View source for A User's Guide to Serre's Arithmetic/p-adic Fields

==The ring <math>\mathbb{Z}_p</math> and the field <math>\mathbb{Q}_p</math>==
The section introduces one of the main players in arithmetic geometry: the p-acids. This chapter studies a few basic properties of the p-adics including their topological structure, multiplicative structure, and solutions of affine polynomials in them.

===(Optional) Advanced Remarks===
For example, if you have an arithmetic scheme <math>X \in \textbf{Sch}/\mathbb{Z}</math> (such as <math>\text{Spec}(\mathbb{Z}[x_1,\ldots,x_n]/(f_1,\ldots,f_k))</math> or <math>\text{Proj}(\mathbb{Z}[x_1,\ldots,x_n]/(f_1,\ldots,f_k))</math>) then you can consider the base change to <math>X_p \in \textbf{Sch}/(\mathbb{Z}/p)</math>. From the inverse system
:<math>\cdots \to \mathbb{Z}/(p^3) \to \mathbb{Z}/(p^2) \to \mathbb{Z}/(p)</math>
there is an associated direct system of schemes
:<math>X_p \to X_{p^2} \to X_{p^3} \to \cdots</math>
which gives <math>\mathfrak{X}_p = \text{Spf}(X_{p^k})</math>. Another example of a system of schemes is in deformation theory. For example, consider a scheme
:<math>
\begin{matrix}
X_1\\
\downarrow \\
\mathbb{Z}/(p)
\end{matrix}
</math>
Deformation theory can be used to ask if there is a scheme <math>X_2 \in \textbf{Sch}/(\mathbb{Z}/(p^2))</math> which fits into a cartesian square
:<math>
\begin{matrix}
X_1 & \to & X_2 \\
\downarrow & & \downarrow \\
\text{Spec}(\mathbb{Z}/p) & \to & \text{Spec}(\mathbb{Z}/p^2)
\end{matrix}
</math>
This question can be repeatedly asked to get a directed system of schemes
:<math>
\begin{matrix}
X_1 & \to & X_2 & \to & X_3 & \to \cdots \\
\downarrow & & \downarrow & & \downarrow & \\
\text{Spec}(\mathbb{Z}/p) & \to & \text{Spec}(\mathbb{Z}/p^2) & \to & \text{Spec}(\mathbb{Z}/p^3) & \to \cdots
\end{matrix}
</math>
where each square is cartesian. It turns out these questions are cohomological. All deformations depend on the cohomology group <math>H^1(X_1,T_{X_1})</math> and all "obstructions" to a deformation live in a group depending on <math>H^2(X_1,T_{X_1})</math>. If we have an algebraic curve <math>X/\mathbb{F}_p</math> then <math>H^2(X,T_X)=0</math> because of dimension reasons. This implies that we can always deform and get a direct system of schemes as above. We can make a minor generalization of this case by considering an arithmetic surface <math>X/\mathbb{Z}</math> which is an algebraic curve over each point <math>(p)</math>. Then, the surface can be deformed into such a system. Deformations then give us another example of constructing a formal scheme <math>\mathfrak{X} = \text{Spf}(X_i)</math>.
===Definitions===
Set <math>A_k = \mathbb{Z}/p^k</math>. You should think of elements in <math>A_k</math> as finite sums
:<math>a_0 + a_1p + a_2p^2 + \cdots + a_{k-1}p^{k-1} + a_kp^k</math> where each <math>a_i \in \{0,1,2,\ldots,p-1\}</math>
There is an obvious morphism <math>\phi_k: A_k \to A_{k-1}</math> with kernel <math>p^{k-1}A_k</math> sending
:<math>\phi_k(a_0 + a_1p + a_2p^2 + \cdots + a_{k-1}p^{k-1} + a_kp^k) = a_0 + a_1p + a_2p^2 + \cdots + a_{k-1}p^{k-1}</math>
We can use these morphisms to construct an inverse system
:<math>\cdots \to A_3 \to A_2 \to A_1</math>
whose inverse limit is defined as the '''p-adic integers''' <math>\mathbb{Z}_p = \lim_{\leftarrow}A_k</math>. Elements in <math>\mathbb{Z}_p</math> should be thought of infinite sums
:<math>a_0 + a_1p + a_2p^2 + \cdots </math> such that <math>a_i \in \{0,1,\ldots,p-1\}</math>
It is sometimes convenient to write these infinite sums as infinite tuples
:<math>(a_0,a_1,a_2,\ldots)</math>
Let's play around with <math>\mathbb{Z}_5</math> to try and get a feel for what the <math>p</math>-adics are about. Since there is a unique morphism <math>\mathbb{Z} \to \mathbb{Z}_p</math> we can ask what the image of elements in <math>\mathbb{Z}</math> look like. If we consider <math>89</math>, then
:<math> 89 = 4 + 2\cdot 5 + 3\cdot 5^2 = (4,2,3,0,0,\ldots)</math> 
So all we did was find the decomposition of the integer in terms of base-<math>5</math>. Negative numbers are a little more tricky since we need to figure out what <math>-1</math> "means" in <math>\mathbb{Z}_p</math>. Notice if we take the sum
:<math>
\begin{align}
(1,0,0,0,\ldots) + (4,4,4,4,\ldots) &= (4+1,4,4,4,4,\ldots) \\
&=(0,1,0,0,\ldots) + (0,4,4,\ldots) & \text{ since we carried the } 1 \\
&= (0,4+1,4,4,\ldots) &\\
&= (0,0,4+1,4,\ldots) &\\
&= (0,0,0,0,\ldots) &
\end{align}
</math>
Then, in <math>\mathbb{Z}_p</math> we can see that
:<math>
-1 = (p-1,p-1,p-1,\ldots)
</math>
In <math>\mathbb{Z}_5</math> we can find that <math>-7</math> is
:<math>(0,3,4,4,4,\ldots)</math>
An interesting set of numbers to look at are the <math>-5^k</math>'s. For example,
:<math> -125 = (0,0,0,1,4,4,4,\ldots)</math>
We can then look to see what the units in <math>\mathbb{Z}_p</math> are like. Observe that for <math>a,b\in \mathbb{Z}_p</math>
:<math>
\begin{align}
a\cdot p &=(a_0 + a_1\cdot p + a_2\cdot p^2 + \cdots)\cdot(b_0 + b_1\cdot p + b_2\cdot p^2 + \cdots) \\
&= (a_0b_0) + (a_0b_1 + a_1b_0)\cdot p + (a_0b_2 + a_1b_1 + a_2b_0)\cdot p^2 + \cdots
\end{align}
</math>
If we have <math>a\cdot b = 1</math> then
:<math>
\begin{align}
a_0b_0 &=1 \\
a_0b_1 + a_1b_0 &=0 \\
a_0b_2 + a_1b_1 + a_2b_0 &= 0 \\
\cdots
\end{align}
</math>
From this we see a <math>p</math>-adic integer <math>a</math> is invertible if and only if the <math>a_0\neq 0</math>.
===Properties of <math>\mathbb{Z}_p</math>===
The previous observations/computations should make the first two propositions easy to parse.

