What is a $p$-adic group

I saw the name $p$-adic group on a book I was reading, so I tried to find some related documents. Although I've found something on this topic, there is no definition.

Would anyone please explain the definition for a $p$-adic group to me? Thanks very much.

Solution 1:

Since the only tag you put is group-theory, it may be the case that you do not have the appropriate background for understanding the following definition. Nonetheless, here is an attempt.

First, let $\mathbb{Q}_p$ be the field of $p$-adic numbers, and let $\overline{\mathbb{Q}}_p$ be an algebraic closure. Let $G=Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$. If you look at the field $\overline{\mathbb{Q}_p}$ itself, you can see that it is the solution set of the zero polynomial, and that the zero polynomial is the only polynomial to vanish identically on $\overline{\mathbb{Q}}_p$. We say that $\overline{\mathbb{Q}}_p$ is an algebraic group, associated to the affine variety corresponding to the vanishing set of the zero polynomial. The algebraic group $\mathbb{Q}_p$ is now the set of fixed points (we call them the $\mathbb{Q}_p$-rational points) for the Galois action of $G$ on $\overline{\mathbb{Q}}_p$. Note that this is well defined because the zero polynomial is defined over $\mathbb{Q}_p$, that is, its coefficients lie in $\mathbb{Q}_p$.

Another example is $\overline{\mathbb{Q}}_p^\times$. This can be identified with the vanishing set for the polynomial $XY-1$ in $\overline{\mathbb{Q}}_p^2$. Again, since $XY-1$ is defined over $\mathbb{Q}_p$, we can see the algebraic group $\mathbb{Q}_p^\times$ as the set of fixed points for the action of $G$ on $\overline{\mathbb{Q}}_p^\times$.

Now, more generally, what you need is an affine variety $X$ over $\overline{\mathbb{Q}}_p$ with a group structure on it which is given by polynomial equations. More precisely, both multiplication and inversion are to be given by polynomial equations. This is an algebraic group. It will be given as the vanishing set of a collection of polynomials. If these polynomials can be chosen over $\mathbb{Q}_p$, as well as the polynomials giving multiplication and inversion, then you have an action of $G$ on $X$, and the fixed points will form an algebraic group over $\mathbb{Q}_p$. Therefore, here is the definition:

A $p$-adic group is an algebraic group over a $p$-adic field, or more precisely, it is the $\mathbb{Q}_p$-rational points of an algebraic group over $\overline{\mathbb{Q}}_p$.

The above is a definition. Here is a very important result:

Theorem: Every (affine) algebraic group is linear, that is, it can realised as a subgroup of a matrix group.

Hence, every $p$-adic group is a group of matrix with entries in the field $\mathbb{Q}_p$ of $p$-adic numbers.

Solution 2:

The following is not a perfect description of a $p$-adic group, but hopefully it will help a bit.

Let $F$ be a non-Archimedean local field: That is you have a valuation $v: F \to \mathbb{Z}\cup \{\infty\}$ which satisfies that $v\lvert_{F^{\times}}: F^{\times} \to \mathbb{Z}$ is a homomorphism. You have a subring of integers $\mathcal{O}$ of $F$ which is defined by $\mathcal{O} = \{r \in F\lvert v(f) \geq 0\}$. You have a unique maximal ideal $\mathfrak{p}$ in $\mathcal{O}$: $\mathfrak{p} = \{r \in \mathcal{O} \lvert v(r) > 0\}$. We have $\mathcal{O} / \mathfrak{p}$ is a field; assume that it is finite or order $q$.

A $p$-adic group is a group over such a field. You can think of for example $GL_n(F)$.

As something a bit more concrete, you can take $\mathbb{Q}$. Here you have a valuation $v$ defined by $v\left(\frac{a}{b}\right) = v(a) - v(b)$, where if for $a\in \mathbb{Q}$ $a = p^nc$, ($p \nmid c$) $v(a) = n$. The completion of $\mathbb{Q}$ with respect to the norm $\lvert x\lvert = \left(\frac{1}{q}\right)^{v(x)}$ is called $\mathbb{Q}_p$.

Note: There are other ways to see this and there are probably people who are going to disagree with the outline of the definition that I have given, but maybe it will be helpful anyway just to illustrate some of the things there are hiding in the background when talking about $p$-adic groups.