# A taster of rigid geometry (Pt I: the Tate perspective)

This is the first in a series of 5 posts whose goal is to briefly introduce rigid geometry with a focus on providing a big picture between the interactions between different perspectives of rigid geometry.

# Motivation

We shall not go into the motivation for rigid geometry in any serious depth here since this has been done ad infinitum in other sources ([Con1], [Nic], [FK1],…) . But, we would be remiss to not say anything at all.

One of the most fruitful interactions in the semi-classical theory of algebraic geometry is that between the theory of schemes and the theory of complex analytic varieties. Namely, there is an analytification functor

$\left\{\begin{matrix}\text{Finite type schemes}\\ \text{over }\mathbb{C}\end{matrix}\right\}\to \left\{\begin{matrix}\text{Complex analytic}\\ \text{spaces}\end{matrix}\right\}$

which one can read about, for example, in the book [Nee]. Intuitively, this analytification functor takes a $\mathbb{C}$-scheme $X$ to $X(\mathbb{C})$ which is endowed with the natural structure of a $\mathbb{C}$-manifold (it’s really only a manifold if $X$ is smooth) by taking the algebraic charts and interpretting them as holomorphic charts.

It turns out that this functor preserves many of the properties of $\mathbb{C}$-schemes algebraic geometers are interested in (cf. [SGA1, Exposé XII)). This statement is no better exemplified than in the GAGA results of Serre et al. (e.g. as in [Ser]) as well as the relationship betwen the étale topology of $X$  and the analytic topology of $X^\mathrm{an}$ exhibited by Grothendieck, Artin, et al. (e.g. see [SGA1, Exposé XII Théorème 5.1] and [SGA4, Exposé XI Théorème 4.4]).

This connection has immense benefits for both the theory of schemes and the theory of complex analytic spaces. For example, one has access to a much wider class of objects in the world of complex analytic spaces that don’t exist in the purely algebro-geometric world. As a simple example of this, in the complex analytic world one can make arguments using discs which are convenient for topological arguments since they’re contractible, whereas such arguments aren’t feasible in the purely scheme theoretic situation. This leads to situations where many results about varieties over characteristic $0$ fields are reduced to statements in the complex analytic setting by the so-called Lefschetz principle which are then proved by complex analytic means. For example, for a long period of time the only known proofs of the degeneration of the Hodge spectral sequence (e.g. see this post) and of the Kodaira-Nakano-Akizuki vanishing theorem (cf. loc. cit.) were only known by an application of the Lefschetz principle.

Now, while the the reduction to $\mathbb{C}$ is useful for proving purely geometric results, the passage from something like $\mathbb{Q}_p$ to $\mathbb{C}$ via the Lefschetz principle completely forgets arithmetic. Namely, if $X$ is a variety over $\mathbb{Q}_p$ then the variety $X_{\overline{\mathbb{Q}_p}}$ has an action of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ which is of great interest to number theorists. Such information is completely lost in the passage from $\mathbb{Q}_p$ to $\mathbb{C}$.

To remedy this one might note that $\mathbb{Q}_p$, like $\mathbb{C}$, is an ‘analytic field’it has a natural metric for which it is a complete topological field. So, one might ask whether or not there is some sort of $p$-adic analytification functor which takes varieties over $\mathbb{Q}_p$ and outputs ‘$p$-adic analytic manifolds’. The hope is that such $p$-adic analytic manifolds would have the benefits of complex analytic manifolds (e.g. access to disks) while retaining the arithmetic information contained in the original variety.

The goal of rigid analytic geometry is, essentially, to create such a theory of $p$-adic analytic manifolds. As it turns out such a theory is fraught with technical difficulties from the start. Most prominently is the fact that the naive $p$-adic disk is totally disconnected which makes naive attempts at defining a theory unwieldy, bordering on usless. Thus, the actual structures involved in the theory of rigid analytic spaces are significantly more nuanced than one first might envisagethey are certainly not just ‘spaces which locally look like open subspaces of $\mathbb{Q}_p^n$‘ or the like (although there are theories of this form which are useful in contexts other than geometry e.g. see [Sch]).

In fact, there are (more than) six theories of rigid analytic geometry, and we hope to touch on each of them in these posts. These are Tate’s theory of rigid analytic varieties, Raynaud’s theory of birational geometry of formal schemes, Berkovich’s theory of Berkovich analytic spaces, and Huber’s theory of adic spaces, and Fujiwara–Kato’s theory, and Scholze’s theory of diamonds (although this last one has a slightly different character than the others). In very broad strokes one can think roughly that

• Tate’s theory is like the classical theory of varieties over an algebraically closed field, and roughly has the same advantages and pitfalls.
• Huber’s theory is like the theory of schemes, and roughly also has the same advantages and pitfalls.
• Berkovich’s theory is a middle ground between Tate’s theory and Huber’s theory which has some of the same pitfalls that Tate’s theory has, but with the advantage of being topologically well-behaved (in large generality Berkovich spaces are Hausdorff and arc connected!).
• Raynaud’s theory tries to make precise the idea that rigid analytic varieties over $\mathbb{Q}_p$ should be thought of as the ‘generic fibers’ of formal schemes over $\mathbb{Z}_p$. This has the advantage of making available the full theory of formal geometry (which has historically been well-developed) but has the disadvantage of being somewhat difficult to talk about simple objectsmany non-isomorphic formal schemes have the same ‘generic fiber’ and so rigid analytic varieties in this perspective are something like equivalence classes of formal schemes.
• Fujiwara–Kato’s theory chooses to take Raynaud’s theory to the next level by constructing something akin to the ‘Zariski–Riemann space’ of a formal scheme. Roughly this is something like an inverse limit over all formal schemes which have isomorphic generic fiber.
• In the p-adic situation, Scholze’s theory of diamonds chooses to view rigid spaces in terms of their interaction with extremely degenerate, but extremely flexible spaces: the perfectoid spaces. This has the advantage of allowing one to reduce many proofs to arguments with a point-set topology theoretic flavor. The disadvantage (besides the inherent complicatedness of the objects involved) is that the theory of diamonds can only remember ‘topological information’ (e.g. the etale site).

All of these theories are relatively different in their presentation even though they generally capture, at their heart, the same theory (except for the theory of diamonds as already mentioned). In fact, one of the many strengths of rigid analytic geometry is that one can move between these various theories to use their different strengths. For example, there might be some powerful argument utilizing the well-behaved topology of Berkovich spaces that is completely invisible (or certainly much less obvious) from the perspective Huber.

# The point of these posts

This series of posts is meant to introduce the reader to rigid geometry in terms of these perspectives: the classical Tate perspective on rigid spaces, the perspective of Raynaud and Fujiwara–Kato in terms of rigid fibers of formal schemes, the perspective of Berkovich spaces, Huber’s perspective in terms of adic spaces, and Scholze’s theory of diamonds.

These posts are mainly meant to be high-level introductions, focusing on giving some major definitions, major results, quirky examples and observations, but no real proofs. In particular, we will (of course) not go into any real detail here. For this, I highly suggest the following sets of notes/books:

#### For Tate perspective:

• Tian’s notes [Tia]. These do a really great job of summarizing the key points of the theory and, in particular, strikes a great balance between terseness and rigor.
• Bosch’s book [Bos]. This is the book Tian’s mostly drew from. It is also wonderful, if a bit longer than Tian’s notes. If one wants more detail than what is in [Tian] I suggest looking here.
• The book [FvdP] by Fresnel and Van de Put is a pretty commonly cited place to look for this theory. It is a bit hard to read in places, but introduces most of the general theory in a good amount of rigor, tries to give some motivational examples first (by considering subspaces of $\mathbb{P}^{1,\mathrm{an}}_k$), and gets into more advanced topics (e.g. rigid and étale cohomology).
• The book [BGR] by Bosch, Guntzer, and Remmert. This is a bit like an EGA for the basic theory of rigid spaces. It’s a great place to look for rigorous statements/proofs of technical results.
• The first chapter of [Ayo] is a great place to get a high-level, rapid, but mostly complete introduction to the technical setup of rigid geometry from the Tate perspective.

#### For Raynaud’s perspective:

• I again suggest [Tia], for the same reasons.
• I again suggest [Bos], for the same reasons.
• The book [FK] is essentially a reimagining/strengthening of the classical perspective of Raynaud. This book is only the first book in a series of books. It is written very, very much in an EGA format.

#### For Berkovich’s perspective:

• I suggest the notes [Jon] as a good place to start.
• Of course, the foundational texts [Ber] and [Ber1] are also a great place to look.
• Part I and Part II of [DT] is also helpful.

#### For Huber’s perspective:

• I suggest Huber’s papers [Hub1] and [Hub2] as well as Huber’s book [Hub] for much of the rigorous foundations. This is where I first learned most of this material, and remains the only place for many fundamental results (especially considering things about étale cohomology).
• The book [SW] has a good introduction to adic spaces in Lecture 2 through Lecture 5.
• The notes [Mor] contain many of the foundational results for adic spaces.
• The series of lectures notes from [Con+] also contain much foundational material.

#### For Fujiwara–Kato’s perspective

• The only real canonical reference is the first book on this topic by the authors. Namely, [FK].
• One can also see the paper [FK1] which is a good conceptual introduction to the ideas which prompt the [FK] theory, and some of its advantages.

#### For Scholze’s theory of diamonds

• The best introduction to these ideas is provided in [SW], where they began to take shape. This lacks many of the deep foundational results though.
• These deep foundational results are then instead supplied in [Scho].
• A nice overview of the theory is also provided in the first several lectures of Scholze’s course on Fargues–Scholze (see here).

#### General overview

• One can look at the notes [Con1] of Conrad from the 2007 Arizona Winter School. They do a great job of introducing all the key players, but with a special emphasis on the Tate perspective.
• One can also look at the notes [Nic] which are similar in scope to [Con1] but have a different set of biases for what to emphasize.
• One can also see the text [FK1] which does a good job of discussing a broad swath of the rigid analytic theory.

# The Tate perspective

As mentioned above, Tate’s theory of rigid analytic geometry is very similar in spirit to the classical theory of varieties over an algebraically closed fields $k$namely spaces locally isomorphic to closed subsets of $k^n$. That said, there is an added twist in the Tate setting since the ‘spaces’ we’ll be considering aren’t actually topological spaces!

For the entirety of this post let us fix $k$ to be a non-archimedean complete (but not necessarily locally compact) field with respect to an absolute value $|\cdot |$. Examples of such fields are

• The field $\mathbb{Q}_p$ of $p$-adic numbers with the usual $p$-adic absolute value $|\cdot |_p$ (which we will just usually abbreviate to $|\cdot |$), or finite extensions thereof (with the unique extension of $|\cdot|_p$).
• The field $\breve{\mathbb{Q}}_p$ which is the completion of the maximal unramified extension $\mathbb{Q}_p^\mathrm{ur}$ of $\mathbb{Q}_p$ (with respect to the unique extension of $|\cdot |_p$).
• The field $\mathbb{C}_p$ which is the completion of $\overline{\mathbb{Q}_p}$ (with respect to the unique extension of $|\cdot|_p$).
• The field $\mathbb{F}_p((t))$ with the usual $t$-adic absolute value , or a finite extension thereof (with the unique extension of the $t$-adic absolute value).
• The field $\mathbb{C}((t))$ with the $t$-adic absolute value.

Let us also fix an element $\pi$ of $k$ satisfying $0<|\pi|<1$ (such an element is called a pseudo-uniformizer since it plays the role for such $k$ that a uniformizer does for discretely valued fields.

## The general setup

### Prologue

As mentioned above, Tate’s perspective is the beginning of rigid geometry which, in essence, tries to capture an ‘algebraic theory of $p$-adic manifolds’. By an ‘algebraic theory’ I mean that we want to consider our building blocks to come from ring theory of analytic functions, and our opens to be somewhat algebraic in nature. The big difference between the algebro-geometric setup and the rigid analytic setup is that in the latter instead of just considering opens to essentially be those generated by non-vanishing loci of analytic functions

$\{x:f(x)\ne 0\}$

we also want to consider opens given by inequalities involving analytic functions

$\{x:|f(x)|\leqslant 1\}$

(which, by the magic of non-archimedean theory, are open in the topology given by $|\cdot|$) and, of course, combinations of these two types of opens (which one might call something like semi-algebraic open subsets).

Remark 1: Note that since this will be an ‘algebraic theory’ of $p$-adic manifolds some sort of surprising things will happen. For example, the closed unit disk over $\mathbb{C}_p$, denoted $\mathbb{B}_{\mathbb{C}_p}$, will be both connected and quasi-compact. These are necessary steps to make a good algebraic theory work. For example, we would imagine that the global ring of functions on $\mathbb{B}_{\mathbb{C}_p}$ is some ring of analytic functions (convergent power series) and, in particular, an integral domain. So, if $\mathbb{B}_{\mathbb{C}_p}$ is to have a sheaf $\mathcal{O}_{\mathbb{B}_{\mathbb{C}_p}}$ of analytic functions, the fact that $\mathcal{O}_{\mathbb{B}_{\mathbb{C}_p}}(\mathbb{B}_{\mathbb{C}_p})$ is an integral domain actually forces connectedness (e.g. see the argument in [Stacks, Tag00EC]). This is in stark contrast to the naive $p$-adic manifold

$\mathbb{B}_{\mathbb{C}_p}(\mathbb{C}_p)=\mathcal{O}_{\mathbb{C}_p}=\{x\in\mathbb{C}_p:|x|\leqslant 1\}$

which, with its native $p$-adic topology, is neither connected nor quasi-compact. $\blacklozenge$

Now, just as one can imagine building the theory of varieties in three steps:

1. (Step 1) Define affine space $\mathbb{A}^n_k$.
2. (Step 2) Define affine varieties as closed subsets $V(I)\subseteq \mathbb{A}^n_k$.
3. (Step 3) Define general varieties as gluings (along affine opens) of affine varieties.

one can also see Tate’s theory of rigid analytic varieties as following the same three step procedure with some differences. We will explain this progression, roughly, below.