The last part of this section shows how to topologize the <math>p</math>-adics. From proposition 2 we know that any <math>p</math>-adic integer is of the form <math>p^ku</math> where <math>u</math> is a unit. We define the <math>p</math>'''-adic valuation''' of this integer as
:<math>v_p:\mathbb{Z}_p \to \mathbb{Z}</math> by <math>v_p(p^ku) = k</math> and <math>v_p(0) = +\infty</math>
For example
:<math>v_p(-25) = v_p(0,0,1,4,4,\ldots) = 2</math> and <math>v_p(-1) = v_p(4,4,4,4,4,\ldots) = 0</math>
Notice that
:<math>v_p(xy) = v_p(x) + v_p(y)</math> and <math>v_p(x+y) \geq \text{inf}(v_p(x),v_p(y))</math>
In particular
:<math>v_p(-x) = v_p((-1)\cdot x) = v_p(-1) + v_p(x) = v_p(x)</math>
The <math>p</math>-adic valuation can be used to topologize <math>\mathbb{Z}_p</math> by defining the metric
:<math>d(x,y) = e^{-v_p(x,y)}</math>
From the definition of the <math>p</math>-adic valuation and it's properties with respect to negatives we can see that
:<math>d(x,x) = e^{-\infty} = 0</math> and <math>d(x,y) = d(y,x)</math>
Since
:<math>v_p(x - z) = v_p((x-y) + (y-z)) \geq \text{inf}(v_p(x-y),v_p(y-z))</math>
and
:<math>-v_p(x - z) \leq -\text{inf}(v_p(x-y),v_p(y-z))</math>
we can see that the triangle inequality holds
:<math>d(x,z) \leq d(x,y) + d(y,z)</math>
We could have also taken the algebraic approach of defining the topology in terms of the neighborhoods <math>p^k\mathbb{Z}_p</math> of <math>0</math>. There are equal to the set
:<math>\{a \in \mathbb{Z}_p : v_p(a) \geq k\}</math>
Finally, we could have given it the topology from the product of the <math>\prod A_k</math> where each <math>A_k</math> is equipped with the discrete topology. From Tynchenoff's theorem, we know that this is a compact space. And since <math>\mathbb{Z}_p \subset \prod A_k</math> is closed it is also compact.

