associated p-adic algebra

Last updated January 27, 2022

Extending definitions

• ( trace) We may extend the bilinear form $Tr_{K/k}(ab)$ from $K$ to $V_\mathfrak{p}$ by $\widehat{k_\mathfrak{p}}$-linearity. Using the notation $\langle \ , \ \rangle$ for the resulting form, $$\langle a,b\rangle=\sum Tr_{K_i/\widehat{k_\mathfrak{p}}}(\pi_i(a)\pi_i(b))$$
• ( Lattice)
• (dual lattice)
• ( index of lattice)

Properties

1. ( extensions of p-adic absolute values)
2. Suppose $M$ and $N$ are $\mathcal{O}$-lattices in $K$. Then $M\widehat{\mathcal{O}}\mathfrak{p}$ and $N\widehat{\mathcal{O}}\mathfrak{p}$ are $\widehat{\mathcal{O}}\mathfrak{p}$-lattices in $\mathbf{V}\mathfrak{p}$ and
   1. $(M:N)\mathcal{O}\widehat{\mathcal{O}}\mathfrak{p}=(M\widehat{\mathcal{O}}\mathfrak{p}:N\widehat{\mathcal{O}}\mathfrak{p}){\widehat{\mathcal{O}}\mathfrak{p}}$
   2. $D_\mathcal{O}(N)\widehat{\mathcal{O}}_\mathfrak{p}=D_{\widehat{\mathcal{O}}_\mathfrak{p}}(N\widehat{\mathcal{O}}_\mathfrak{p})$
   3. $\mathfrak{d}\mathcal{O}(N)\widehat{\mathcal{O}}\mathfrak{p}=\mathfrak{d}{\widehat{\mathcal{O}}\mathfrak{p}}(N\widehat{\mathcal{O}}_\mathfrak{p})$
3. Suppose $M$ and $N$ are $\widehat{\mathcal{O}}_\mathfrak{p}$-lattices in $\mathbf{V}_p$ such that $M=\bigoplus\pi_i(M)$ and $N=\bigoplus\pi_i(N)$. Then
   1. $(M:N){\widehat{\mathcal{O}}\mathfrak{p}}=\prod(\pi_i(M):\pi_i(N)){\widehat{\mathcal{O}}\mathfrak{p}}$
   2. $D_{\widehat{\mathcal{O}}_\mathfrak{p}}(N)=\bigoplus D_{\widehat{\mathcal{O}}_\mathfrak{p}}(\pi_i(N))$
   3. $\mathfrak{d}{\widehat{\mathcal{O}}\mathfrak{p}}(N)=\prod\mathfrak{d}{\widehat{\mathcal{O}}\mathfrak{p}}(\pi_i(N))$
4. Let $\mathcal{O}i$ be the Valuation ring in $K_i$, $|\cdot|$ being the unique absolute value on $K_i$ which extends the $\mathfrak{p}$-adic absolute value on $\widehat{k\mathfrak{p}}$. Then
   1. $\mathcal{O}i$ is the integral closure of $\widehat{\mathcal{O}}\mathfrak{p}$ in $K_i$.
   2. The topological closure of $\mathcal{O}K\mathcal{O}\mathfrak{p}$ in $\mathbf{V}_\mathfrak{p}$ is $\mathcal{O}K\widehat{\mathcal{O}}\mathfrak{p}=\bigoplus\mathcal{O}_i$.
5. Suppose $I$ is an $\mathcal{O}_K$- fractional ideal in $K$.
   1. $I\widehat{\mathcal{O}}_\mathfrak{p}$ fits the assumptions of (3)
   2. $N_{K/k}(I)\widehat{\mathcal{O}}_\mathfrak{p}=\prod N_{K_i/\widehat{k_\mathfrak{p}}}(\pi_i(I)\mathcal{O}_i)$
   3. $N_{K/k}:\mathcal{F}(\mathcal{O_K})\to\mathcal{F}(\mathcal{O})$ is a group homomorphism.
   4. $\pi_i(\mathfrak{D}{K/k})\mathcal{O}i=\mathfrak{D}{K_i/\widehat{k\mathfrak{p}}}$ (see different)
   5. $\mathfrak{d}{K/k}\widehat{\mathcal{O}}\mathfrak{p}=\prod \mathfrak{d}{K_i/\widehat{k\mathfrak{p}}}$ (see discriminant)
   6. $N_{K/k}(\mathfrak{D}_{K/k})=\mathfrak{d}_{K/k}$

Related

• ring of integers