### Step 1

The first major difference is that instead of our building blocks being affine space, which intrinsically doesn’t capture our desired notion of inequalities, we instead use closed polydisks $\mathbb{B}^n_k$ which one imagines as

$\mathbb{B}^n_k:=\{(x_1,\ldots,x_n):|x_i|\leqslant 1\}$

(we will shorten $\mathbb{B}_k^1$ to $\mathbb{B}_k$) as our building blocks in Step 1.

Of course, the naive guess for what the closed unit polydisk $\mathbb{B}_k^n$ should be is that it’s, roughly, the set

$\{(x_1,\ldots,x_n)\in k^n:|x_i|\leqslant 1\}$

Of course, if $k$ is not algebraically closed this has the exact same pitfalls as if one considers $k$ to be the ‘affine line’ over $k$. For example, the closed subspace $V(x^2-\alpha)$, where $\sqrt{\alpha}\notin k$, is empty. Thus, as in the case of classical varieties, one first wants to figure things out over an algebraic closure and then ‘descend’ back down.

Now, since $k$ is complete we know that there is a unique extension of the absolute value on $k$ to the field $\overline{k}/k$ (e.g. see [Iya, Chapter II Theorem 4.4] or [Bos, Appendix A Theorem 3]). We can thus imagine that, at least over $\overline{k}$, the ‘closed unit polydisk’ has a clear meaing. Namely, let us make the following definition:

$\mathbb{B}^n_k(\overline{k}):=\left\{(x_1,\ldots,x_n)\in \overline{k}^n:|x_i|\leqslant 1\right\}$

The question then becomes the following: how do we descend back down over $k$?

Well, if $X$ is a finite type $k$-scheme then we can recover $X$ (or, at least, the closed points of $X$) as the quotient $X(\overline{k})/\mathrm{Aut}(\overline{k}/k)$ (e.g. see [GW, Proposition 5.4]). This suggests, perhaps, that we can make the definition

$\mathbb{B}^n_k:=\mathbb{B}^n_k(\overline{k})/\mathrm{Aut}(\overline{k}/k)$

Of course, this is a bit unwieldy, and so we’d like to give an alternative description of the set $\mathbb{B}^n_k$.

If we were back in the situation of a finite type affine $k$-scheme $X=\mathrm{Spec}(A)$ then it is true that the closed points of $X$ are in bijection with $X(\overline{k})/\mathrm{Aut}(\overline{k}/k)$ but we can also describe the closed points of $X$ as $\mathrm{Max}(A)$-the set of maximal ideals of $A$ where $A$ is thought of as the ring of functions on $X$. Maybe something similarly can be done for $\mathbb{B}^n_k$.

To begin to do this we need to answer the question “what should the ring of analytic functions on the closed unit polydisk over $k$ disk be?” Well the only reasonable possible answer is

\begin{aligned} k\langle t_1,\ldots,t_n\rangle &= \left\{f(t_1,\ldots,t_n)\in k[[t_1,\ldots,t_n]]: f(t_1,\ldots,t_n)\text{ converges everywhere on }\mathbb{B}^n_k(\overline{k})\right\}\\ &=\left\{\sum_{I\in\mathbb{N}^n}^\infty a_I t^I \in k[[t_1,\ldots,t_n]]: \lim_{|I|\to\infty}|a_I|=0\right\}\end{aligned}

where $|(i_1,\ldots,i_n)|=i_1+\cdots+i_n$. Thus, a naive hope would be then that the natural map

$\mathbb{B}^n_k(\overline{k})/\mathrm{Aut}(\overline{k}/k)\to \mathrm{Max}(k\langle t_1,\ldots,t_n\rangle)\qquad (1)$

(which sends $\alpha\in \mathbb{B}_k(\overline{k})$ to the kernel of the evaluation-at-$\alpha$ map $k\langle t_1,\ldots,t_n\rangle\to\overline{k}$) is a bijection. This, in fact, is the case (e.g. see [BGR, §7.1.1 Proposition 1]).

So, we have that $\mathbb{B}^n_k$ is describable, at least as a set, in algebraic termsit is$\mathrm{Max}(k\langle t_1,\ldots,t_n\rangle)$. But, of course, we would like to have $\mathbb{B}^n_k$ as more than just as a setwe’d like to have it as some sort of locally ringed space (the correct setting to do ‘modern geometry’).

To this end, let us note that $\mathbb{B}^n_k$ has a topology coming from $(1)$ since $\mathbb{B}^n_k(\overline{k})$ itself has a natural topologythat induced by a basis of open subsets of the form

$\{x\in \mathbb{B}^n_k(\overline{k}): |f(x)|\leqslant \varepsilon \}$

for $\varepsilon$ a positive real number and $f(t_1,\ldots,t_n)\in k\langle t_1,\ldots,t_n\rangle$. This (quotient) topology on $\mathbb{B}_k$ is called the canonical topology. Note that in the canonical topology there are many other open sets not of the above formyou can have multiple functions defining your inequalities, non-vanishing loci, strict inequality, ‘circles’ (e.g. see [Bos, §3.3]).

Unfortunately, the canonical topology is far too fine. The classical example is to note that $\mathbb{B}_k$ has a natural open cover as the $\mathrm{Aut}(\overline{k}/k)$-orbits of the following decomposition of $\mathbb{B}_k(\overline{k})$

$\displaystyle \mathbb{B}_k(\overline{k})=\underbrace{\bigcup_{n\geqslant 1}\left\{x\in \overline{k}:|x|\leqslant |\pi|^{\frac{1}{n}}\right\}}_{\mathbb{D}_k(\overline{k})}\sqcup \underbrace{\{x\in \overline{k}:|x|=1\}}_{\mathbb{T}_k(\overline{k})}$

where $\mathbb{D}_k$ is the open disk over $k$ and $\mathbb{T}_k$ is the unit circle (or analytic $1$-torus). This decomposition shows that with the canonical topology $\mathbb{B}_k$ is neither connected nor quasi-compact in contrast to our stated desires. Another way to think about this is that the sheaf theory doesn’t work for this cover. Namely, again we have that

$\mathbb{B}_k(\overline{k})=\mathbb{D}_k(\overline{k})\sqcup \mathbb{T}_k(\overline{k})$

but certainly one can check that

$\mathcal{O}(\mathbb{B}_k)\ne \mathcal{O}(\mathbb{D}_k)\times \mathcal{O}(\mathbb{T}_k)$

(where $\mathcal{O}(-)$ is something like the ‘convergent power series on $-$‘).

### Step 2

Before we discuss the fix to this topological problem we actually first discuss the analogue of Step 2  in the theory of rigid spaces. Namely, if we are to think of $\mathbb{B}^n_k$ as $\mathrm{Max}(k\langle t_1,\ldots,t_n\rangle)$ then the analogue of Step 2 should be the following.

Let us first define the analogue of finite type $k$-algebras. Namely, let $A$ be a $k$-algebra. We say that $A$ is an affinoid $k$-algebra if it is a quotient of $k\langle t_1,\ldots,t_n\rangle$ for some $n\geqslant 0$.

The following properties of affinoid algebras show that they are somewhat similar to polynomial rings:

Lemma 2: Let $A$ be an affinoid algebra. Then:

1. Any choice of surjection $k\langle t_1,\ldots,t_n\rangle\to A$ defines a quotient topology on $A$ (using the norm $\displaystyle \left|\sum_I a_I t^I\right|:=\sup|a_I|$). This topology is actually independent of presentation.
2. The ring $A$ is Noetherian.
3. The ring $A$ is Jacobson.
4. A prime ideal $\mathfrak{p}$ of $A$ is maximal if and only if $A/\mathfrak{p}$ is finite-dimensional over $k$.
5. Every ideal of $A$ is closed (in the topology from 1.).
6. Every homomorphism of $k$-algebras between affinoid $k$-algebras is continuous.

Proof: All of these are contained [Tia, Chapter 1] and mostly in [Tia, §1.3]. $\blacksquare$

Now, for any ideal $I\subseteq k\langle t_1,\ldots,t_n\rangle$ we get the subset

$V(I):=\mathrm{Max}(k\langle t_1,\ldots,t_n\rangle/I)\cong \{\mathfrak{m}\in\mathbb{B}_k^n:\mathfrak{m}\supseteq I\}\subseteq \mathbb{B}^n_k$

Of course, one can easily check that we can also identify $V(I)$ as the quotient by $\mathrm{Aut}(\overline{k}/k)$ of the set

$V(I)(\overline{k}):=\{(x_1,\ldots,x_n)\in \mathbb{B}^n_k(\overline{k}):f(x_1,\ldots,x_n)=0\text{ for all }f\in I\}$

In particular, for an affinoid algebra $A$ the choice of a surjection $k\langle t_1,\ldots,t_n\rangle$ identifies $\mathrm{Max}(A)$ concretely in terms of vanishing loci in $\mathbb{B}^n_k(\overline{k})$ and thus endows it with a canonical topology.

Note that by Lemma 2 part 4. that if one has a ring map

$\varphi:A\to B$

of affinoid algebras it induces a map

$\mathrm{Max}(\varphi):\mathrm{Max}(B)\to\mathrm{Max}(A)$

which one can show is continuous with the canonical topology for any presentation of $A$ and $B$ as quotients of some $k\langle t_1,\ldots,t_n\rangle$ (e.g. see [Bos, §3.1, Proposition 6]). In particular, we see that the canonical topology on $\mathrm{Max}(A)$ doesn’t depend on the presentation of $A$. We call sets of the form $\mathrm{Max}(A)$ affinoid sets (it will be these sets with more structure that are really the affinoid spaces which form the building blocks of rigid analytic varieties).

Moreover, note that a presentation allows us to easily talk about inequalities in an affinoid set. Namely, if we realize $A=k\langle t_1,\ldots,t_n\rangle/I$ then we get an embedding

$\mathrm{Max}(A)\hookrightarrow \mathbb{B}^n_k$

and in particular for a set elements $f_1,\ldots,f_m,g\in A$ we can make senset of the set

$\left\{x\in\mathrm{Max}(A):|f_i(x)|\leqslant |g(x)|\right\}$

Indeed, we can think of the set $\{f_1,\ldots,f_m,g\}$ as a set of functions in $k\langle t_1,\ldots,t_n\rangle$ and take the $\mathrm{Aut}(\overline{k}/k)$-orbit of the set

\left\{(x_1,\ldots,x_n)\in \mathbb{B}^n_k(\overline{k}): \begin{aligned}(1)&\quad f(x_1,\ldots,x_n)=0\text{ for all }f\in I\\ (2)&\quad f_i(x_1,\ldots,x_n)\leqslant g(x_1,\ldots,x_n)\text{ for all }i\end{aligned}\right\}\quad (2)

which is clearly independent of the lifts of $f_1,\ldots,f_m,g\in A$ along the surjection $k\langle t_1,\ldots,t_n\rangle\to A$.

But, there is an instrinic way to describe these sets as well. Namely, for any $x=\mathfrak{m}\in\mathrm{Max}(A)$ we know that (again by [Bos,  §3.1 Proposition 4]) $A/\mathfrak{m}$ is a finite extension of $k$ and therefore carries a unique absolute value extending that of $k$. One can then define $|f(x)|$ to be the real number given by evaluating this absolute on $A/\mathfrak{m}$ on the element $f\mod \mathfrak{m}$. One can then check that the set from $(2)$ agrees with the set

$\{x\in\mathrm{Max}(A):|f_i(x)|\leqslant |g(x)|\text{ for all }i\}$

(e.g. see [BGR, §7.1.4 Lemma 2]).

### Step 1 and Step 2 (cont.)

Of course, the canonical topology on $\mathrm{Max}(A)$ is too fine as in the special case of $\mathbb{B}^n_k$. So, let’s return to trying to fix this topology or, what amounts to the same thing geometrically, fixing the sheaf theory. By reexamining the classical the classical theory of the structure sheaf $\mathcal{O}_{\mathrm{Spec}(A)}$ on an affine scheme $\mathrm{Spec}(A)$ Tate was able to see how to fix the sheaf theory on $\mathbb{B}^n_k$. Namely, how does the structure sheaf $\mathcal{O}_{\mathrm{Spec}(A)}$ work? Well, one begins by working with the distinguished base $\{D(f)\}$ of $\mathrm{Spec}(A)$ (e.g. see [Vak,§2.5] and [Vak, Exercise 3.5.A]) and defining a presheaf on this base as follows:

$\mathcal{O}_{\mathrm{Spec}(A)}(D(f)):=A_f$

One then shows that this is a sheaf on a base (e.g. see [Vak, Theorem 4.1.2]). A pivotal step in this procedure is essentially to use the quasi-compactness of $\mathrm{Spec}(A)$ to reduce to showing that a finite distinguished open cover $\{D(f_i)\}$ satisfies the sheaf property.

