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Multi-variable admissible distributions

Kengo Fukunaga, Tadashi Ochiai

TL;DR

This work extends the classical one-variable theory of admissible distributions (due to Amice–Vélu and Vishik) to a multi-variable weight-space, addressing the construction and interpolation of $p$-adic $L$-functions in the non-ordinary setting within multi-variable Iwasawa theory. It introduces a comprehensive framework of multi-variable spaces $oldsymbol{H}_{oldsymbol{h}/oldsymbol{K}}$, $oldsymbol{I}_{oldsymbol{h}}^{[oldsymbol{d},oldsymbol{e}]}(M)$, and $oldsymbol{ m D}^{[oldsymbol{d},oldsymbol{e}]}_{oldsymbol{h}}(oldsymbol{ m\Gamma},M)$, establishing isomorphisms and interpolation properties that connect projective limit modules with distribution spaces. A key contribution is the Banach-space-valued extension of the one-variable theory, including Weierstrass-type results and integral lattice control, enabling a robust multi-variable interpolation framework. The results facilitate the construction of multi-variable $p$-adic $L$-functions for general non-ordinary Galois deformations and yield a concrete application to a two-variable Rankin–Selberg $L$-series in Hida families, highlighting the approach’s relevance to deformation spaces and Panchishkin-type phenomena.

Abstract

The theory of admissible distributions over a weight-space of one-variable was studied by Amice--Vélu and played important roles in the cyclotomic Iwasawa theory of non-ordinary p-adic Galois representations. In this article, we discuss the multi-variable generalization of the theory of admissible distributions over a weight-space of several variables.

Multi-variable admissible distributions

TL;DR

This work extends the classical one-variable theory of admissible distributions (due to Amice–Vélu and Vishik) to a multi-variable weight-space, addressing the construction and interpolation of -adic -functions in the non-ordinary setting within multi-variable Iwasawa theory. It introduces a comprehensive framework of multi-variable spaces , , and , establishing isomorphisms and interpolation properties that connect projective limit modules with distribution spaces. A key contribution is the Banach-space-valued extension of the one-variable theory, including Weierstrass-type results and integral lattice control, enabling a robust multi-variable interpolation framework. The results facilitate the construction of multi-variable -adic -functions for general non-ordinary Galois deformations and yield a concrete application to a two-variable Rankin–Selberg -series in Hida families, highlighting the approach’s relevance to deformation spaces and Panchishkin-type phenomena.

Abstract

The theory of admissible distributions over a weight-space of one-variable was studied by Amice--Vélu and played important roles in the cyclotomic Iwasawa theory of non-ordinary p-adic Galois representations. In this article, we discuss the multi-variable generalization of the theory of admissible distributions over a weight-space of several variables.

Paper Structure

This paper contains 11 sections, 90 theorems, 769 equations.

Table of Contents

  1. Introduction
  2. Preparation on the precise notation
  3. One-variable power series of logarithmic order with values in Banach spaces
  4. Proof of the main result for the case of the multi-variable Iwasawa algebra
  5. Proof of the main result for the case of the deformation space
  6. Applications
  7. Review of modular forms
  8. A two-variable admissible distribution for the case of $\Lambda$-adic cusp forms
  9. Construction of a two-variable $p$-adic Rankin-Selberg $L$-series
  10. Two-variable $p$-adic $L$-function constructed by Panchishkin
  11. Appendix

Key Result

Theorem 1

If $f\in\mathcal{H}_{h/\mathcal{K}}$ satisfies $f(u^{i}\epsilon-1)=0$ for every $i\in [d,d+\lfloor h\rfloor]$ and for every $\epsilon\in \mu_{p^{\infty}}$, then $f$ is zero.

Theorems & Definitions (168)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Remark 1.1
  • Theorem 1: Theorem \ref{['main theorem 1 and proof']}
  • Theorem 2: Theorem \ref{['main thm 2 and proof']}
  • Proposition 3: Proposition \ref{['multivariable iwasawa I(ii) sufficient']}
  • Theorem 4: Theorem \ref{['multi-variable results on admissible distributions']}
  • Remark 1.2
  • ...and 158 more