation formula in motivic integration, established by Ehud Hrushovski and David Kazhdan (Moscow Math. J, 2009).

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Full text arXiv 1302.2077: pdf, ps.

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