Tate then realized that he should figure out what the analogue of a ‘distinguished open’ $D(f)\subseteq \mathrm{Spec}(A)$ inside of $\mathbb{B}^n_k$ is and try to prove that a finite distinguished cover satisfies the sheaf axiom. The analogue of a distinguished open on an affinoid $\mathrm{Max}(A)$ is a so-called rational domain. Namely, for elements $g,f_1,\ldots,f_m\in A$ such that $(g,f_1,\ldots,f_m)$ is the unit ideal one defines the associated rational domain as follows:

$\displaystyle U\left(\frac{f_1,\ldots,f_m}{g}\right)=\{x\in \mathrm{Max}(A):|f_i(x)|\leqslant |g(x)|\}$

which one can pretty easily check are open for the canonical topology on $\mathrm{Max}(A)$.

Now that we have singled out the correct analogue of $D(f)$ for $\mathrm{Max}(A)$ we want to say what the value of our presheaf $\mathcal{O}_{\mathrm{Max}(A)}$ on these distinguished opens is. The guess is quite clear: the ring $\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right)$ should be the ring of analytic functions on $\mathrm{Max}(A)$ that are convergent on $\displaystyle U\left(\frac{f_1,\ldots,f_m}{g}\right)$. Namely, we could embed $\mathrm{Max}(A)$ inside of $\mathbb{B}^n_k$ and say that $\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right)$ is

$\displaystyle \left\{f\mid_{\mathrm{Max}(A)}:f(t_1,\ldots,t_n)\in k[[t_1,\ldots,t_n]]\text{ which converges everywhere on }U\left(\frac{f_1,\ldots,f_m}{g}\right)\right\}$

(where we have implicitly chosen liftings of $f_1,\ldots,f_m,g$ to $k\langle t_1,\ldots,t_n\rangle$ but it’s clearly independent of such liftings).

But, a coordinate-free (but still a priori dependent on the choice of the set $f_1,\ldots,f_m,g$) way to describe this ring is as follows:

$\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right):=A\langle s_1,\ldots,s_m\rangle/(f_1 s_1-g,\ldots,f_ms_m-g)$

where

$\displaystyle A\langle s_1,\ldots,s_m\rangle := \left\{\sum_{J\in\mathbb{N}^m}a_J s^J\in A[[s_1,\ldots,s_m]]: \lim_{|J|\to\infty}a_J=0\right\}$

where the topology we are using on $A$ (as in Lemma 2 part 1.). In particular, note that $\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right)$ is still an affinoid $k$-algebra.

Remark 3: Note that this ring enlarges $k\langle t_1,\ldots,t_n\rangle$ in more ways than adding in inverses. Namely, in normal algebraic geometry essentially if you have an open subset $U\subseteq X$ (let’s say of integral schemes for simplicity) the map $\mathcal{O}(X)\to \mathcal{O}(U)$ allows one to view $\mathcal{O}(U)$ as an enlargement of $\mathcal{O}(X)$ by algebraically adjoining inverses of functions not invertible on $X$ but invertible on $U$a sort of generalized localization. In the rigid setting there is not only localization but also completion. The result is that one doesn’t just algebraically adjoin new things but also forces new series to be convergent.

For example $\displaystyle U\left(\frac{t}{\pi}\right)\subseteq \mathbb{B}_k$ has ring of sections $\displaystyle k\langle t\rangle\left\langle\frac{t}{\pi}\right\rangle$. Now, algebraically adjoining $\displaystyle \frac{t}{\pi}$ does noting (it’s already in the ring) but this ‘completed adjoinment’ forces certain power series in $\displaystyle \frac{t}{\pi}$ to convergeany series of the form $\displaystyle \sum_{i=0}a_i \left(\frac{t}{\pi}\right)^i$ with $\lim a_i=0$ converges in $\displaystyle k\langle t\rangle\left\langle\frac{t}{\pi}\right\rangle$ but needn’t converge in $k\langle t\rangle$. For a specific example one has that

$\displaystyle \sum_{i=0}^\infty \pi^i \left(\frac{t}{\pi}\right)^i=\sum_{i=0}^\infty t^i$

is in $\displaystyle k\langle t\rangle\left\langle\frac{t}{\pi}\right\rangle$ but not in $k\langle t\rangle$. $\blacklozenge$

Finally, one can show that for an open subset $U$ of $\mathrm{Max}(A)$ which is a rational domainwhich can be presented as $\displaystyle U\left(\frac{f_1,\ldots,f_m}{g}\right)$ for some set $f_1,\ldots,f_m,g\in A$ generating the unit idealthat the ring $\mathcal{O}_{\mathrm{Max}(A)}(U)$ doesn’t depend on the choice of presentation (i.e. on the set $f_1,\ldots,f_m,g$)see [Tia, Example 1.5.12 part 3.]).

We then have the following theorem of Tate:

Theorem 4(Tate): Let $A$ be an affinoid algebra. Then, for any rational domain $U\subseteq \mathrm{Max}(A)$ and any finite covering $\{U_1,\ldots,U_\ell\}$ of $U$ by rational domains (cf. [Bos, §3.3 Proposition 17]) the sequence

$\displaystyle 0\to\mathcal{O}_{\mathrm{Max}(A)}(U)\to \prod_i \mathcal{O}_{\mathrm{Max}}(A_i)\to \prod_{i,j}\mathcal{O}_{\mathrm{Max}(A)}(U_i\cap U_j)$

is exact.

In other words, the presheaf $\mathcal{O}_{\mathrm{Max}(A)}$ is a sheaf with respect to rational domains and finite coverings by rational domains. The idea of proof is to do explicit computations much as in the case of [Vak, Theorem 4.1.2] (for a full proof see the references later on).

In fact, two stronger versions of this theorem can be shown to hold which are often stated instead of Theorem 4. Namely, one can replace rational domains and finite rational coverings by affinoid domains (resp. admissible opens) and finite affinoid coverings (resp. admissible coverings).

The idea of an affinoid subdomain is that it’s basically the analogue of an ‘affine open subscheme’ in the rigid geometry setting. In particular, using this post one is tempted to define a subset $U\subseteq\mathrm{Max}(A)$ to be an affinoid domain if there exists an affinoid algebra $B$ and a map of $k$-algebras $j^\ast:B\to A$ such that the induced map $j:\mathrm{Max}(B)\to\mathrm{Max}(A)$ has image $U$ and for which $j$ is universal for such a property: if $h^\ast:A\to C$ is a $k$-algebra map of affinoid algebras such that $h:\mathrm{Max}(C)\to\mathrm{Max}(A)$ has image landing in $U$ then $h^\ast$ unique factors through $j^\ast$. One can show that such $U$ are open for the canonical topology (see [Tia, Corollary 1.5.22]) and that the $k$-algebra $B$ is unique up to unique isomorphism.

As an example, one can show that rational domains are affinoid domains (e.g. see [Tia, Example 1.5.12 part 3.]). In fact, the super important Gerritzen–Grauert theorem gives a partial converse to this statement: every affinoid domain is a finite union of rational domains. Now, given the unicity of $B$ (as in the last paragraph) one can associate to any affinoid domain $U\subseteq \mathrm{Max}(A)$ the ring $\mathcal{O}_{\mathrm{Max}(A)}(U):=B$.

One then has the following strengthening of Theorem 4:

Theorem 5: Let $A$ be an affinoid algebra. Then, for any affinoid domain $U\subseteq \mathrm{Max}(A)$ and any finite covering $\{U_1,\ldots,U_\ell\}$ of $U$ by affinoid domains (cf. [Bos, §3.3 Proposition 12]) the sequence

$\displaystyle 0\to\mathcal{O}_{\mathrm{Max}(A)}(U)\to \prod_i \mathcal{O}_{\mathrm{Max}}(A_i)\to \prod_{i,j}\mathcal{O}_{\mathrm{Max}(A)}(U_i\cap U_j)$

is exact.

Proof: See [Tia, Theorem 1.6.3]. The idea is to use the Gerritzen–Grauert theorem to reduce this to rational domains and rational covers, then even simpler types of opens and covers and finally show the sheaf condition explicitly by hand. $\blacksquare$

The final strengthening allows one to replace affinoid domains $U\subseteq\mathrm{Max}(A)$, which (again) one should think of as just open affinoid subspaces, with certain open subspaces and certain covers. The idea is that whatever general open subset $U$ of $\mathrm{Max}(A)$ we encounter, one shouldn’t be able to produce using $U$ another affinoid $\mathrm{Max}(B)$ (not necessarily an affinoid open in $\mathrm{Max}(A)$ and an affinoid cover of $\mathrm{Max}(B)$ which does not have a finite subcoverfor this is precisely the case in which we have constructed our theory thusfar to avoid. How does one make this rigorous?

Well, let us call an open subset $U\subseteq\mathrm{Max}(A)$ admissible if there exists an affinoid (in $\mathrm{Max}(A)$) cover (possibly infinite) $\{U_i\}$ of $U$ such that for any map of affinoid $k$-algebras $\varphi^\ast:A\to B$ such that the induced map $\varphi:\mathrm{Max}(B)\to\mathrm{Max}(A)$ has $\varphi(\mathrm{Max}(B))\subseteq U$ one has that the affinoid cover $\{\varphi^{-1}(U_i)\}$ of $\mathrm{Max}(B)$ has a refinement (cf. [Bos, Pg. 83]) by a finite affinoid open cover.  Similarly, an admissible covering of the admissible open $U$ is a collection of admissible open subsets $\{V_j\}$ of $\mathrm{Max}(A)$ which cover $U$ and have the property that for any $k$-algebra map of $k$-affinoids $\varphi^\ast:A\to B$ such that $\varphi(\mathrm{Max}(B))\subseteq U$ one has that the cover $\{\varphi^{-1}(V_j)\}$ has a refinement by a finite affinoid open cover.

One should interpret non-admissible covers of an affinoid $\mathrm{Max}(A)$ as ‘bad’ and, in particular, should be disregarded.

Example 6: The cover the open unit disk $\mathbb{D}_k$ is an admissible open subset of $\mathbb{B}_k$ (e.g. see [Tia, Example 2.1.6]) and $\mathbb{T}_k$ is admissible (it’s even rationalit’s $\displaystyle U\left(\frac{1}{t}\right)$) but the cover $\{\mathbb{D}_k,\mathbb{T}_k\}$ is not admissible (e.g. see [Con1, Example 2.2.8]). We will later give a good interpretation for this in terms of adic spaces. $\blacklozenge$

We then have the following general strengthening of Theorem 4 and Theorem 5:

Theorem 7:Let $A$ be an affinoid algebra. Then, for any admissible open $U\subseteq \mathrm{Max}(A)$ and any admissible covering $\{V_i\}$ of $U$ by admissible opens the sequence

$\displaystyle 0\to\mathcal{O}_{\mathrm{Max}(A)}(U)\to \prod_i \mathcal{O}_{\mathrm{Max}}(V_i)\to \prod_{i,j}\mathcal{O}_{\mathrm{Max}(A)}(V_i\cap V_j)$

is exact.

### Step 3

Now, we have singled out three (increasingly general) situations where our sheaf theory has worked:

1. It works for rational domains in $\mathrm{Max}(A)$ and finite rational covers.
2. It works for affinoid domains in $\mathrm{Max}(A)$ and finite affinoid covers (this is the ‘obvious’ version of 1.it’s just easier to work with 1. than general finite affinoid covers)
3. It works for admissible opens in $\mathrm{Max}(A)$ and admissible covers (this is the finest replacement $G$-topology [see below] which ‘acts the same’ as the $G$-topology from 2.see [BGR, §9.1.2] for more details).

In all three cases we see that we are in the situation of having some set of distinguished open subsets of $\mathrm{Max}(A)$ (rational domains, affinoid domains, admissible opens) and for each distinguished open subset a ‘permissible’ set of covers of that distinguished open subset by distinguished open subsets (finite rational covers, finite affinoid covers, admissible covers). Now, none of these are an acutal topology on $\mathrm{Max}(A)$ (e.g. they aren’t closed under arbitrary unions) but they act similarly enough like a topology to make sheaf theory work. In fact, they are an example of a $G$-topology.

Roughly, a $G$ topology on a topological space $X$ is

• A (usually proper) collection $\mathrm{Perm}(X)$ of open subsets of $X$ ‘the permissible opens’.
• For each permissible open subset $U\in\mathrm{Perm} (X)$ a set $\mathrm{Cov}(U)$ of open covers of $U$ by permissible opens‘the permissible covers’ (NB: not every open cover by permissible opens is assumed permissiblejust a subset of such covers).

One needs to assume that the above colletions have reasonable enough properties to talk about sheaf theory (for a rigorous definition see [Bos, §5.1] and [Bos, §5.2]). Namely, one calls a contravariant functor

$\mathcal{F}:\left\{\text{Permissible opens}\right\}\to\mathsf{Ab}$

(where $\mathsf{Ab}$ is the category of abelian groups and the set of permissible opens is considered a category where the maps are the inclusions of permissible open sets) an abelian presheaf on the $G$-space $X$. One calls an abelian presheaf $\mathcal{F}$ on the $G$-space $X$ an abelian sheaf if for all $U\in\mathrm{Perm}(X)$ and all $\{U_i\}\in\mathrm{Cov}(U)$ the presheaf $\mathcal{F}$ satisfies the sheaf conditions for the cover in other words the sequence

$\displaystyle 0\to\mathcal{F}(U)\to \prod_i \mathcal{F}(U_i)\to \prod_{i,j}\mathcal{F}(U_i\cap U_j)$

is exact. Of course, one can define sheaves with values in any reasonable category (e.g. see [Stacks, 00VL] or [Bos, §5.1] and [Bos, §5.2]).