# edit/reorganize
# show density is obvious
# http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf for hensel's lemma
# Completeness of compact metric space - https://math.stackexchange.com/questions/627667/every-compact-metric-space-is-complete

===The field <math>\mathbb{Q}_p</math>===
From the computation earlier, if we wanted to invert an element <math>p^ku</math> we would have to find <math>v = u^{-1}</math> but also invert the <math>p^k</math>. This should give us the hint that the fraction field <math>\mathbb{Q}_p</math> of <math>\mathbb{Z}_p</math> is isomorphic to
:<math>\mathbb{Z}_p\left[\frac{1}{p}\right]</math>
This is called the '''field of <math>p</math>-adic numbers'''. A <math>p</math>-adic number should be thought of as an infinite sum of the form
:<math>\frac{a_{-k}}{p^k} + \frac{a_{-k+1}}{p^{k-1}} + \cdots + \frac{a_{-1}}{p} + a_0 + a_1\cdot p + a_2\cdot p^2 + \cdots</math>
A useful tool for computing inverses is the formal power series
:<math>\frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - \cdots</math>
For example, setting <math>x=p</math> we find that the inverse of <math>1+p</math> in <math>\mathbb{Z}_5</math> is
:<math> (1,4,1,4,1,4,1,4,\ldots)</math>
and the inverse of <math>1-p</math> is
:<math>(1,1,1,1,1,1)</math>
In general, you have to use [https://math.stackexchange.com/a/1187037/251222 iterated long division] to find the <math>p</math>-adic expansion of a rational number.

We can extend the <math>p</math>-adic valuation to <math>v_p:\mathbb{Q}_p \to \mathbb{Z}\cup\{\infty\}</math> by
:<math>v_p(p^ku) = k</math> and <math>v_p(0) = +\infty</math>
The metric constructed previously on <math>\mathbb{Z}_p</math> extends to <math>\mathbb{Q}_p</math> and defines a locally compact topology. In addition, <math>\mathbb{Q}</math> is dense in <math>\mathbb{Q}_p</math> using a similar kind of argument as before.
==== Absolute Values (Extra) ====
There is an alternative construction of the p-adic numbers using a valuation on <math>\mathbb{Q}</math>. Given a rational number <math>a/b \in \mathbb{Q}</math> such that <math>gcd(a,b) = 1</math> we can construct the <math>p</math>-adic absolute value
:<math>|\cdot|_p:\mathbb{Q} \to \mathbb{R}_{\geq 0}</math> defined by <math>|a/b|_p = p^{-v_p(a/b)} = p^{v_p(b) - v_p(a)}</math>
using the <math>p</math>-adic valuation on <math>\mathbb{Q}_p</math>. This absolute value satisfies the following axioms
# <math>|x|_p = 0</math> if and only if <math>x=0</math>
# <math>|xy|_p = |x|_p\cdot|y|_p</math>
# <math>|x+y|_p \leq |x|_p + |y|_p</math>
In addition, it satisfies a stronger version of 3. called the '''non-archimedian property'''
# <math>|x+y|_p \leq \text{max}\{|x|_p, |y|_p\}</math>

A natural question to ask then is if there exists a classification scheme for absolute values on <math>\mathbb{Q}</math>. This turns out to be true and is called '''Ostrowski's Theorem'''. These [https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec5.pdf notes] give an introduction and proof to this theorem. In addition, there is a generalization to a number field <math>K/\mathbb{Q}</math> (meaning it is a finite field extension of <math>\mathbb{Q}</math>) which shows that the isomorphism classes of absolute values on <math>K</math> are classified by the closed points of <math>\text{Spec}(\mathcal{O}_K)</math>. This is discussed in these [http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/ostrowskinumbfield.pdf notes] by Keith Conrad.

