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Note on \(p\)-Adic Local Functional Equation

Abstract

Given primes \(\ell\ne p\), we record here a \(p\)-adic valued Fourier theory on a local field over \(\mathbf{Q}_\ell\), which is developed under the perspective of group schemes. As an application, by substituting rigid analysis for complex analysis, it leads naturally to the \(p\)-adic local functional equation at \(\ell\), which strongly resembles the complex one in Tate’s thesis.

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Acknowledgments

The author is very thankful to Antonio Lei, for his constant encouragement and for reading the draft and providing many helpful suggestions on the organization and the grammar; to Yiannis Sakellaridis for the many inspirational discussions in Newark. We would also like to thank Professor John Coates for kindly pointing out an important application of a potential global functional equation in Iwasawa theory. Finally, we thank the referees for their careful reading and helpful suggestions.

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Correspondence to Luochen Zhao.

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Zhao, L. Note on \(p\)-Adic Local Functional Equation. P-Adic Num Ultrametr Anal Appl 14, 238–264 (2022). https://doi.org/10.1134/S2070046622030037

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  • DOI: https://doi.org/10.1134/S2070046622030037

Keywords

  • local functional equation
  • Fourier transform
  • group scheme
  • Cartier duality