Remark 8: The reason for the name ‘$G$-topology’ is that a $G$-topology on $X$ is, essentially, a Grothendieck subtopology of $\mathrm{Open}(X)$ for those familiar with Grothendieck topologies (e.g. from étale cohomology). This allows one to use all the theory one has learned from the theory of cohomology on sites etc. (e.g. [Stacks, 00UZ] and [Stacks, 01FQ]) $\blacklozenge$

Note that what we are calling here ‘permissible opens’ and ‘permissible covers’ is usually called ‘admissible opens’ and ‘admissible covers’, but so as to not confuse these with our explicit admissible covers for affinoid sets we have elected to use this non-standard terminology.

Given a $G$-space $X$ and an abelian sheaf $\mathcal{F}$ on $X$ one can define the stalk of $\mathcal{F}$ at a point $x\in X$. Namely, one can define

$\displaystyle \mathcal{F}_x:=\varinjlim_{U\in N(x)}\mathcal{F}(U)$

where $N(x)$ is the set of permissible open subsets of $X$ containing $x$.

A locally ringed $G$-space is a $G$-space $X$ with a sheaf $\mathcal{O}_X$ of rings whose stalks are local rings. If, in addition, $\mathcal{O}_X$ is a sheaf of $k$-algebras we call $X$ a locally ringed $G$-spaces over $k$. A morphism of locally ringed $G$-spaces and/or locally locally ringed $G$– space over $k$ is the obvious one (e.g. again see [Stacks,00VL] or [Bos, §5.1] and [Bos, §5.2]).

Now, from the discussion in the previous subsections we see that for every affinoid algebra $A$ one gets two (major) locally ringed $G$-space over $k$ given as the pair

$\mathrm{Sp}(A):=(\mathrm{Max}(A),\mathcal{O}_{\mathrm{Max}(A)})$

where the permissible opens are the admissible opens (resp. affinoid domains), and the permissible covers are the admissible ones (resp. finite affinoid covers). We call this the strong (resp. weak) $G$-topology on $\mathrm{Max}(A)$. Unless specified otherwise, the $G$-topology on $\mathrm{Max}(A)$ will always taken to be the strong one. We call a locally ringed $G$-space over $k$ of the form $(\mathrm{Max}(A),\mathcal{O}_{\mathrm{Max}(A)})$ (with the strong $G$-topology) an affinoid space over $k$. While $\mathrm{Sp}(A)$ is really a pair, we shall, in practice conflate $\mathrm{Sp}(A)$ and $\mathrm{Max}(A)$.

This pair satisfied the expected property:

Lemma 9: Let $A$ and $B$ be affinoid $k$-algebras. Then, the map

$\mathrm{Hom}(\mathrm{Sp}(B),\mathrm{Sp}(A))\to \mathrm{Hom}_k(A,B)$

given by the map on global sections is an isomorphism. In other words the functor

$\mathrm{Sp}:\left\{\begin{matrix}\text{Affinoid algebras}\\ \text{over }k\end{matrix}\right\}\to \left\{\begin{matrix}\text{Locally ringed }G-\text{spaces}\\ \text{over }k\end{matrix}\right\}$

is fully faithful.

Proof: See [Bos, §5.3 Corollary 7]. $\blacksquare$

Now, given $(X,\mathcal{O}_X)$ any locally ringed $G$-space over $k$ and any open permissible subset $U\subseteq X$ one naturally gets a $G$-topology on $U$ (e.g. $\mathrm{Perm}(U)$ is the set of elements of $\mathrm{Perm}(X)$ contained in $U$) and one can restrict $\mathcal{O}_X$ to get a sheaf $\mathcal{O}_U:=\mathcal{O}_X\mid_U$ on $U$ so that the pair $(U,\mathcal{O}_U)$ is also a locally ringed $G$-space over $k$.

So, with all of this setup we can finally defined rigid analytic varieties a la Tate. Namely, a rigid analytic variety is a pair $(X,\mathcal{O}_X)$ which is a locally ringed $G$-space over $k$ such that there is a permissible open cover $\{U_i\}$ of $X$ such that $(U_i,\mathcal{O}_{U_i})$ is isomorphic to an affinoid space. A morphism of rigid spaces is a morphism of locally ringed $G$-spaces over $k$.

One has a natural generalization of Lemma 9:

Lemma 10: Let $X$ and $Y$ be rigid spaces with $Y$ affinoid. Then, the natural map

$\mathrm{Hom}(X,Y)\to \mathrm{Hom}_k(\mathcal{O}(Y),\mathcal{O}(X))$

is a bijection.

Proof: See [Bos, §5.3 Corollary 7]. $\blacksquare$

Let us finally note that the set of affinoid open subspaces of a rigid space $X$ form a distinguished base for $X$ (in the sense of [Vak, §2.5]) and thus if one wants to define a sheaf on $X$ it suffices to define a sheaf on the distinguished base of affinoid opens (e.g. by [Vak, Theorem 2.5.1]).

## The examples

Now with all of this setup we can finally discuss some simple examples of rigid spaces. We shall be introducing some more theory along the way.

### The closed unit disk

We start with the basic object which we have already seen many times above. Namely, the closed unit disk (or closed unit ball) over $k$ is the rigid analytic variety $\mathbb{B}_k:=\mathrm{Sp}(k\langle t\rangle)$.  Let us start to prove some of the basic properties of $\mathbb{B}_k$.

#### Topological properties

Perhaps the first thing to check is that $\mathbb{B}_k$, as we envisioned it, is quasi-compact and connected. Of course, since $\mathbb{B}_k$ is actually a $G$-space (in particular not a topological space) we need to define what this means. But, of course, it’s the obvious thing. Namely a $G$-space $X$ is quasi-compact if every permissible cover of $X$ has a finite refinement. And, a $G$-space $X$ is connected if there does not exist a permissible cover $\{U,V\}$ of $X$ such that $U\cap V$ is empty and $U,V\ne\varnothing$

To prove that $\mathbb{B}_k$ is quasi-compact and connected we make the following general observation:

Proposition 11: Let $A$ be an affinoid $k$-algebra. Then, $\mathrm{Sp}(A)$ is quasi-compact. Moreover, $\mathrm{Sp}(A)$ is connected if and only if $A$ has no non-trivial idempotents.

Proof: We tailor build admissible covers so to force the quasi-compactness. Namely, if $\{U_i\}$ is an admissible cover of $\mathrm{Sp}(A)$ then letting $\varphi^\ast:A\to A$ be the identity map (in the definition of admissible) one sees by definition that $\{\varphi^{-1}(U_i)\}=\{U_i\}$ must admit a finite affinoid refinement.

To see the second claim we proceed as usual. If $X=U\sqcup V$ with $\{U,V\}$ admissible then $\mathcal{O}(X)=\mathcal{O}(U)\times \mathcal{O}(V)$ and so $\mathcal{O}(X)$ has non-trivial idempotents as soon as $\mathcal{O}(U)$ and $\mathcal{O}(V)$ are not the zero ring. To see this note that if $U$ is a non-empty rigid space then $\mathcal{O}(U)$ is non-zero. Indeed, let $x\in U$ and let $W$ be an affinoid neighborhood of $x$. Then, we have a ring map $\mathcal{O}_U(U)\to\mathcal{O}_W(W)$ and, in particular, if $\mathcal{O}_U(U)$ is the zero ring this forces $\mathcal{O}_W(W)$ to be the zero ring which forces $W=\mathrm{Max}(\mathcal{O}_W(W))$ to be empty, which is a contradiction.

Conversely, suppose that $\mathcal{O}(X)$ has non-trivial idempotents so that $\mathcal{O}(X)=R\times S$ with $R,S$ not zero rings. Let $e:=(1,0)$ and $f:=(0,1)$. Then I leave it to the reader to check that the non-vanishing locus of $e$ and $f$, called $U$ and $V$, are open, non-intersecting, and that $\{U,V\}$ is an admissible cover of $X$. $\blacksquare$

That said, let’s notice our first descrepancy with the scheme picture (although it is consonant with the classical picture of varieties). Namely, recall that if $A$ is an integral domain that $\mathrm{Spec}(A)$ is more than just connected it’s actually irreducible. This irreducibility, and the existence of generic points, are a key component of what makes scheme-theoretic algebraic geometry tick. This completely fails for objects in the rigid setting. Namely, let us make the following non-standard definition. Namely, let us say that a $G$-space is irreducible if for all admissible opens $U,V\in\mathrm{Perm}(X)$ we have that $U\cap V\ne\varnothing$. Essentially no rigid space is irreducible.

For example, certainly $\mathbb{B}_k$ is not irreducible since, for example,

$\mathbb{D}_k=\{x\in \mathbb{B}_k:|x|<1\}$

and

$\mathbb{T}_k=\{x\in\mathbb{B}_k:|x|=1\}$

are admissible opens (we leave this to the reader, or see [Con1, Example 2.2.8])  which intersect disjointly. If one wants to soup this up to an example consisting only of affinoid opens one can take two non-equal points $x_0,x_1\in\mathbb{B}_k(k)$ and let $|x_0-x_1|=\varepsilon>0$. Then,

$\displaystyle U_i:=\left\{x\in\mathbb{B}_k:|x-x_i|\leqslant \frac{\varepsilon}{2}\right\}$

are disjoint admissible opens. Thus, $\mathbb{B}_k$ is far from being ‘irreducible’.

Remark 12: One should take this reducibility of (most) rigid spaces as both a blessing and curse. Namely, the fact that our spaces are reducible means that they are closer to the normal topological spaces one deals with in complex geometry and therefore we can often times perform natural operations in the rigid setting which are impossible to perform in the scheme setting (e.g. see the discussion on Tate elliptic curves below).

But, on the other hand, we are now robbed of one of the most powerful tools at the disposal of algebraic geometers: the ubiquitous generic point. Since most reasonable schemes we work with are irreducible they have generic points and, in particular, one can often use this generic point to simplify a huge amount of technical assumptions. I didn’t realize how important generic points were to my understanding of scheme theory until I tried to do rigid geometry where such points were suddenly missing.

Let us also note here by that ‘irreducible’ we mean irreducible in the naive senseirreducible as topological spaces. This, of course, is the type of irreducibility we usually mean in the context of schemes. That said, there is a more ‘algebraic’ notion of irreducibility which does work well in the context of rigid spaces. Essentially, one replaces the usual $G$-topology for a coarser onethe Zariski topology. For this, one can see [Con3]. $\blacklozenge$

#### Functor of points

Let us now contemplate what the functor of points of $\mathbb{B}_k$ is. Namely, we have the following observation:

Proposition 13: For any rigid space $X$ over $k$ is a natural isomorphism of sets

$\mathrm{Hom}(X,\mathbb{B}_k)\xrightarrow{\approx}\mathcal{O}_X^+(X)$

Before we prove this let us define what $\mathcal{O}_X^+$ is. This is a presheaf of $\mathcal{O}_k$-algebras on $X$ defined as follows. For an affinoid open $\mathrm{Sp}(A)\subseteq X$ we define

$\mathcal{O}_X^+(\mathrm{Sp}(A)):=A^\circ$

where $A^\circ$ is the ring of power bounded elements of $A$ (i.e. those elements $a\in A$ such that the set $\{a^n\}$ is boundedsee [Bos, §3.1 Theorem 17], [Tia, Proposition 1.4.13], or [BGR, §1.2.5]). Note that this is a sheaf with respect to the distinguished base of affininoid opens (this is easy and left to the readeruse the equivalences given in loc. cit.). Then, as previously remarked, $\mathcal{O}_X^+$ extends uniquely to a sheaf on the $G$-space $X$.

One then sees $\mathcal{O}_X^+(X)$ as the global sections of this sheaf. Intuitively one imagines that $\mathcal{O}_X^+(X)$ is the set of elements $\alpha$ of $\mathcal{O}_X(X)$ such that $|\alpha|\leqslant 1$ (e.g. see [Tia, Proposition 1.4.13] for a rigorous statement) from which Proposition 13 then makes intuitive sense.

Finally we can prove Proposition 13:

Proof( Proposition 13): Note that since $\mathrm{Hom}(-,\mathbb{B}_k)$ and $\mathcal{O}_X^+$ are sheaves for the $G$-topology it suffices to prove this natural isomorphism when $X$ is affinoid, say $X=\mathrm{Sp}(A)$. Note then that by Lemma 9 there is a natural bijection

$\mathrm{Hom}(\mathrm{Sp}(A),\mathbb{B}_k)=\mathrm{Hom}_k(k\langle t\rangle, A)$

Now, note that since any $k$-homomorphism $f:k\langle t\rangle\to A$ is continuous (e.g. see Lemma 38) that such a homomorphism is uniquely determined by where it sends $t$. Namely, if $t\mapsto \alpha\in A$ then

\begin{aligned}f\left(\sum_i a_i t^i\right) &=f\left(\lim \sum_{i=0}^N a_i t^i\right)\\ &=\lim f\left(\sum_{i=0}^N a_i t^i\right)\\ &= \lim \sum_{i=0}^N a_i f(t)^i\\ &= \sum_{i=0}^\infty a_i \alpha^i\end{aligned}

Moreover, this also easily shows that $\alpha$ is of the form $f(t)$ if and only if $\displaystyle \sum_{i=0}^\infty a_i \alpha^i$ converges for every sequence $\{a_i\}$ in $k$ such that $\lim a_i=0$. But it’s easy to see that this is equivalent to $\alpha\in A^\circ$. The conclusion follows. $\blacksquare$

In particular, let us note that since $\mathcal{O}_X^+(X)$ is a ring (as we leave to the reader to check) one has that $\mathbb{B}_k$ is a ring object in the category of rigid spaces over $k$. This is perhaps not too surprising since $\mathbb{B}_k$ is meant to be the rigid geometry avatar of $\mathcal{O}_k$ which is itself a ring.

