Special values of $p$-adic $L$-functions and Iwasawa $λ$-invariants of Dirichlet characters
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910647236689920 |
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| author | Knospe, Heiko |
| author_facet | Knospe, Heiko |
| contents | We study the Iwasawa $λ$-invariant of Dirichlet characters $χ$ of arbitrary order for odd primes $p$. From special values of the $p$-adic $L$-function and its derivative we derive several novel and easily computable criteria to distinguish between the cases $λ= 0$, $λ= 1$, $λ= 2$ and $λ\geq 3$. In particular, we look at the case when the $p$-adic $L$-function vanishes at $s=0$. Using formulas of Ferrero-Greenberg and Gross-Koblitz, we give conditions for $λ_p(χ) >1 $ and $λ_p(χ) > 2$. Furthermore, we extend methods of Ernvall-Metsänkylä and Dummit et al. to calculate the $λ$-invariant by twisting $χ$ with characters $ψ$ of the second kind and using the values of the $p$-adic $L$-function at $s=2-p, \dots, 0$. In addition, we leverage the value at $s=1$ to compute $λ_p(χ)$. The formulas are also used to obtain numerical data on the distribution of $λ$-invariants, where either the prime $p$ or the Dirichlet character $χ$ is fixed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06100 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Special values of $p$-adic $L$-functions and Iwasawa $λ$-invariants of Dirichlet characters Knospe, Heiko Number Theory 11R23 (Primary) 11R42, 11S80, 11M41 (Secondary) We study the Iwasawa $λ$-invariant of Dirichlet characters $χ$ of arbitrary order for odd primes $p$. From special values of the $p$-adic $L$-function and its derivative we derive several novel and easily computable criteria to distinguish between the cases $λ= 0$, $λ= 1$, $λ= 2$ and $λ\geq 3$. In particular, we look at the case when the $p$-adic $L$-function vanishes at $s=0$. Using formulas of Ferrero-Greenberg and Gross-Koblitz, we give conditions for $λ_p(χ) >1 $ and $λ_p(χ) > 2$. Furthermore, we extend methods of Ernvall-Metsänkylä and Dummit et al. to calculate the $λ$-invariant by twisting $χ$ with characters $ψ$ of the second kind and using the values of the $p$-adic $L$-function at $s=2-p, \dots, 0$. In addition, we leverage the value at $s=1$ to compute $λ_p(χ)$. The formulas are also used to obtain numerical data on the distribution of $λ$-invariants, where either the prime $p$ or the Dirichlet character $χ$ is fixed. |
| title | Special values of $p$-adic $L$-functions and Iwasawa $λ$-invariants of Dirichlet characters |
| topic | Number Theory 11R23 (Primary) 11R42, 11S80, 11M41 (Secondary) |
| url | https://arxiv.org/abs/2401.06100 |