Existential uniform p-adic integration and descent for integrability and largest poles

• Raf Cluckers ^[1]
  1. [1] Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000, Lille, France
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 2, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01023-y
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• Resumen

  • Since the work by Denef, p-adic cell decomposition provides a well-established method to study p-adic and motivic integrals. In this paper, we present a variant of this method that keeps track of existential quantifiers. This enables us to deduce descent properties for p-adic integrals. In particular, we show that integrability for ‘existential’ functions descends from any p-adic field to any p-adic subfield. As an application, we obtain that the largest pole of certain Poincaré series, which are generating series of p-adic point counts, can only increase when passing to field extensions. As a side result, we prove a relative quantifier elimination statement for Henselian valued fields of characteristic zero that preserves existential formulas.

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