Bibliographic Details
| Title: |
Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing |
| Authors: |
Burungale, Ashay A., He, Wei, Kobayashi, Shinichi, Ota, Kazuto |
| Publication Year: |
2024 |
| Collection: |
Mathematics |
| Subject Terms: |
Mathematics - Number Theory |
| More Details: |
Let $\lambda$ 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 $6N_{K/\mathbb{Q}}({\mathrm cond}(\lambda))$. We determine the $\ell$-adic valuation of Hecke $L$-values $L(1,\lambda\chi)/\Omega_K$ as $\chi$ varies over $p$-power order anticyclotomic characters over $K$. As an application, for $p$ inert in $K$, we prove the vanishing of the $\mu$-invariant of Rubin's $p$-adic $L$-function, leading to the first results on the $\mu$-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. |
| Document Type: |
Working Paper |
| Access URL: |
http://arxiv.org/abs/2408.13932 |
| Accession Number: |
edsarx.2408.13932 |
| Database: |
arXiv |
