The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes
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| Format: | Preprint |
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2025
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| _version_ | 1866909684089225216 |
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| author | Burungale, Ashay Kobayashi, Shinichi Ota, Kazuto |
| author_facet | Burungale, Ashay Kobayashi, Shinichi Ota, Kazuto |
| contents | We prove an asymptotic formula for the $p$-adic valuation of Hecke $L$-values of an imaginary quadratic field at an inert prime $p$ along the anticyclotomic $\mathbb{Z}_p$-tower. The key is determination of the $p$-adic valuation of generalized Gauss sums defined using Coates-Wiles homomorphism, and of local resolvents in $\mathbb{Z}_p$-extensions. This answers a question of Rubin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08221 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes Burungale, Ashay Kobayashi, Shinichi Ota, Kazuto Number Theory We prove an asymptotic formula for the $p$-adic valuation of Hecke $L$-values of an imaginary quadratic field at an inert prime $p$ along the anticyclotomic $\mathbb{Z}_p$-tower. The key is determination of the $p$-adic valuation of generalized Gauss sums defined using Coates-Wiles homomorphism, and of local resolvents in $\mathbb{Z}_p$-extensions. This answers a question of Rubin. |
| title | The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.08221 |