On the $p$-adic $L$-function and Iwasawa Main Conjecture for an Artin motive over a CM field
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914151667859456 |
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| author | Hara, Takashi Ochiai, Tadashi |
| author_facet | Hara, Takashi Ochiai, Tadashi |
| contents | For an algebraic Hecke character defined on a CM field $F$ of degree $2d$, Katz constructed a $p$-adic $L$-function of $d+1+δ_{F,p}$ variables in his innovative paper published in 1978, where $δ_{F,p}$ denotes the Leopoldt defect for $F$ and $p$. In the present article, we generalise the result of Katz under several technical conditions (containing the absolute unramifiedness of $F$ at $p$), and construct a $p$-adic Artin $L$-function of $d+1+δ_{F,p}$ variables, which interpolates critical values of the Artin $L$-function associated to a $p$-unramified Artin representation of the absolute Galois group $G_F$. Our construction is an analogue over a CM field of Greenberg's construction over a totally real field, but there appear new difficulties which do not matter in Greenberg's case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06983 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $p$-adic $L$-function and Iwasawa Main Conjecture for an Artin motive over a CM field Hara, Takashi Ochiai, Tadashi Number Theory 11R23 (Primary), 11G15, 11L05, 11R42 (Secondaries) For an algebraic Hecke character defined on a CM field $F$ of degree $2d$, Katz constructed a $p$-adic $L$-function of $d+1+δ_{F,p}$ variables in his innovative paper published in 1978, where $δ_{F,p}$ denotes the Leopoldt defect for $F$ and $p$. In the present article, we generalise the result of Katz under several technical conditions (containing the absolute unramifiedness of $F$ at $p$), and construct a $p$-adic Artin $L$-function of $d+1+δ_{F,p}$ variables, which interpolates critical values of the Artin $L$-function associated to a $p$-unramified Artin representation of the absolute Galois group $G_F$. Our construction is an analogue over a CM field of Greenberg's construction over a totally real field, but there appear new difficulties which do not matter in Greenberg's case. |
| title | On the $p$-adic $L$-function and Iwasawa Main Conjecture for an Artin motive over a CM field |
| topic | Number Theory 11R23 (Primary), 11G15, 11L05, 11R42 (Secondaries) |
| url | https://arxiv.org/abs/2407.06983 |