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

The ring and the field Edit

The section introduces one of the main players in arithmetic geometry: the p-adics. 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 RemarksEdit

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  .

DefinitionsEdit

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

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.

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

The field  Edit

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

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

  1.   if and only if  
  2.  
  3.  

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

  1.  

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 EquationsEdit

This section gives us the criterion for finding  -adic varieties, or even better, schemes in  .

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

SolutionsEdit

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 $f_1,\ldots,f_k \in \mathbb{Z}_p[x_1,\ldots,x_n]$: 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 SolutionsEdit

=ApplicationsEdit

In the next chapter Serre will be applying the tools here to study the polynomial

  in  

The Multiplicative Group of Edit

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 UnitsEdit

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

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.

Squares in  Edit