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Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing

Ashay A. Burungale, Wei He, Shinichi Kobayashi, Kazuto Ota

TL;DR

This work develops an automorphic, CM-driven framework to establish mod $\ell$ non-vanishing of central Hecke $L$-values for self-dual characters over imaginary quadratic fields, connecting CM periods with automorphic periods on definite Shimura sets via explicit Waldspurger formulas. A key innovation is the construction of $\ell$-optimal test vectors for supercuspidal representations and an $\ell$-integral comparison of quaternionic and CM periods, enabling precise control of $\ell$-adic valuations across split and inert primes. The authors apply these results to Rubin’s $p$-adic $L$-function, proving $\mu$-invariant vanishing in the anticyclotomic CM setting and deriving consequences for CM Iwasawa theory at inert primes and for CM Selmer groups. The paper also provides a novel perspective on Rubin’s theory by linking automorphic and $p$-adic constructions, with potential extensions to CM fields and broader self-dual families. Overall, it advances the understanding of mod $\ell$ phenomena in CM and anticyclotomic Iwasawa theory through a robust automorphic–Shimura set approach.

Abstract

Let $λ$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}({\mathrm cond}(λ))$. We determine the $\ell$-adic valuation of Hecke $L$-values $L(1,λχ)/Ω_K$ as $χ$ varies over $p$-power order anticyclotomic characters over $K$. As an application, for $p$ inert in $K$, we prove the vanishing of the $μ$-invariant of Rubin's $p$-adic $L$-function, leading to the first results on the $μ$-invariant of imaginary quadratic fields at non-split primes. Our approach and results complement the work of Hida and Finis. The approach is rooted in the arithmetic of a CM form on a definite Shimura set.The application to Rubin's $p$-adic $L$-function also relies on the proof of his conjecture. Along the way, we present an automorphic view on Rubin's theory.

Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing

TL;DR

This work develops an automorphic, CM-driven framework to establish mod non-vanishing of central Hecke -values for self-dual characters over imaginary quadratic fields, connecting CM periods with automorphic periods on definite Shimura sets via explicit Waldspurger formulas. A key innovation is the construction of -optimal test vectors for supercuspidal representations and an -integral comparison of quaternionic and CM periods, enabling precise control of -adic valuations across split and inert primes. The authors apply these results to Rubin’s -adic -function, proving -invariant vanishing in the anticyclotomic CM setting and deriving consequences for CM Iwasawa theory at inert primes and for CM Selmer groups. The paper also provides a novel perspective on Rubin’s theory by linking automorphic and -adic constructions, with potential extensions to CM fields and broader self-dual families. Overall, it advances the understanding of mod phenomena in CM and anticyclotomic Iwasawa theory through a robust automorphic–Shimura set approach.

Abstract

Let be a self-dual Hecke character over an imaginary quadratic field of infinity type . Let and be primes which are coprime to . We determine the -adic valuation of Hecke -values as varies over -power order anticyclotomic characters over . As an application, for inert in , we prove the vanishing of the -invariant of Rubin's -adic -function, leading to the first results on the -invariant of imaginary quadratic fields at non-split primes. Our approach and results complement the work of Hida and Finis. The approach is rooted in the arithmetic of a CM form on a definite Shimura set.The application to Rubin's -adic -function also relies on the proof of his conjecture. Along the way, we present an automorphic view on Rubin's theory.
Paper Structure (92 sections, 72 theorems, 344 equations, 1 table)

This paper contains 92 sections, 72 theorems, 344 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Setting
  4. $(\ell,p)$ non-vanishing
  5. Application to CM Iwasawa theory at inert primes
  6. Relation to prior work
  7. Strategy
  8. Definite Shimura set
  9. Ancillary results
  10. About Theorem \ref{['thmD']}
  11. About Theorem \ref{['thmE']}
  12. Newforms as $\ell$-optimal test vectors for supercuspidal representations
  13. About Theorems \ref{['thmB']} and \ref{['thmC']}
  14. Further remarks on the work of Hida and Finis
  15. Organisation
  16. ...and 77 more sections

Key Result

Theorem 1.1

Let $\lambda$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be two different primes which are coprime to $2{\rm{N}}_{K/{\mathbb {Q}}}(\mathrm{cond^r}{\lambda})$. If $\epsilon(\lambda)=-1$, suppose that $\ell \geq 5$. Then for all bu

Theorems & Definitions (153)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • proof
  • Remark 1.8
  • Corollary 1.9
  • ...and 143 more