Airy Functions Over Local Fields
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Abstract
Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated with compact p-adic Lie groups also have these properties.
Mathematics Subject Classification (2000)
11S80 33E20Keywords
Airy functions Airy integrals local fields compact p-adic Lie groups quaternion algebrasNotes
Acknowledgments
R. N. F. would like to thank Alexander Bobenko for inviting him to visit the Technische Universität, Berlin; this visit was supported by an ENIGMA fellowship. He would also like to thank the Max-Planck-Institut für Mathematik, Bonn, for their hospitality during his visit in 2008. V. S. V. is grateful to the INFN, Genova, Italy for their hospitality in September 2008 for a stay during which a part of the results of this paper were worked out. D. W. would like to thank the Department of Mathematics at UCLA for their hospitality. We also like to thank Professor Pierre Deligne for his comments.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.