# A User's Guide to Serre's Arithmetic/p-adic Fields

## The ring and the field

[edit | edit source]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

[edit | edit source]For example, if you have an arithmetic scheme (such as or ) then you can consider the base change to . From the inverse system

there is an associated direct system of schemes

which gives . Another example of a system of schemes is in deformation theory. For example, consider a scheme

Deformation theory can be used to ask if there is a scheme which fits into a cartesian square

This question can be repeatedly asked to get a directed system of schemes

where each square is cartesian. It turns out these questions are cohomological. All deformations depend on the cohomology group and all "obstructions" to a deformation live in a group depending on . If we have an algebraic curve then 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 which is an algebraic curve over each point . Then, the surface can be deformed into such a system. Deformations then give us another example of constructing a formal scheme .

### Definitions

[edit | edit source]Set . You should think of elements in as finite sums

- where each

There is an obvious morphism with kernel sending

We can use these morphisms to construct an inverse system

whose inverse limit is defined as the **p-adic integers** . Elements in should be thought of infinite sums

- such that

It is sometimes convenient to write these infinite sums as infinite tuples

Let's play around with to try and get a feel for what the -adics are about. Since there is a unique morphism we can ask what the image of elements in look like. If we consider , then

So all we did was find the decomposition of the integer in terms of base-. Negative numbers are a little more tricky since we need to figure out what "means" in . Notice if we take the sum

Then, in we can see that

In we can find that is

An interesting set of numbers to look at are the 's. For example,

We can then look to see what the units in are like. Observe that for

If we have then

From this we see a -adic integer is invertible if and only if the .

### Properties of

[edit | edit source]The previous observations/computations should make the first two propositions easy to parse.

The last part of this section shows how to topologize the -adics. From proposition 2 we know that any -adic integer is of the form where is a unit. We define the **-adic valuation** of this integer as

- by and

For example

- and

Notice that

- and

In particular

The -adic valuation can be used to topologize by defining the metric

From the definition of the -adic valuation and it's properties with respect to negatives we can see that

- and

Since

and

we can see that the triangle inequality holds

We could have also taken the algebraic approach of defining the topology in terms of the neighborhoods of . There are equal to the set

Finally, we could have given it the topology from the product of the where each is equipped with the discrete topology. From Tynchenoff's theorem, we know that this is a compact space. And since 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

[edit | edit source]From the computation earlier, if we wanted to invert an element we would have to find but also invert the . This should give us the hint that the fraction field of is isomorphic to

This is called the **field of -adic numbers**. A -adic number should be thought of as an infinite sum of the form

A useful tool for computing inverses is the formal power series

For example, setting we find that the inverse of in is

and the inverse of is

In general, you have to use iterated long division to find the -adic expansion of a rational number.

We can extend the -adic valuation to by

- and

The metric constructed previously on extends to and defines a locally compact topology. In addition, is dense in using a similar kind of argument as before.

#### Absolute Values (Extra)

[edit | edit source]There is an alternative construction of the p-adic numbers using a valuation on . Given a rational number such that we can construct the -adic absolute value

- defined by

using the -adic valuation on . This absolute value satisfies the following axioms

- if and only if

In addition, it satisfies a stronger version of 3. called the **non-archimedian property**

A natural question to ask then is if there exists a classification scheme for absolute values on . This turns out to be true and is called **Ostrowski's Theorem**. These notes give an introduction and proof to this theorem. In addition, there is a generalization to a number field (meaning it is a finite field extension of ) which shows that the isomorphism classes of absolute values on are classified by the closed points of . This is discussed in these notes by Keith Conrad.

## p-adic Equations

[edit | edit source]This section gives us the criterion for finding -adic varieties, or even better, schemes in .

- add section with discussion of Hensel's lemma in both the simple and general cases
- given which is square free, we can show that the vanishing locus of has no rational points.

### Solutions

[edit | edit source]This section starts out with a useful technical lemma: a projective system

of finite non-empty sets has a non-empty inverse limit . This is directly applied to the case of considering a finite set of polynomials : they have a non-empty vanishing locus in if and only if their reductions have a solution in for each . 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 ? This is answered in the next subsection where Serre proves Hensel's lemma.

### Amelioration of Approximate Solutions

[edit | edit source]### =Applications

[edit | edit source]In the next chapter Serre will be applying the tools here to study the polynomial

- in

## The Multiplicative Group of

[edit | edit source]The section studies the various multiplicative groups we have encountered so far: 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

[edit | edit source]This subsection determines some of the roots of unity containted in , hence . Serre does this through a filtration on the group of units

given by

where

Notice that each is the kernel of the morphism

- sending

We can see that

- since

There is a short exact sequence

since contains the -adic integers of the form while can have any . Furthermore, there are short exact sequences of the form

This is because if we take two elements we can multiply them together to get

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

### Structure of the Group

[edit | edit source]This subsection determines the structure of the group . It uses the observation that an is equal to and , hence we can decompose this group as the product . Now we are reduced to determining the group structure of — this is done in proposition 8.