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$p$-adic $L$-functions for Hecke characters of totally imaginary fields

Guido Kings, Johannes Sprang

Abstract

We construct $p$-adic $L$-functions interpolating critical $L$-values of algebraic Hecke characters for arbitrary unramified primes $p$ and any totally imaginary field. For non-ordinary primes, the only previously known case was that of imaginary quadratic extensions of $\mathbb{Q}$. One of the main ingredients is a new $p$-adic Fourier theory relating generic fibers of $p$-divisible groups to a general class of character varieties. Combining this with equivariant cohomology classes constructed in a previous paper allows us to construct the $p$-adic $L$-function.

$p$-adic $L$-functions for Hecke characters of totally imaginary fields

Abstract

We construct -adic -functions interpolating critical -values of algebraic Hecke characters for arbitrary unramified primes and any totally imaginary field. For non-ordinary primes, the only previously known case was that of imaginary quadratic extensions of . One of the main ingredients is a new -adic Fourier theory relating generic fibers of -divisible groups to a general class of character varieties. Combining this with equivariant cohomology classes constructed in a previous paper allows us to construct the -adic -function.
Paper Structure (18 sections, 36 theorems, 224 equations)

This paper contains 18 sections, 36 theorems, 224 equations.

Table of Contents

  1. $p$-adic Fourier theory
  2. $W$-analytic character groups
  3. $W$-analytic character varieties
  4. The Fourier transform
  5. $W$-analytic character varieties for abelian $p$-adic Lie groups
  6. Abelian Galois groups and their character varieties
  7. $p$-divisible groups over $\mathscr{O}_{\mathbb{C}_p}$
  8. Character varieties and $p$-divisible groups
  9. $p$-adic $L$-functions of Hecke characters
  10. Hecke $L$-functions of totally imaginary fields
  11. $p$-adic Hecke characters
  12. The main result
  13. Periods, complex and $p$-adic
  14. Serre twists and decompositions
  15. Eisenstein-Kronecker classes and the class $\vartheta_{\mathfrak{f},\mathfrak{c}}(\mathfrak{b})$
  16. ...and 3 more sections

Key Result

Theorem A

There is a a rigid analytic function $L_p(\cdot)$ with the following interpolation property: For every critical algebraic Hecke character $\chi$ of conductor dividing $\mathfrak{f} p^\infty$ and infinity type $-\alpha\in \mathbb{Z}^\Sigma$ with $\alpha\geq \underline{1}$, we have where $L_\mathfrak{f}(\chi,0)$ is the $L$-function of $\chi$ at $0$ and $\Omega$ (resp. $\Omega_p$) are ($p$-adic) per

Theorems & Definitions (92)

  • Theorem A: see Theorem \ref{['thmA']}
  • Theorem B: see Theorem \ref{['thm:FT-abelian']}
  • Theorem C: see Theorem \ref{['thm:UniformizationOfCharVar']}
  • Definition C
  • Definition C
  • Remark C
  • Example C
  • Proposition C
  • proof
  • Theorem C
  • ...and 82 more