#### Picard group

Let us also try and say something slightly more interesting about $\mathbb{B}_k$. Namely, let us define a line bundle on a rigid space $X$ as a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules which locally (for the $G$-topology) is isomorphic to $\mathcal{O}_X$ (cf. [FdvP, Definition 4.5.1]). As per usual the group of (isomorphism classes of) line bundles on $X$ forms a group under tensor product which is called the Picard group of $X$ and written $\mathrm{Pic}(X)$. What is the Picard group of $\mathbb{B}_k$?

To give us some intuition, let us first try and understand a closely related question over the complex numberswe will later see this is actually a bit deceiving, but let’s use it anyways:

Example 14: Let $\mathbb{D}_\mathbb{C}$ be the open unit disk thought of as a complex manifold. We claim that $\mathrm{Pic}(\mathbb{D}_\mathbb{C})=0$. Indeed, we have the exponential sequence

$0\to \underline{2\pi i \mathbb{Z}}\to \mathcal{O}_{\mathbb{D}_\mathbb{C}}\to \mathcal{O}^\times_{\mathbb{D}_{\mathbb{C}}}\to 0$

Taking the long exact sequence in cohomology, using the fact that

$H^i(\mathbb{D}_\mathbb{C},\underline{2\pi i\mathbb{Z}})\cong H^i_\mathrm{sing}(\mathbb{D}_{\mathbb{C}},\mathbb{Z})$

(e.g. see [Wed, Theorem 11.13]), and the fact that $\mathbb{D}_\mathbb{C}$ is contractible we get a natural isomorphism

$H^1(\mathbb{D}_\mathbb{C},\mathcal{O}_{\mathbb{D}_\mathbb{C}})\cong H^1(\mathbb{D}_{\mathbb{C}},\mathcal{O}_{\mathbb{D}_{\mathbb{C}}}^\times)$

and of course we have natural isomorphisms

$\mathrm{Pic}(\mathbb{D}_{\mathbb{C}})\cong H^1(\mathbb{D}_{\mathbb{C}},\mathcal{O}_{\mathbb{D}_{\mathbb{C}}}^\times)$

Thus, it suffices to show that $H^1(\mathbb{D}_{\mathbb{C}},\mathcal{O}_{\mathbb{D}_{\mathbb{C}}})=0$. This last equality can be shown to be true by the Mittag-Leffler theorem. $\blacklozenge$

So, a guess might be that since the open disk and closed disk are not too far apart that the Picard group of $\mathbb{B}_k$ is also trivial. This, as it turns out, is correct:

Proposition 15: Let $A$ be an affinoid $k$-algebra. Then, there is a natural isomorphism

$\mathrm{Pic}(\mathrm{Spec}(A))\xrightarrow{\approx}\mathrm{Pic}(\mathrm{Sp}(A))$

given by associating a finitely generated projective $A$-module $M$ with the sheaf $\widetilde{M}$ on $\mathrm{Sp}(A)$ which associates to an affinoid open $\mathrm{Sp}(B)\subseteq\mathrm{Sp}(A)$ the tensor product $M\otimes_A B$ .

Proof: See [Bos, §6.1 Theorem 4] or [FvdP, Proposition 4.7.2]. $\blacksquare$

In particular, we see that proving that $\mathrm{Pic}(\mathbb{B}_k)=0$ is equivalent to proving that $\mathrm{Pic}(\mathrm{Spec}(k\langle t\rangle))=0$. But, to do this it suffices (e.g. see [Stacks, 0BCH]) to show that $k\langle t\rangle)$ is a PID. But, this is well-known (e.g. see [Bos, §2.2 Corollary 10]).

#### Dimension

Defining the dimension for a rigid space $X$ in the classical Tate perspective is necessarily ad-hoc. Namely, one is tempted to try and define it as in the case of schemesthe dimension is the maximal chain of specializations. But, of course, this cannot work well since we have only closed points! One might then try to make an adapatation of what we do in classical variety language (e.g. as in [Har, Chapter 1]) where we use chains of irreducible closed subsets. But, as we saw above (around Remark 12) that irreducible analytic spaces are quite rare. In the Huber and/or Raynaud setting there is a natural fix to these issues, but for now we instead take the following as our definition of dimension. The dimension of a rigid space $X$ is the supremum of $\mathrm{dim}(\mathcal{O}_{X,x})$ as $x$ ranges over the points of $x$.

So, what is the dimension of $\mathbb{B}_k$? Of course the answer is $1$, but we need to verify this. But, as observed above, since $\mathbb{B}_k$ is a group under addition it suffices to check $\dim\mathcal{O}_{\mathbb{B}_k,0}$. What is this ring? Well, almost by definition (since closed subdisks around $0$ are clearly a basis for $N(0)$cf. our definition of stalk) that

\begin{aligned}\mathcal{O}_{\mathbb{B}_k,0} &=\varinjlim_{n\to\infty }k\langle t,s\rangle/(\pi^n s-t)\\ &\cong \left\{\sum_{i=0}^\infty a_i t^i \in k[[t]]: \lim_{i\to\infty} |\pi^{ni} a_i|=0\text{ for some }n\geqslant 0\right\}\end{aligned}

In other words, $\mathcal{O}_{\mathbb{B}_k,0}$ is nothing but the ring of power series convergent in some neighborhood $0$. This is, by inspection, a DVR. Indeed, it’s easy to see that $(t)$ is the unique maximal ideal of $\mathcal{O}_{\mathbb{B}_k,0}$ (e.g. it’s the same proof as here) from where one can easily deduce the conclusion.

Thus,

$\dim\mathbb{B}_k=\dim \mathcal{O}_{\mathbb{B}_k,0}=1$

as desired.

### The punctured closed disk

We now would like to consider the spaces $\mathbb{B}_k-\{p_1,\ldots,p_m\}$ where $p_i$ are any points of $\mathbb{B}_k$. We call such a space a (several times) punctured disk. For sake of simplicity we assume that $k$ is algebraically closed (or that the points $p_1,\ldots,p_m$ are $k$-points). This does not change the story in any substantive way, it just allows one to not have to worry about an $L$-point, for $L/k$ finite, splitting in to several points over $\overline{k}$.

#### Topological properties

Let us begin with the most basic case: the $k$-analytic variety $\mathbb{B}_k-\{p_1\}$. There is evidently an automorphism of $\mathbb{B}_k$ which takes $p_1$ to $0$. Indeed, this follows from the aforementioned fact that $\mathbb{B}_k$ is a ring object over $\mathrm{Sp}(k)$just take the translation by $-p_1$ automorphism of $\mathbb{B}_k$. Thus, we may assume without loss of generality that $p_1=0$.

Remark 16: Note that this observation already dispels an easy-to-make misconception about $\mathbb{B}_k$. Namely, if one draws $\mathbb{B}_k$ then one has a tendecy to treat points in the ‘boundary’ $\mathbb{T}_k$ as being different than points in the interior $\mathbb{D}_k$. But, the above shows that points in $\mathbb{T}_k$ really don’t look any different than points in $\mathbb{D}_k$.

For example, a picture of the closed disk lends one to believe that points on the ‘boundary’ don’t have closed disk neighborhoods still contained in the disk. This, of course, is false. If $|x|=1$ then the set $\{y:|x-y|\leqslant \frac{1}{2}\}$ is, in fact, contained in $\mathbb{T}_k$, let alone $\mathbb{B}_k$. This is one of the reasons that notions of ‘boundary’ (e.g. as in Berkovich’s theory of rigid spaces) are more subtle than one might first imagine. $\blacklozenge$

We begin with the following observation:

Proposition 17: The punctured disk $\mathbb{B}_k-\{0\}$ is an admissible open subset of $\mathbb{B}_k$. It is connected and not quasi-compact.

Proof: We can essentially kill three birds with one stone. Namely, for each $n\geqslant 1$ let us set

$U_n:=\{x\in\mathbb{B}_k:|\pi^n|\leqslant |x|\}$

(this is an example of an annulus). Note that each $U_n$ is an affinoid domain in $\mathbb{B}_k$. In fact, $U_n$ is a rational domain. Indeed, we have that

$\displaystyle U_n=U\left(\frac{t}{\pi^n}\right)$

and $\{\pi^n,t\}$ evidently generate the unit ideal in $k\langle t\rangle$ since $\pi^n$ is a unit. The affinoid domain claim then follows from previous comment (cf. [Tia, Example 1.5.12 part 3.]).

Note that $\{U_n\}$ is an open cover of $\mathbb{B}_k$ because $|\pi^n|\to 0$ and so if $x\ne 0$ then $|x|\ne 0$ and thus $|x|\geqslant |\pi^n|$ for some $n\geqslant 1$. We claim that $\{U_n\}$ is an admissible cover. To do this, let $\varphi:\mathrm{Sp}(A)\to\mathbb{B}_k$ be a morphism such that $\varphi(\mathrm{Sp}(A))\subseteq \mathbb{B}_k-\{0\}$. Let $\varphi^\ast:k\langle t\rangle\to A$ be the associated map of affinoid algebras and let $f:=\varphi^\ast(t)$. Note that since $\varphi(\mathrm{Sp}(A))\subseteq \mathbb{B}_k-\{0\}$ that $f$ is a unit in $A$. Indeed, note that we have a factorization of maps of rings

$\begin{matrix}k\langle t\rangle & \to & \mathcal{O}(\mathbb{B}_k-\{0\})\\ &\searrow & \downarrow\\ & & A\end{matrix}$

and since $t$ is invertible in $\mathcal{O}(\mathbb{B}_k-\{0\})$ the invertibility of $\alpha$ follows. Let us now consider $f^{-1}\in A$. Note then that by the maximum modulus principle (e.g. see [BGR, §6.2.1 Proposition 4]) that there exists some $c>0$ such that $|f(x)^{-1}|\leqslant c$ for all $x\in\mathrm{Sp}(A)$ and so $|f(x)|\geqslant c^{-1}$ for all $x$. Let $n$ be large enough so that $|\pi^n|. Then, it’s clear by this that $\varphi(\mathrm{Sp}(A))\subseteq U_n$. So then evidently $\{\varphi^{-1}(U_n)\}$ has a finite refinement as desired.

Now, to see that $\mathbb{B}_k-\{0\}$ is connected it suffices to note that $U_n$ is connected. Indeed this would follow from the obvious $G$-topology analogue of the classical statement “if a topological space $X$ has an open cover by connected subspaces all of which intersect then $X$ is connected”. To see that $U_n$ is connected we can use Proposition 11 and merely note that

$U_n=\mathrm{Sp}\left(k\langle t,s\rangle/(st-\pi^n)\right)$

and I leave to the reader to check that $k\langle t,s\rangle/(st-\pi^n)$ is a domain.

Finally, to see that $\mathbb{B}_k-\{0\}$ is not quasi-compact it suffices to note that $\{U_n\}$ has no finite refinement which is easy and left to the reader. $\blacksquare$

One thing that follows from the above is the following simple observation:

Corollary 18: The $G$-space $\mathbb{B}_k$ is not Noetherian.

By Noetherian we mean that not every admissible open subset is quasi-compact. This is, of course, in stark constrast to the algebraic setting where an affine scheme with Noetherian global sections is Noetherian. This is a subtelty that can often lead one astray when applying algebraic geometry intuition to the rigid setting (we’ll see an example of this in another post).

#### Functor of points

We can now try and understand the functor of points of $\mathbb{B}_k-\{0\}$. But, this has essentially been observed in the process of proving Proposition 17:

Proposition 19: There is a functorial isomorphism

$\mathrm{Hom}(X,\mathbb{B}_k-\{0\})\xrightarrow{\approx}\mathcal{O}_X^+(X)^\times$

In particular, we see that $\mathbb{B}_k-\{0\}$ is a group object in the category of rigid analytic varieties over $k$.

### The multi-punctured disk

Let us now turn our attention to the case of a several times punctured closed disk. Namely, let $m\geqslant 1$ be an integer and consider $\mathbb{B}_k-\{p_1,\ldots,p_m\}$.

#### Topological properties

We would first like to give the analogue of Proposition 17 for these multi-punctured spaces:

Proposition 20: The multi-punctured closed disk $\mathbb{B}_k-\{p_1,\ldots,p_m\}$ is an admissible open subset of $\mathbb{B}_k$. It is connected and not quasi-compact.

Proof: To show that $\mathbb{B}_k-\{p_1,\ldots,p_k\}$ is an admissible open subset which is not quasi-compact, one can proceed essentially as in the proof of Lemma 17 (see [BGR, §9.1.4 Proposition 5] for more details).