==p-adic Equations==
This section gives us the criterion for finding <math>p</math>-adic varieties, or even better, schemes in <math>\textbf{Sch}/\mathbb{Z}_p</math>.
# add section with discussion of Hensel's lemma in both the simple and general cases
# given <math>a\in \mathbb{Z}</math> which is square free, we can show that the vanishing locus of <math>ay^2 = f(x)</math> has no rational points.
===Solutions===
This section starts out with a useful technical lemma: a projective system
:<math>\cdots\to D_3 \to D_2 \to D_1</math>
of finite non-empty sets has a non-empty inverse limit <math>D = \lim_{\leftarrow}D_i</math>. This is directly applied to the case of considering a finite set of polynomials <math>f_1,\ldots,f_k \in \mathbb{Z}_p[x_1,\ldots,x_n]</math>: they have a non-empty vanishing locus in <math>(\mathbb{Z}_p)^m</math> if and only if their reductions <math>\text{mod }p^k</math> have a solution in <math>(\mathbb{Z}/p^k)^m</math> for each <math>k</math>. This proposition can be considered in the homogeneous case as well.

We should then be asking ourselves: how can we guarantee that there is a solution in each <math>(\mathbb{Z}/p^k)^m</math>? This is answered in the next subsection where Serre proves Hensel's lemma.

===Amelioration of Approximate Solutions===

====Applications===
In the next chapter Serre will be applying the tools here to study the polynomial
:<math>z^2 = ax^2 + by^2</math> in <math>\mathbb{Z}_p</math>

==The Multiplicative Group of <math>\mathbb{Q}_p</math>==
The section studies the various multiplicative groups we have encountered so far: <math>\mathbb{F}_p^*,\mathbb{Z}_p^*, \mathbb{Q}_p^*</math> and the squares of these groups. This tools in this section will be useful in the next chapter when Serre discusses the Hilbert symbol.
===The Filtration of the Group of Units===
This subsection determines some of the roots of unity containted in <math>\mathbb{Z}_p</math>, hence <math>\mathbb{Q}_p</math>. Serre does this through a filtration on the group of units
:<math>U = \mathbb{Z}_p^* = \{ a_0 + a_1\cdot p + a_2\cdot p^2 + \cdots |  a_0 \neq 0 \}</math>
given by
:<math>U \supseteq U_1 \supseteq U_2 \supseteq U_3 \supseteq \cdots</math>
where
:<math>U_n = 1 + p^n\mathbb{Z}_p = \{1 + 0\cdot p +\cdots + 0 \cdot p^{n-1} + a_n\cdot p^n + \cdots \}</math>
Notice that each <math>U_n</math> is the kernel of the morphism
:<math>\varepsilon_n: \mathbb{Z}_p^* \to (\mathbb{Z}/p^n)^*</math> sending <math>a_0 + a_1\cdot p + a_2\cdot p^2 + \cdots \mapsto a_0 + a_1\cdot p + \cdots + a_{n-1}\cdot p^{n-1}</math>
We can see that
:<math> U \cong \lim_{\leftarrow}\frac{U}{U_n}</math> since <math> \frac{U}{U_n} = \{a_0 + a_1\cdot p + \cdots + a_{n-1}\cdot p^{n-1} | a_0 \neq 0 \}</math>
There is a short exact sequence
:<math>1 \to U_1 \to U \to \mathbb{F}_p^* \to 1</math>
since <math>U_1</math> contains the <math>p</math>-adic integers of the form <math>1+a_1\cdot p + a_2\cdot p^2 + \cdots </math> while <math>U</math> can have any <math>a_0\neq 0</math>. Furthermore, there are short exact sequences of the form
:<math>1 \to U_{n+1} \to U_n \to \mathbb{Z}/p \to 0</math>
This is because if we take two elements <math>(1+p^nx), (1 + p^ny) \in U_n</math> we can multiply them together to get
:<math> (1+p^nx)\cdot(1 + p^ny) = 1 + p^n(x+y) + p^{2n}xy \equiv 1 + p^n(x+y) \text{ } (\text{mod } p^{n+1})</math>

Serre then introduces a useful auxillary lemma to analyze the following direct system of short exact sequences

===Structure of the Group <math>U_1</math>===
This subsection determines the structure of the group <math>\mathbb{Q}_p^*</math>. It uses the observation that an <math>x\in \mathbb{Q}_p^*</math> is equal to <math>p^n\cdot u<math> where <math>n\in \mathbb{Z}</math> and <math> u \in U</math>, hence we can decompose this group as the product <math>\mathbb{Z}\times U</math>. Now we are reduced to determining the group structure of <math>U</math> — this is done in proposition 8.

===Squares in <math>\mathbb{Q}_p^*</math>===

:<math>
\frac{\mathbb{Q}_p^*}{(\mathbb{Q}_p^*)^2} = \begin{cases}
(\mathbb{Z}/2)^3 & p = 2 \\
(\mathbb{Z}/2)^2 & p\text{ odd}\\
\text{ }\mathbb{Z}/2 & p = \infty 
\end{cases}
</math>

{{BookCat}}