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class number formula corrected

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edit approved May 28, 2014 at 15:26

GNiklasch

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Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The class number formula says that the leading term of $\zeta_F$ at $s=0$ equals $$ \zeta_F^\ast(0):=\lim_{s \to 0} \zeta_F(s)s^{-(r_1+r_2-1)}=-\frac{w}{h} R, \quad (1) $$$$ \zeta_F^\ast(0):=\lim_{s \to 0} \zeta_F(s)s^{-(r_1+r_2-1)}=-\frac{h}{w} R, \quad (1) $$ where $R$ is the covolume of the image of Dirichlet's regulator $ \mathcal{O}_F^\ast \to \mathbb{R}^{r_1+r_2}. $ By$\mathcal{O}_F^\ast \to \mathbb{R}^{r_1+r_2}.$ By the functional equation, this statement is equivalent to the computation of the residue of $\zeta_F$ at $s=1$.
Borel introduced higher regulators $$ K_{2n-1}(\mathcal{O}_F) \to \mathbb{R}^{d_n} $$ where $n \geq 1$ and $d_n=r_1+r_2$ if $n$ is odd and $r_2$ if $n$ is even. When $n=1$, $K_1(\mathcal{O}_F)\sim \mathcal{O}_F^\ast$ and one recovers the first regulator. Then he proved that the leading term of $\zeta_F(s)$ at $s=1-n$ is $$ \zeta_F^\ast(1-n)=\alpha \pi^{d_n} R_n \quad (2) $$ where $\alpha \in \mathbb{Q}^\times$ and $R_n$ is the covolume of the image of $K_{2n-1}(\mathcal{O}_F)$ in $\mathbb{R}^{d_n}$.
Given a prime number $p$, one has at disposal a $p$-adic Dedekind zeta function $\zeta_{F, p}$ and an equivalent of Dirichlet's regulator, constructed by Leopoldt. It is known that the equivalent of (1) holds in this case (I think for arbitrary $F$ this is due to Colmez).

Question: Does there exist a $p$-adic version of (2)?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The class number formula says that the leading term of $\zeta_F$ at $s=0$ equals $$ \zeta_F^\ast(0):=\lim_{s \to 0} \zeta_F(s)s^{-(r_1+r_2-1)}=-\frac{w}{h} R, \quad (1) $$ where $R$ is the covolume of the image of Dirichlet's regulator $ \mathcal{O}_F^\ast \to \mathbb{R}^{r_1+r_2}. $ By the functional equation, this statement is equivalent to the computation of the residue of $\zeta_F$ at $s=1$.
Borel introduced higher regulators $$ K_{2n-1}(\mathcal{O}_F) \to \mathbb{R}^{d_n} $$ where $n \geq 1$ and $d_n=r_1+r_2$ if $n$ is odd and $r_2$ if $n$ is even. When $n=1$, $K_1(\mathcal{O}_F)\sim \mathcal{O}_F^\ast$ and one recovers the first regulator. Then he proved that the leading term of $\zeta_F(s)$ at $s=1-n$ is $$ \zeta_F^\ast(1-n)=\alpha \pi^{d_n} R_n \quad (2) $$ where $\alpha \in \mathbb{Q}^\times$ and $R_n$ is the covolume of the image of $K_{2n-1}(\mathcal{O}_F)$ in $\mathbb{R}^{d_n}$.
Given a prime number $p$, one has at disposal a $p$-adic Dedekind zeta function $\zeta_{F, p}$ and an equivalent of Dirichlet's regulator, constructed by Leopoldt. It is known that the equivalent of (1) holds in this case (I think for arbitrary $F$ this is due to Colmez).

Question: Does there exist a $p$-adic version of (2)?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The class number formula says that the leading term of $\zeta_F$ at $s=0$ equals $$ \zeta_F^\ast(0):=\lim_{s \to 0} \zeta_F(s)s^{-(r_1+r_2-1)}=-\frac{h}{w} R, \quad (1) $$ where $R$ is the covolume of the image of Dirichlet's regulator $\mathcal{O}_F^\ast \to \mathbb{R}^{r_1+r_2}.$ By the functional equation, this statement is equivalent to the computation of the residue of $\zeta_F$ at $s=1$.
Borel introduced higher regulators $$ K_{2n-1}(\mathcal{O}_F) \to \mathbb{R}^{d_n} $$ where $n \geq 1$ and $d_n=r_1+r_2$ if $n$ is odd and $r_2$ if $n$ is even. When $n=1$, $K_1(\mathcal{O}_F)\sim \mathcal{O}_F^\ast$ and one recovers the first regulator. Then he proved that the leading term of $\zeta_F(s)$ at $s=1-n$ is $$ \zeta_F^\ast(1-n)=\alpha \pi^{d_n} R_n \quad (2) $$ where $\alpha \in \mathbb{Q}^\times$ and $R_n$ is the covolume of the image of $K_{2n-1}(\mathcal{O}_F)$ in $\mathbb{R}^{d_n}$.
Given a prime number $p$, one has at disposal a $p$-adic Dedekind zeta function $\zeta_{F, p}$ and an equivalent of Dirichlet's regulator, constructed by Leopoldt. It is known that the equivalent of (1) holds in this case (I think for arbitrary $F$ this is due to Colmez).

Question: Does there exist a $p$-adic version of (2)?

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asked May 28, 2014 at 14:17

Borp

is there a p-adic Borel theorem?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The class number formula says that the leading term of $\zeta_F$ at $s=0$ equals $$ \zeta_F^\ast(0):=\lim_{s \to 0} \zeta_F(s)s^{-(r_1+r_2-1)}=-\frac{w}{h} R, \quad (1) $$ where $R$ is the covolume of the image of Dirichlet's regulator $ \mathcal{O}_F^\ast \to \mathbb{R}^{r_1+r_2}. $ By the functional equation, this statement is equivalent to the computation of the residue of $\zeta_F$ at $s=1$.
Borel introduced higher regulators $$ K_{2n-1}(\mathcal{O}_F) \to \mathbb{R}^{d_n} $$ where $n \geq 1$ and $d_n=r_1+r_2$ if $n$ is odd and $r_2$ if $n$ is even. When $n=1$, $K_1(\mathcal{O}_F)\sim \mathcal{O}_F^\ast$ and one recovers the first regulator. Then he proved that the leading term of $\zeta_F(s)$ at $s=1-n$ is $$ \zeta_F^\ast(1-n)=\alpha \pi^{d_n} R_n \quad (2) $$ where $\alpha \in \mathbb{Q}^\times$ and $R_n$ is the covolume of the image of $K_{2n-1}(\mathcal{O}_F)$ in $\mathbb{R}^{d_n}$.
Given a prime number $p$, one has at disposal a $p$-adic Dedekind zeta function $\zeta_{F, p}$ and an equivalent of Dirichlet's regulator, constructed by Leopoldt. It is known that the equivalent of (1) holds in this case (I think for arbitrary $F$ this is due to Colmez).

Question: Does there exist a $p$-adic version of (2)?