To prove connectivity one makes the following general topological observation. Let $X$ be a connected topological space and let $p\in X$ be a closed point. Suppose that $p$ has a connected neighborhood $U$ of $p$ such that $U-\{p\}$ is connected, then $X-\{p\}$ is connected. Using this, and the fact that every point of $\mathbb{B}_k$ has a neighborhood isomorphic to $\mathbb{B}_k$, one reduces the question (inductively) to the case when $k=1$. The claim then follows from Proposition 17. $\blacksquare$

#### Functor of points

We would now like to compute the functor of points of $\mathbb{B}_k-\{p_1,\ldots,p_k\}$. But, again, this follows exactly as Proposition 19:

Proposition 21: There is a functorial isomorphism

$\mathrm{Hom}(X,\mathbb{B}_k-\{p_1,\ldots,p_m\})\xrightarrow{\approx}\left\{\alpha\in\mathcal{O}_X^+(X):(\alpha-p_i)\in \mathcal{O}_X^+(X)^\times\text{ for all }i\right\}$

where note that $p_i\in \mathbb{B}_k(k)=\mathcal{O}_k\subseteq \mathcal{O}_X^+(X)$ so that the above makes sense.

#### The Picard group

Finally, we would like to discuss what the Picard group $\mathrm{Pic}(\mathbb{B}_k-\{p_1,\ldots,p_m\})$ is. To do this, we assume that $k$ is actually spherically complete (also called maximally complete)for a discussion of spherically complete fields see [BGR, §2.4.4]. Then, one can show that $\mathrm{Pic}(\mathbb{B}_k-\{p_1,\ldots,p_m\})=0$. This is substantially harder for $m>0$ than the case $m=0$ discussed in the last section. For a proof one can see [FvdP, Theorem 2.7.6].

Remark 22: I’m actually not sure if the Picard group of the multi-punctured disk is non-trivial if $k$ is just assumed algebraically closed and/or complete. See Theorem 25 below for why this is not an unreasonable uncertainty. $\blacklozenge$

#### Dimension

Computing the dimension of the multi-punctured disk is easy. Namely, since $\mathbb{B}_k$ is a group object we have, as observed before, that for all points $p\in\mathbb{B}_k$ we have that

$\dim\mathcal{O}_{\mathbb{B}_k,p}=\dim\mathcal{O}_{\mathbb{B}_k,0}=1$

in particular since for any point $p\in\mathbb{B}_k-\{p_1,\ldots,p_m\}$ the local ring agrees with $\mathcal{O}_{\mathbb{B}_k,p}$ we deduce that

$\dim\mathbb{B}_k-\{p_1,\ldots,p_m\}=1$

as one would expect.

### The open disk

The next example we want to consider is the open disk $\mathbb{D}_k\subseteq \mathbb{B}_k$. More rigorously, we consider the open subset

$\mathbb{D}_k:=\{x\in\mathbb{B}_k:|x|<1\}$

which we can think of as an open rigid subspace of $\mathbb{D}_k$.

#### Topological properties

We start our discussion of $\mathbb{D}_k$ as in the case of the closed disk/multi-punctured closed disk:

Proposition 23: The open disk $\mathbb{D}_k$ is an admissible open subset of $\mathbb{B}_k$ which is connected and not quasi-compact.

Proof: As in the proof of Propsition 17 we can kill all three birds with one stone. Namely, one can readily see that

$\displaystyle \mathbb{D}_k=\bigcup_{n\geqslant 1}\underbrace{\left\{x\in \mathbb{B}_k:|t|\leqslant |\pi|^{\frac{1}{n}}\right\}}_{U\left(\frac{t^n}{\pi}\right)}$

Let us shorten $\displaystyle U\left(\frac{t^n}{\pi}\right)$ to $U_n$. Then it’s easy to see that each $U_n$ is an affinoid open subset of $\mathbb{B}_k$ (it’s a rational open). We claim that this is an admissible cover, but this follows exactly as in Proposition 17 (it follows from the Maximum Modulus Principle). Moreover, each $U_n$ is connected (use Proposition 11 and the fact that $\mathcal{O}(U_n)=k\langle t,s\rangle/(\pi s-t^n)$). Since $U_n\subseteq U_{n+1}$ this implies that $\mathbb{D}_k$ is connected.  Finally, since $U_n\subsetneq\mathbb{D}_k$ for all $n\geqslant 1$ (e.g. think about $\overline{k}$-points) we see that we have given an infinite cover of $\mathbb{D}_k$ with no finite refinement.  $\blacksquare$

#### Functor of points

We can, similar to Proposition 13, describe the functor of points of $\mathbb{D}_k$. Namely, we have the following:

Proposition 24: There is a functorial isomorphism

$\mathrm{Hom}(X,\mathbb{D}_k)\xrightarrow{\approx}\mathcal{O}_X^{++}(X)$

As in Proposition 13 we need to explain our (non-standard) notation here. Namely, we want to define a sheaf $\mathcal{O}_X^{++}$ on $X$ and to do this it suffices, as mentioned before, to define a sheaf on the Grothendieck topology of affinoid opens and affinoid open covers. So, let $\mathrm{Sp}(A)\subseteq X$ be an affinoid open. Then, we set

$\mathcal{O}_X^{++}(\mathrm{Sp}(A)):=A^{\circ\circ}$

where $A^{\circ\circ}$ is the (additive) group of topologically nilpotent elements of $A$ (e.g. the set $a\in A$ such that $\lim a_n=0$see [Bos, §3.1 Corollary 18] for more details). One can check that this satisfies the sheaf conditions and thus extends to a sheaf $\mathcal{O}_X^{++}$ on $X$. Of course, we then have that $\mathcal{O}_X^{++}(X)$ is the group of global sections of $X$.

Proof(of Proposition 24): Since both sides are sheaves for the $G$-topology on $X$ it suffices to check functorial agreeance on affinoids. So, let $\mathrm{Sp}(A)\subseteq X$ be affinoid open. Then, let us note that we have an injection

$\mathrm{Hom}(X,\mathbb{D}_k)\hookrightarrow \mathrm{Hom}(X,\mathbb{B}_k)=\mathcal{O}_X^+(X)$

where the last identification is by Proposition 13. So then, we need to figure out what subset this is. But, let us note that if $\alpha\in\mathcal{O}_X^+(X)$ corresponds to a map with image in $\mathbb{D}_k$ then $|\alpha|_\mathrm{sup}<1$ (in the language of loc. cit.see also [Bos, §3.1 Proposition 7]) and so $\alpha A^{\circ\circ}$. The converse is also clear since if $\alpha\in A^{\circ\circ}\subseteq A^\circ$ then $\alpha$ defines a map $\mathrm{Sp}(A)\to \mathbb{B}_k$. But, note that since $\alpha\in A^{\circ\circ}$ that $\frac{\alpha^n}{\pi}\in A^\circ$ for some $n\geqslant 1$ (since $\lim |\pi|^{\frac{1}{n}}=1$). Then, evidently (given Proposition 13 and the decomposition of $\mathbb{D}_k$ in Proposition 13) that $\varphi(\mathrm{Sp}(A))\subseteq U_n\subseteq \mathbb{D}_k$ and thus $\alpha$ deos define a map $\mathrm{Sp}(A)\to \mathbb{D}_k$. $\blacksquare$

#### The Picard group

The Picard group of $\mathbb{D}_k$ is surprisingly, surprisingly subtle. In particular, our Example 14 was a bit of a ruse. Namely, we have the following:

Theorem 25: The following are equivalent:

1. $\mathrm{Pic}(\mathbb{D}_k)=0$.
2. $k$ is spherically complete.

Proof: See [Gru, V, Proposition 2]. $\blacksquare$

In particular, since $\mathbb{C}_p$ is not spherically complete we have that $\mathrm{Pic}(\mathbb{D}_{\mathbb{C}_p})\ne 0$! Maybe I’ll discuss an example in the futureuntil then I suggest looking at loc. cit.

#### Dimension

Again, for any $p\in\mathbb{D}_k$ we have

$\dim\mathcal{O}_{\mathbb{D}_k,p}=\dim\mathcal{O}_{\mathbb{B}_k,p}=\dim\mathcal{O}_{\mathbb{B}_k,0}=1$

so that

$\dim\mathbb{D}_k=1$

as expected.

### Affine space (and analytification)

#### Analytification

We would now like to describe the rigid space $\mathbb{A}^{n,\mathrm{an}}_k$ which one can think of as the analytification (a la Serre’s GAGA) of the $k$-scheme $\mathbb{A}^n_k$. In some sense one can think of it as $\mathrm{Max}(k[t_1,\ldots,t_n])$ (e.g. see [Bos, §5.4 Lemma 1] for a discussion in this direction), but we shall not do so.

In general, we define the analytification of a locally of finite type $k$-scheme $X$ exactly as in complex geometry. Namely, an analytification of $X$ is a rigid space $X^\mathrm{an}$ and a map of locally ringed $G$-spaces over $k$

$(X^\mathrm{an},\mathcal{O}_{X^\mathrm{an}})\to (X,\mathcal{O}_X)$

(where $X$ has the $G$-toplogy given by its usual Zariski topology) such that for any other rigid analytic variety $Z$ one has that the natural map

$\text{Hom}((Z,\mathcal{O}_Z),(X^\mathrm{an},\mathcal{O}_{X^\mathrm{an}})\to \mathrm{Hom}((X^\mathrm{an}),\mathcal{O}_{X^\mathrm{an}},(X,\mathcal{O}_X))$

is a bijection (here both sets of maps are of locally $G$-ringed spaces over $k$).

We then have the usual proposition:

Proposition 26: For every $X$ locally of finite type $k$-scheme there exists a unique analytifiation.

Proof: See [Bos, §5.4 Definition/Proposition 3]. $\blacksquare$

The idea is somewhat clear though. Namely, since

$\displaystyle k^n=\bigcup_{n\geqslant 1}\{(x_1,\ldots,x_n): |x_i|\leqslant |\pi|^{-n}\}$

one can set

$\displaystyle \mathbb{A}^{n,\mathrm{an}}_k:=\bigcup_{n\geqslant 1}U_n$

where $U_n:=\mathbb{B}_k$ and we open embed $U_n\hookrightarrow U_{n+1}$ by identifying $U_n$ with $U\left(\frac{t}{\pi}\right)\subseteq \mathbb{B}_k=U_{n+1}$. In other words, we think of $\mathbb{A}^{n,\mathrm{an}}_k$ as the union of increasing closed disks. For an affine finite type $k$-scheme $\mathrm{Spec}(k[x_1,\ldots,x_n]/I)$ we then set

\begin{aligned}\mathrm{Spec}(k[x_1,\ldots,x_n]/I)^\mathrm{an}:= V(I\mathcal{O}_{\mathbb{A}^{n,\mathrm{an}}_k}) & =\bigcup_n V(I,U_n)\end{aligned}

where

$V(I,U_n):=\{x\in U_n: f(x)=0\text{ for all }f\in I\}$

which, as one can check, is affinoid.

For a general locally of finite type $k$-scheme $X$ one covers by finite type affines, shows that the above definition for affine finite type $k$-schemes turns distinguished open embeddings into affinoid open embeddings, then glues. Again, for more details see [Bos, §5.4 Definition/Proposition 3].

We are in this subsection particularly interested in just $\mathbb{A}^{n,\mathrm{an}}_k$ but it can’t hurt to list some basic properties of the analytification functor

$(-)^\mathrm{an}:\left\{\begin{matrix}\text{Locally of finite type}\\ k\text{-schemes}\end{matrix}\right\}\to\left\{\begin{matrix}\text{Rigid analytic varieties}\\ \text{over }k\end{matrix}\right\}$

(it is clearly a functor by the universal property of analytification) in general. Namely we have the following:

Proposition 27: Let $X$ and $Y$ be locally of finite type $k$-schemes and let $\varphi:X\to Y$ be a map of $k$-schemes.

1. There is a canonical bijection of $\mathrm{Aut}(\overline{k}/k)$-sets between $X(\overline{k})$ and $X^\mathrm{an}(\overline{k})$.
2. $X$ is connected if and only if $X^\mathrm{an}$ is connected.
3. For $x\in X(\overline{k})$ The natural map $\mathcal{O}_{X,x}\to \mathcal{O}_{X^\mathrm{an},x}$ is faithfully flat and, moreover, induces an isomorphism on completions.
4. $X$ is smooth over $k$ if and only if $X^\mathrm{an}$ is smooth over $k$. More generally, $\varphi$ is smooth if and only if $\varphi^\mathrm{an}$ is smooth.
5. $\varphi$ is finite if and only if $\mathrm{\varphi}^\mathrm{an}$ is finite.
6. $\dim X=\dim X^\mathrm{an}$.
7. $X$ is reduced if and only if $X^\mathrm{an}$ is reduced.
8. If $Z$ is any rigid space and $Y$ is affine, then the natural map $\mathrm{Hom}(Z,Y^\mathrm{an})\to \mathrm{Hom}(\mathcal{O}_Y(Y),\mathcal{O}_Z(Z))$ is a bijection.

In the above we used three  notions which have not been defined: finite morphisms, smooth morphisms, and reduced rigid analytic varieties.

So, let us say that a morphism $f:X_1\to X_2$ of rigid analytic varieties is finite if for all affinoid opens $\mathrm{Sp}(A)\subseteq X_2$ we have that $f^{-1}(\mathrm{Sp}(A))$ is affinoid open, say equal to $\mathrm{Sp}(B)$, and the induced map $A\to B$ is finite.

We will say that $f$ is smoothly factorizable if we can factorize $X\to Y$ as $X\to \mathbb{B}_k^n\times Y\to Y$ where the first map is étale (this means the usual thingit’s flat and unramified as a map of locally ringed spaces) and the second map is the projection (cf. the definition of smooth given in [Fu, §2.4]) and we say that $f$ is smooth if it’s smoothly factorizable locally on source and target. In particular, $X_1$ is smooth over $X_2=\mathrm{Sp}(k)$ if $\mathcal{O}_{X_1,x}$ is regular local for all $x\in X_1$.

