Closure of an ideal with respect to p-adic valuation

Question

Goro Shimura, Euler Products and Eisenstein Series,

Chapter II (Adelization of algebraic groups and automorphic forms)

"Let $\mathbf{F}$ be an algebraic number field of finite degree. We denoe by $\mathfrak{a}$ and $\mathfrak{h}$ the sets of archimedean primes and nonarchimedean primes of $\mathbf{F}$; Further we denote by $\mathfrak{g}$ the maximal order of $\mathbf{F}$. For every $\nu \in \mathbf{a} \cup \mathbf{h}$ we denote by $\mathbf{F}_\nu$ the $\nu$-completion of $\mathbf{F}$."

What does the followin mean? What about if we set $\mathbf{F}=\mathbb{Q}$, and special simple ideal?

"In particular, for $\nu \in \mathbf{h}$ and a $\mathfrak{g}$-ideal $\mathfrak{a}$ we denote by $\mathfrak{a}_\nu$ the $\nu$-closure of $\mathfrak{a}$ in $\mathbf{F}_\nu$, which coinsides with the $\mathfrak{g}_\nu$-linear span of $\mathfrak{a}$ in $\mathbf{F}_\nu$."

Is there some introductory note or book, which can explain this??

Answer 1

You probably want to read Serre's Local Fields.

In any case, let me briefly explain everything over $F = \mathbb{Q}$:

• $\mathbf{a}$, the set of archimedean places of $\mathbb{Q}$, corresponds to the real place $\mathbb{R}$ of $\mathbb{Q}$.
• $\mathbf{h}$, the set of nonarchimedean places of $\mathbb{Q}$, is just the set of all prime ideals of $\mathbb{Z} = \mathcal{O}_{\mathbb{Q}}$, namely the ideals $p\mathbb{Z}$ with $p$ a prime.
• $\mathfrak{g}$, the maximal order of $\mathbb{Q}$, is the ring of integers $\mathbb{Z} = \mathcal{O}_{\mathbb{Q}}$ of $\mathbb{Q}$.

Now let me describe the completion of $F = \mathbb{Q}$ at a place:

• At the real place, the completion (as a metric space) of $\mathbb{Q}$ with respect to the usual (archimedean) absolute value is $\mathbb{R}$.
• At a nonarchimedean place corresponding to a prime $p$, this completion is denoted $\mathbb{Q}_p$. This is a local field that contains $\mathbb{Q}$. It is the completion (as a metric space) of $\mathbb{Q}$ with respect to the $p$-adic absolute value on $\mathbb{Q}$. $\mathbb{Q}_p$ contains $\mathbb{Z}_p$, which in turn has a maximal ideal $p\mathbb{Z}_p$.
• Given an ideal $n\mathbb{Z} = \mathfrak{a}$ of $\mathbb{Z} = \mathfrak{g}$, we let $\mathfrak{a}_p$ denote the $p$-closure of $\mathfrak{a}$ in $\mathbb{Q}_p$, which coincides with the $\mathbb{Z}_p$-linear span of $\mathfrak{a}$ in $\mathbb{Q}_p$.

Many texts in algebraic number theory discuss local fields in great detail. My favourite is Goldstein's Analytic Number Theory (in spite of its name, it's really about algebraic number theory).

A good introduction to the adèlisation of automorphic forms is Goldfeld and Hundley's Automorphic Representations and $L$-Functions for the General Linear Group; Chapter 1 covers the local fields $\mathbb{Q}_p$ and the adèle ring $\mathbb{A_Q}$ of $\mathbb{Q}$ in a very readable fashion.

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To expand on the last point: recall that $\mathbb{Q}$ is a subset of $\mathbb{Q}_p$, and so an ideal $\mathfrak{a} = n\mathbb{Z}$ of $\mathbb{Z}$ is also a subset of $\mathbb{Q}_p$. However, it is not a closed subset of $\mathbb{Q}_p$ with respect to the topology induced by the $p$-adic absolute value, just as $\mathbb{Q}$ is not complete with respect to this metric. We denote by $\mathfrak{a}_p$ the closure of $\mathfrak{a}$ in $\mathbb{Q}_p$ with respect to this topology.

On the other hand, since $\mathfrak{a}$ is a subset of $\mathbb{Q}_p$, we can create a new subset of $\mathbb{Q}_p$ consisting of all linear combinations of elements of $\mathfrak{a}$ with coefficients in $\mathbb{Z}_p$ (which is a subset of $\mathbb{Q}_p$); that is, elements of the form $x_1 \alpha_1 + \cdots + x_k \alpha_k$ with $x_1,\ldots,x_k \in \mathbb{Z}_p$ and $\alpha_1, \ldots, \alpha_k \in \mathfrak{a}$. Then this is again a subset of $\mathbb{Q}_p$, and it is a theorem that this subset is equal to $\mathfrak{a}_p$.