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<p>Since the work by Denef, <i>p</i>-adic cell decomposition provides a well-established method to study <i>p</i>-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 <i>p</i>-adic integrals. In particular, we show that integrability for ‘existential’ functions descends from any <i>p</i>-adic field to any <i>p</i>-adic subfield. As an application, we obtain that the largest pole of certain Poincaré series, which are generating series of <i>p</i>-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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年2月15日

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  • Raf Cluckers,
  • Mathias Stout

摘要

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.