Finally let us say that a rigid analytic variety $X_1$ is reduced if $\mathcal{O}_{X_1,x}$ is reduced for all $x\in X_1$.

Proof (of Proposition 27): See [Con3, Lemma 5.1.1], [Con3, Lemma 5.1.3], [Con3, Theorem 5.1.3], and [Con3, Theorem 5.2.1]. $\blacksquare$

Of course, one also has a GAGA theorem as in the complex analytic setting:

Theorem 28 (Kopf—rigid analytic GAGA): Let $X$ be a projective $k$-scheme. Then, there is an equivalence of categories

$(-)^\mathrm{an}:\left\{\begin{matrix}\text{Coherent}\\ \mathcal{O}_X\text{-modules}\end{matrix}\right\}\to \left\{\begin{matrix}\text{Coherent}\\ \mathcal{O}_{X^\mathrm{an}}\text{-modules}\end{matrix}\right\}$

such which induces natural isomorphisms

$H^i(X,\mathcal{F})\cong H^i(X^\mathrm{an},\mathcal{F}^\mathrm{an})$

for all $i\geqslant 0$.

Proof: The original attribution is [Kop] but an alternative proof, in English, can be seen in [Con2]. $\blacksquare$

In the usual way we deduce the following corollary:

Corollary 29: The analytification functor

$(-)^\mathrm{an}:\left\{\begin{matrix}\text{Projective}\\ k\text{-schemes}\end{matrix}\right\}\to\left\{\begin{matrix}\text{Rigid analytic varieties}\\ \text{over }k\end{matrix}\right\}$

is fully faithful.

Of course, we remark that the analytification functor is always faithful, but rarely full for non-projective schemes. For example the natural map

$\mathrm{Hom}(\mathbb{A}^1_k,\mathbb{A}^1_k)\hookrightarrow \mathrm{Hom}(\mathbb{A}^{1,\mathrm{an}}_k,\mathbb{A}^{1,\mathrm{an}}_k)$

is not surjective since the latter contains something like

$\displaystyle f(t):=\sum_{i=0}^\infty \pi^{i!} t^i$

which converges everywhere on $\mathbb{A}^{1,\mathrm{an}}_k$ but is not a polynomial. We will see an even more extreme example of this later on with our discussion of Tate curves.

It is also true that the analytification functor is far from essentially surjective. For example we have the following nice proposition:

Proposition 30: Let $X$ be a reduced connected finite type $k$-scheme and $A$ an affinoid $k$-algebra. Then, there is no non-constant map $X^\mathrm{an}\to \mathrm{Sp}(A)$.

In particular, the above implies that any reduced and connected rigid space which admits a non-constant map to an affinoid $\mathrm{Sp}(A)$ (e.g. an affinoid or the open disk or…) is not the analytification of a $k$-scheme.

Proof (of Proposition 30): I don’t know a canonical reference for this. See [HL, Lemma 5.4] or [JV, Proposition 2.8]. $\blacksquare$

#### Topological properties

Let us now get in to the discussion of $\mathbb{A}^{n,\mathrm{an}}_k$ proper. We begin with the following:

Proposition 31: The rigid analytic space $\mathbb{A}^{n,\mathrm{an}}_k$ is connected and not quasi-compact.

Proof: One could use Lemma 27 part 2. to show that $\mathbb{A}^{n,\mathrm{an}}_k$ is connected, but that’s too complicated. Namely, one can kill both birds with one stone by observing that, by definition, we have that

$\displaystyle \mathbb{A}^{n,\mathrm{an}}_k=\bigcup_{n\geqslant 1}U_n,\qquad U_n\subseteq U_{n+1}$

and $U_n\cong\mathbb{B}_k^n$. To see that $\mathbb{B}_k^n=\mathrm{Sp}(k\langle t_1,\ldots,t_n\rangle)$ is connected we merely use Proposition 11. Since no $U_n$ is equal to $\mathbb{A}^{n,\mathrm{an}}_k$ (by definition) we see that $\mathbb{A}^{n,\mathrm{an}}_k$ is not quasi-compact. $\blacksquare$

#### Functor of points

One can cheat and figure out the functor of points by using Proposition 27 part 8. But, we can prove this directly (and thus prove that $\mathbb{A}^{n,\mathrm{an}}_k$ really is the analytification):

Proposition 32: There is a functorial identification

$\mathrm{Hom}(X,\mathbb{A}^{n,\mathrm{an}}_k)\xrightarrow{\approx}\mathcal{O}_X(X)^n$

Proof: It suffices to check that this equality holds for $X=\mathrm{Sp}(A)$ affinoid since both sides are sheaves for the strong $G$-topology. But, then

$\displaystyle \mathrm{Hom}(\mathrm{Sp}(A),\mathbb{A}^{n,\mathrm{an}}_k)=\varinjlim_m \mathrm{Hom}(\mathrm{Sp}(A),U_m)$

where we have used the quasi-compactness of $\mathrm{Sp}(A)$ to commute this limit (since the image of $\mathrm{Sp}(A)$ must land in some $U_n$). But, since $U_n\cong \mathbb{B}_k^n$ we have (exactly as in Proposition 13) an identification

$\mathrm{Hom}(\mathrm{Sp}(A),U_m)\cong \mathcal{O}_X^+(X)^n$

but given the definition of our transition maps we then see that

$\displaystyle \varinjlim_m \mathrm{Hom}(\mathrm{Sp}(A),U_m)=\bigcup_{m\geqslant 1}\pi^{-m}\mathcal{O}_X^+(X)^n$

where the union is taken in $\mathcal{O}_X(X)$. We leave it to the reader to verify that

$\displaystyle \bigcup_{m\geqslant 1}\pi^{-m}\mathcal{O}_X^+(X)^n=\mathcal{O}_X(X)^n$

as desired (hint: show both sides are sheaves and so it suffices to show the result on affinoids and then use the definition in [Tia, Proposition 1.4.13], the fact that $|\pi^n|\to 0$, and the fact that $|\cdot|_\mathrm{sup}$ is submultaplicative). $\blacksquare$

#### The Picard group

Let us now discuss what the Picard group of $\mathbb{A}^{n,\mathrm{an}}_k$ is. One should be leery about guessing that it’s $0$ since, in some sense, $\mathbb{A}^{n,\mathrm{an}}_k$ is like an ‘infinite radius’ version of the open (poly)disk $\mathbb{D}_k^n$ since it’s an infinite increasing union of closed disks. Thanks to Theorem 25 this might make one guess that perhaps the Picard group is non-trivial. Thankfully this is not the case:

Theorem 33: The group $\mathrm{Pic}(\mathbb{A}^{n,\mathrm{an}}_k)$ is trivial.

Proof: See [Gru, V, Proposition 2] for a nice discussion of this fact. $\blacksquare$

Remark 34: Just as an interesting side note it’s actually still an open question as to whether or not the ‘Bass-Quillen conjecture’ is true for $\mathbb{A}^{n,\mathrm{an}}_k$in other words whether all vector bundles are trivial. See for instance see [KST] or [Sig]. $\blacklozenge$

#### Dimension

The dimension of $\mathbb{A}^{n,\mathrm{an}}_k$ is what one would expect. Namely, from Proposition 27 part 6. we have that

$\dim \mathbb{A}^{n,\mathrm{an}}_k=\dim\mathbb{A}^n_k=n$

as desired. Of course, one can also verify this by hand essentially by showing that each $U_n\cong\mathbb{B}^n_k$ is $n$-dimensional. This follows from generalizing our dimension arguments in the case when $n=1$ from above.

### Projective space

We are now interested in studying $\mathbb{P}^{n,\mathrm{an}}_k$ which, by deifnition for us, is $(\mathbb{P}^n_k)^\mathrm{an}$.

#### Topological properties

We would now like to state the basic topological properties of $\mathbb{P}^{n,\mathrm{an}}_k$. Namely, we have the following:

Proposition 34: The space $\mathbb{P}^{n,\mathrm{an}}_k$ is connected and quasi-compact.

The proof of Proposition 34 can be proven by giving a decomposition of $\mathbb{P}^{n,\mathrm{an}}_k$ into certain open affinoids. Since this decomposition is independently interesting, we discuss it outside the context of the proof of Proposition 34.

A naive guess is that one can decompose $\mathbb{P}^{n,\mathrm{an}}_k$ in to open subspace isomorphic to $\mathbb{A}^{n,\mathrm{an}}_k$. This is, in fact, true but is not as useful since it’s not a decomposition into affinoids. To decompose in to affinoids we can first remember how the decomposition of $\mathbb{P}^n_k$ into open copies of $\mathbb{A}^n_k$ works. Namely, this decomposition comes by the observation that an element $[x_0:\cdots:x_n]\in\mathbb{P}^n_k$ with $x_i\ne 0$ can be scaled as follows:

$\displaystyle [x_0:\cdots: x_n]=\left[\frac{x_0}{x_i}:\cdots :\underbrace{1}_{i^\text{th spot}}:\cdots:\frac{x_n}{x_i}\right]$

One then notes that such a representation is unique and, in fact, the map

$\displaystyle \{[x_0:\cdots :x_n]: x_i\ne 0\}\to\mathbb{A}^n_k:[x_0:\cdots:x_n]\mapsto \left(\frac{x_0}{x_i},\ldots,\frac{x_n}{x_i}\right)$

(where in the last tuple we omit the entry $\displaystyle \frac{x_i}{x_i}$) is a bijection.

In the rigid setting we can in fact do better. Namely, let us say that a $(x_0,\ldots,x_n)\in k^{n+1}$ is unidmodular if $x_i\in \mathcal{O}_k$ for all $i$ and $x_i\in\mathcal{O}_k^\times$ for some $i$. Note that by scaling one can assume that every $[x_0:\cdots:x_n]\in\mathbb{P}^n_k$ is represented such that $(x_0,\ldots,x_n)$ is unimodular. One then defines a bijection

\left\{[x_0:\cdots :x_n]: \begin{aligned}(1)&\quad (x_0,\ldots,x_n)\text{ is unimodular}\\ (2)&\quad x_i\in \mathcal{O}_k^\times\end{aligned}\right\}\to\mathbb{B}^n_k

given by

$\displaystyle [x_0:\cdots : x_n]\mapsto \left(\frac{x_0}{x_i},\dots,\frac{x_n}{x_i}\right)$

(where again we’ve omitted $\displaystyle \frac{x_i}{x_i})$).

In more rigorously term we have an isomorphism

$\displaystyle \mathbb{P}^{n,\mathrm{an}}_k=\left(\bigsqcup_{i=0}^n U_i\right)/\sim$

with (following the nice notation of [Vak, Exercise 4.4.9])

$U_i\cong \mathrm{Sp}\left(k\left\langle t_{\frac{0}{i}},\ldots,t_{\frac{n}{i}}\right\rangle\right)$

(where again we are omitting $t_{\frac{i}{i}}$) and we are making the identifications

$U_i\supsetneq \mathrm{Sp}\left(k\left\langle t_{\frac{0}{i}},\ldots,t_{\frac{n}{i}}\right\rangle\left\langle t_{\frac{j}{i}}^{-1}\right\rangle\right) \cong \mathrm{Sp}\left(k\left\langle t_{\frac{0}{j}},\ldots,t_{\frac{n}{j}}\right\rangle\left\langle t_{\frac{i}{j}}^{-1}\right\rangle\right)\subseteq U_j$

given by $t_{\frac{j}{i}}\mapsto t_{\frac{i}{j}}^{-1}$. In fact, one could (perhaps should) take this as our definition of $\mathbb{P}^{n,\mathrm{an}}_k$ using our description of the functor of points of $\mathbb{P}^{n,\mathrm{an}}_k$ (using this gluing description as our definition!) to verify it really is an analytification of $\mathbb{P}^n_k$.

Remark 35: Since we were working at the level of points the above is only really valid over $\overline{k}$. That said, one can easily check that this decomposition is $\mathrm{Aut}(\overline{k}/k)$-invariant and so descends to a decomposition over $k$.

Regardless, this decomposition immediately implies Proposition 34 since each $U_i$ (where we are imprecisely aslo denoting by $U_i$ image under the quotient map $\displaystyle \bigsqcup_i U_i\to \mathbb{P}^{n,\mathrm{an}}_k$) is connected and quasi-compact, and $U_i\cap U_j\ne \varnothing$ for all $i,j$.

#### Functor of points

The functor of points of $\mathbb{P}^{n,\mathrm{an}}_k$ is what one expects from its corresponding description in algebraic geometry. Namely:

Theorem 36: There is a functorial bijection

\mathrm{Hom}(X,\mathbb{P}^{n,\mathrm{an}}_k)\xrightarrow{\approx} \left\{(\mathscr{L},s_0,\ldots,s_n):\begin{aligned}(1)&\quad \mathscr{L}\text{ is a line bundle on }X\\ (2)&\quad s_i\in\mathscr{L}(X)\text{ globally generated }\mathscr{L}\end{aligned}\right\}/\sim

Here we say, as in algebraic geometry, a set $\{s_i\}$ of $\mathscr{L}(X)$ is globally generating if for all $x$ in $X$ one has that the image of $\{s_i\}$ under the map

$\mathscr{L}(X)\to\mathscr{L}_x$

is a generating set of the $\mathcal{O}_{X,x}$-module $\mathscr{L}_x$. Moreover, we say that

$(\mathscr{L},s_0,\ldots,s_n)\sim (\mathscr{L}',s_0',\ldots,s_n')$

if there exists an isomorphism $\mathscr{L}\to\mathscr{L}'$ carrying $s_i$ to $s_i'$.

