Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing
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| Format: | Preprint |
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2024
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| author | Burungale, Ashay A. He, Wei Kobayashi, Shinichi Ota, Kazuto |
| author_facet | Burungale, Ashay A. He, Wei Kobayashi, Shinichi Ota, Kazuto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing Burungale, Ashay A. He, Wei Kobayashi, Shinichi Ota, Kazuto Number Theory 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. |
| title | Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.13932 |