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Fourier transform of the additive group in algebraically closed valued fields

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Abstract

We continue the study of the Hrushovski–Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier transform in our integration theory and establish some fundamental properties of it. Thereafter, a basic theory of distributions is also developed. We construct the Weil representations in the end as an application. The results are completely parallel to the classical ones.

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Acknowledgments

I would like to thank Thomas Hales for many hours of stimulating discussions during the preparation of this paper, in particular, for pointing out to me the possibility of constructing a motivic version of the Weil representations. The research reported in this paper has been partially supported by the ERC Advanced Grant NMNAG.

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  1. Institut Mathématique de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, 75252 , Paris Cedex 05, France

    Yimu Yin

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Yin, Y. Fourier transform of the additive group in algebraically closed valued fields. Sel. Math. New Ser. 20, 1111–1157 (2014). https://doi.org/10.1007/s00029-014-0153-y

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