People have been trying to find an analogue of complex analysis in non-archimedean fields. But these spaces have some weird properties,
They are totally disconnected. This means the intuitive notion of continuity in classical analysis will not work here, analytic continuation along ‘paths’ will not work.
The construction of reasonable theory of analytic functions in which local properties of functions determine their global properties is not straightforward.
M. Krasner come up with an elementary approach in 1940s
An analytic function on \(U\subset k\), \(f:U\to k\) is a limit of rational functions regular on \(U\).
Next he introduces a class of quasi-connected subsets of \(k\) that have the analytic continuation property, for any analytic element \(f\) on \(U\), if \(f|_V=0\) for any open subset \(V\subset U\), then \(f=0\) on \(U\).
Example: closed disks minus finite number of open disks are quasi-connected. These are the sets that although are not topologically connected, they are connected in the analytic sense.
Remark: this concept corresponds to the connected affinoid domains in Tate’s approach.
Finally, a function on an open subset is \(U\) analytic if \(U\) can be covered by quasi-connected opens \(U_i\) on which \(f|_{U_i}\) is analytic.
Tate’s approach is inspired by Krasner’s elementary approach, but he uses less elementary languages that generalizes.
First introduce a class of commutative Banach algebras over \(k\) called affinoid algebras, of the form \(A=k\{T_1,\dots,T_n\}/I\).
The maximal spectrum \(X=\mathrm{mSpec}(A)\) has a canonical topology induced by \(k\), but this topology is not very interesting.
Instead of the canonical topology Tate introduces a class of open subsets of \(X\) and a class of coverings which gives a Grothendieck topology on \(X\).
Then a structure sheaf of rings \(\mathcal{O}_X\) on this Grothendieck topology can be defined. The pair \((X,\mathcal{O}_X)\) is now an affinoid space.
Berkovich considered an approach that generalizes the definition of analytic functions and analytic spaces over \(\mathbb{C}\). The goal is to get a good definition which makes the analytic functions on the unit disk \(D(0,1)\) coincide with \(k\{T\}\).
A naive definition of analytic function \(f:k\to k\) using locally analytic function is, \(f\) is analytic iff for any \(x\in k\) there is a closed ball \(D(x,r)\) such that \(f\) is an analytic power series on \(D(x,r)\).
The problem with this definition is that the class of functions is too large since \(k\) is totally disconnected.
This similar problem occurs if you consider the set of ‘continuous’ functions on another familiar totally disconnected space \(f:\mathbb{Q}\to \mathbb{C}\), the class of ‘continuous’ function using the induced topology obtained is larger than the original class of continuous functions on \(\mathbb{R}\to \mathbb{C}\).
We know we can use completion to obtain the continuous functions on \(\mathbb{R}\), but for our \(k\), it is already complete, instead we need a different notion of ‘completion’ that solves the issue and retains the \(\mathbb{Q}\leadsto \mathbb{R}\) property.
Suppose we have an algebra of \(\mathscr{C}\) continuous functions on \(I=[0,1]\), how do we find the process \(\mathscr{C}\leadsto I\)? The answer is Gel’fand space of maximal ideals of \(\mathscr{C}\). This process is valid for any commutative Banach algebra over the complex numbers.
This motivates we do the same for non-archimedean \(k\) and apply it to the commutative Banach algebra \(k\{T\}\), though we cannot expect in this case \(k\{T\}\leadsto k\).
There are two goals:
Find the non-archimedean analogue of the Gel’fand space of maximal ideals.
Understand what should be the non-archimedean affine line \(k\{T\}\leadsto \mathbb{A}^1_k\).
The answer is the spectrum \(A\leadsto \mathscr{M}(A)\) of a commutative Banach algebra, which is the set of bounded multiplicative seminorms equipped with natural topology induced by evaluation maps (the weakest topology that makes all evaluation maps continuous).
In the classical situation it gives back to the space of maximal ideals.
\(\mathcal{M}(A)\) is always nonempty, compact and Hausdorff.
Every point \(x\in \mathscr{M}(A)\) determines a character \(A\to \mathscr{H}(x) : f\mapsto f(x) = [|f|_x+\mathfrak{p}_x]\).
Let’s briefly examine what the affine line is and their analytic functions.
\(\mathbb{A}^1\) is the set of multiplicative seminorms on \(k[T]\) extending the valuation of \(k\) (so that it is based over \(k\)). It is equipped with the weakest topology that makes all evaluation maps \(\mathrm{ev}_f:\mathbb{A}^1\to \mathbb{R}_{\ge 0}\) continuous. This is very natural and intuitive: points close to each other should map (evaluate) the same function to close values.
Every point \(x\in \mathbb{A}^1\) determines a character \(k[T]\to \mathscr{H}(x): f\mapsto f(x)\).
An analytic function \(\varphi\) on \(\mathcal{U}\subset \mathbb{A}^1\) is a function mapping each point \(x\in \mathcal{U}\) to the value field \(\mathscr{H}(x)\) at this point, such that it is locally limits of rational functions in \(k(T)\) (limits measured using local valuations on \(\mathscr{H}(x)\)).
This \(\mathbb{A}^1\) constructed is a \(1\)-d locally compact, contractible and locally contractible space. Moreover,
\(k \hookrightarrow \mathbb{A}^1\), defined as \(a\mapsto |\cdot|_k\circ \mathrm{ev}_a\), using the norm on \(k\).
In this way the topology on \(\mathbb{A}^1\) induces the canonical topology on \(k\).
The set of analytic functions on \(D(0,1):=\{x\in \mathbb{A}^1: |T(x)|\le 1\}\) is exactly what we expect, \(k\{T\}\). We also have \(\mathscr{M}(k\{T\})=D(0,1)\).