The proof of Proposition 36 is exactly in the scheme theoretic situation, and we leave it to the reader to check this.

#### Picard group

One can use Rigid Analytic GAGA to see that

$\mathrm{Pic}(\mathbb{P}^{n,\mathrm{an}}_k)\cong \mathbb{Z}$

and, moreover, that a generator for $\mathrm{Pic}(\mathbb{P}^{n,\mathrm{an}}_k)$ is given by $\mathcal{O}(1)^\mathrm{an}$. Concretely one can think about this as the line bundle obtained on

$\displaystyle \mathbb{P}^{n,\mathrm{an}}_k\cong \left(\bigsqcup_{i=0}^n U_i\right)/\sim$

which is given by the trivial line bundle on each $U_i$ and which is glued via the overlap isomorphisms $t_{\frac{i}{j}}\mapsto t_{\frac{j}{i}}^{-1}$.

If one doesn’t want to use the full power of Rigid Analytic GAGA one might try to use the isomorphism

$\mathrm{Pic}(\mathbb{P}^{n,\mathrm{an}}_k)\cong H^1(\mathbb{P}^{n,\mathrm{an}}_k,\mathcal{O}_{\mathbb{P}^{\mathrm{an}}_k}^\times)\cong \check{H}^1(\mathbb{P}^{n,\mathrm{an}}_k,\mathcal{O}_{\mathbb{P}^{\mathrm{an}}_k}^\times)$

(which is true for all locally $G$-ringed spacee.g. see [Stacks, Tag040E] and [Stacks, Tag0A6G]) together with the non-trivial fact that $\{U_i\}$ constitutes a Leray cover (i.e. a covering as in [Stacks,03F7]) to deduce that

$\mathrm{Pic}(\mathbb{P}^{n,\mathrm{an}}_k)\cong \check{H}^1(\{U_i\},\mathcal{O}_{\mathbb{P}^{n,\mathrm{an}}_k}^\times)$

from which one really can deduce that $\mathcal{O}(1)^{\mathrm{an}}$ is a generator by the usual computation. The fact that $\{U_i\}$ really is a Leray cover, the fact that $\mathbb{B}^n_k$ has vanishing higher coherent cohomology, is merely a souping up of Theorem 7 (e.g. see [BGR, §8.2.1 Theorem 1]).

#### Dimension

Given our decomposition

$\mathbb{P}^{n,\mathrm{an}}_k=\bigcup_{i=0}^n U_i$

we know that

$\dim\mathbb{P}^{n,\mathrm{an}}_k=\sup_i \dim U_i=\sup_i n=n$

as one would expect.

### The multiplicative group

We now move on the multiplicative group $\mathbb{G}_{m,k}^\mathrm{an}$ which, by definition for us, is the analytification of the multiplicative group $\mathbb{G}_{m,k}:=\mathrm{Spec}(k[t,t^{-1}])$.

#### Topological properties

We start with the following topological description of $\mathbb{G}_{m,k}^\mathrm{an}$:

Theorem 37: The topological space $\mathbb{G}_{m,k}^\mathrm{an}$ is connected but not quasi-compact.

Again, we might try to apply Proposition 27 part 2., but this is wholly overkill. In fact, both of these properties follow from the following generally useful decomposition of $\mathbb{G}_{m,k}^\mathrm{an}$. Namely, recall that we have the decomposition

$\displaystyle \mathbb{A}^{1,\mathrm{an}}_k=\bigcup_{n\geqslant 1}U_n$

with each $U_n\cong \mathbb{B}_k$. From this, we see that we have the decomposition

$\displaystyle \mathbb{G}_{m,k}^\mathrm{an}=\bigcup_{n\geqslant 1}(U_n-\{0\})$

where one can check that $U_n-\{0\}\cong \mathbb{B}_k-\{0\}$. Since each $U_n-\{0\}$ intersects all the other $U_m-\{0\}$, and each are connected (by Proposition 17) we deduce that $\mathbb{G}_{m,k}^\mathrm{an}$ is connected. It’s clear that no finite subset of $\{U_n-\{0\}\}$ covers $\mathbb{G}_{m,k}^\mathrm{an}$ and thus we also deduce that $\mathbb{G}_{m,k}^\mathrm{an}$ is not quasi-compact.

#### Functor of points

We now move on to the description of the functor of points of $\mathbb{G}_{m,k}^\mathrm{an}$. Of course, we expect that it spits out the units of the global sections of a rigid analytic variety and this actually follows from Proposition 27 part 8. But, again, we can see this in a much more down-to-earth fashion:

Theorem 38: There is a functorial bijection

$\mathrm{Hom}(X,\mathbb{G}_{m,k}^\mathrm{an})\xrightarrow{\approx} \mathcal{O}_X(X)^\times$.

Proof: The proof of this is essentially exactly as in Proposition 32. Namely, we see that it suffices to check this for an affinoid space $\mathrm{Sp}(A)$. From there we use the quasi-compactness of $\mathrm{Sp}(A)$ to write

$\displaystyle \mathrm{Hom}(\mathrm{Sp}(A),\mathbb{G}_{m,k}^\mathrm{an})=\varinjlim\mathrm{Hom}(\mathrm{Sp}(A),U_n-\{0\})$

to which one then applies Propositition 19 and the decomposition

$\displaystyle \bigcup_{n\geqslant 1}\pi^{-n}(\mathcal{O}_X^+(X)^n)^\times=\mathcal{O}_X(X)^\times$

$\blacksquare$

#### The Picard group

The Picard group of $\mathbb{G}_{m,k}^\mathrm{an}$ is again slightly more subtle than one might bargain for. That said, we do have the following result:

Theorem 39: If $k$ is spherically complete then $\mathrm{Pic}(\mathbb{G}_{m,k}^\mathrm{an})=0$.

Proof: This follows from [FvdP, Theorem 2.7.6]. $\blacksquare$

It’s likely that this theorem holds without the assumption that $k$ is sphereically complete, but I’m not sure.

#### Dimension

The dimension of $\mathbb{G}_{m,k}^\mathrm{an}$ is $1$ as one would expect. This follows since each $U_n-\{0\}$ has dimension $1$ by our previous discussion about the dimension of multi-punctured disks.

### Tate elliptic curves

So, let us consider the meromorphic function field

$\mathcal{M}(\mathbb{G}_{m,k}^\mathrm{an}):=\mathrm{Frac}(\mathcal{O}(\mathbb{G}_{m,k}^\mathrm{an}))$

where more explicitly one can write

\displaystyle \mathcal{O}(\mathbb{G}_{m,k}^\mathrm{an})=\left\{\sum_{n\in\mathbb{Z}}a_n t^n:\begin{aligned}(1)&\quad a_n\in k\\ (2)&\quad \lim_{|n|\to \infty}|a_n|r^n=0\text{ for all }r>0\end{aligned}\right\}

which is a domain as one can easily check. Let us now fix $q$ in $k$ with $|q|<1$. We can then consider the subset

$\mathcal{M}_q(\mathbb{G}_{m,K}^\mathrm{an}):=\{f(t)\in \mathcal{M}(\mathbb{G}_{m,k}^\mathrm{an}): f(qt)=f(t)\}$

of $\mathcal{M}(\mathbb{G}_{m,k}^\mathrm{an})$ which is, in fact, is a subfield as one can easily check.

The realization that Tate had, which was one of the first motivating reasons to want to create a theory of rigid geometry, was the following beautiful result:

Theorem 40 (Tate): There exists a unique elliptic curve $E_q$ over $k$ such that $K(E_q)\cong \mathcal{M}_q(\mathbb{G}_{m,k}^\mathrm{an})$. Moreover, one has that

$\displaystyle j(E_q) =j(q):=1728 \frac{\displaystyle \left(1+240\sum_{n\geq 1}\sigma_3(n)q^n\right)^3}{\displaystyle \left(1+240\sum_{n\geq 1} \sigma_3(n)q^n\right)^3 - \left(1-504\sum_{n\geq 1}\sigma_5(n)q^n \right)^2}$

Moreover, one has that there is a functorial isomorphism

$E_q(L)\cong \mathbb{G}_{m,k}^\mathrm{an}(L)/q^\mathbb{Z}$

for every extension $L$ of $k$. Moreover, the map

$j:\{q\in k^\times:0<|q|<1\}\to \{x\in k^\times: |x|>1\}$

is a bijection with analytic inverse.

Proof: See [Lut, Theorem 1.1] and [Lut, Theorem 1.2] and the references therein. $\blacksquare$

Now, one is tempted to think that the isomorphism

$E_q(\overline{k})\cong \mathbb{G}_{m,k}^\mathrm{an}(\overline{k})/q^\mathbb{Z}$

could be upgraded to an isomorphism

$E_q\cong \mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$

in some sense. Of course, we have to make sense of what this quotient means on the right-hand side.

To do this, we need to give a decomposition of $\mathbb{G}_{m,k}^\mathrm{an}$ which is shuffled around by the action of $q^\mathbb{Z}$. Namely, let us define the following affinoid domains inside of $\mathbb{B}_k$

$\displaystyle U_1:=U\left(\frac{q}{t^2}\right),\qquad U_2:=U\left(\frac{q}{t}\right)\cap U\left(\frac{t^2}{q}\right)$

One note that

$U_1(\overline{k})=\{x\in\mathbb{B}_k(\overline{k}):|q|^{\frac{1}{2}}\leqslant |x|\leqslant 1\}$

and

$U_2(\overline{k})=\{x\in\mathbb{B}_k(\overline{k}):|q|\leqslant |x|\leqslant |q|^{\frac{1}{2}}\}$

Both of these spaces are annuli and they have an outer circle and an inner circle defined as further rational domains inside of $\mathbb{B}_k$. We denote the outer circles of $U_i$ by $U_i^+$ and the inner circles by $U_i^-$. Evidently we have that $U_1^-=U_2^+$.

Note that for each $n\in\mathbb{Z}$ we have a natural multiplication-by-$q^n$-map

$q^n:\mathbb{G}_{m,k}^\mathrm{an}\to\mathbb{G}_{m,k}^\mathrm{an}$

which for an affinoid algebra $A$ is just the map

$A^\times\to A^\times:x\mapsto q^nx$

This is an isomorphism of rigid analytic varieties. We note then that one has $q(U_1^+)=U_2^-$. Moreover

$\displaystyle \mathbb{G}_{m,k}^\mathrm{an}=\bigcup_{n\in\mathbb{Z}}q^n(U_1\cup U_2)$

gives an open cover of $\mathbb{G}_{m,k}^\mathrm{an}$.

From this, it seems reasonable to define $\mathbb{G}_{m,k}^\mathrm{an}$ to be

$(U_1\cup U_2)\sqcup_q (U_1\cup U_2)$

which which we mean the rigid space obtained by gluing $U_1\cup U_2$ to itself along the isomorphism $q:U_1^+\to U_2^-$ (gluings along open subspaces clearly exist in the category of rigid analytic spaces).

We then have the following justification for writing this as a quotient

Proposition 41: The obvious map

$\mathbb{G}_{m,k}^\mathrm{an}\to\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$

is $q^n$-invariant for all $n\in\mathbb{Z}$ and is universal for such.

Proof: Obvious. $\blacksquare$

The beautiful tie-in with Theorem 40, as we’d hoped, is the following:

Theorem 42(Tate): There is an isomorphism of rigid analytic varieties $E_q^\mathrm{an}\cong \mathbb{G}_{m,k}/q^\mathbb{Z}$.

Proof: See [Lut, Theorem 1.2] and the references therein. $\blacksquare$

We call rigid analytic varieties of the form $\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ Tate elliptic curves.

Note that we really needed to be working in the rigid category to have an isomorphism

$E_q^\mathrm{an}\cong\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$

for two distinct, but equally important, reasons:

1. The map $\mathbb{G}_{m,k}^\mathrm{an}\to\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ has infinite fibers. Note that any infinite subset of $\mathbb{G}_{m,k}$ is dense, and thus one cannot even have a non-constant map from $\mathbb{G}_{m,k}$ with infinite fibers.
2. We get from Theorem 41, in particular, a surjective map $\mathbb{G}_{m,k}^\mathrm{an}\to E_q^\mathrm{an}$. Note that any algebraic map $\mathbb{G}_{m,k}\to E$ (for any elliptic curve $E$ over $k$) is constant since otherwise it would give rise to a surjective map $\mathbb{P}^1_k\to E$ which is impossible by the Riemann–Hurwitz formula (e.g. see [Har, Example 2.5.4]).

Remark 43: As mentioned earlier we see from 2. the striking non-fullness of the map

$\mathrm{Hom}_k(\mathbb{G}_k,E_q)\to \mathrm{Hom}(\mathbb{G}_{m,k}^\mathrm{an},E_q^\mathrm{an})$

since the former set is consists only of constant maps but the latter set contains a topological cover!.

#### Topological properties

The topological properties of Tate elliptic curves are quite simple. Namely, they are quasi-compact and connected. The quasi-compactness comes from the fact that they are a quotient of the compact space

$(U_1\cup U_2)\sqcup(U_1\cup U_2)$

and they are connected since the images of each copy of $U_1\cup U_2$ in $\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ are connected and intersect non-trivial.

Functor of points

The functor of points of $\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ is not particularly easy to describe since it is, after all, related to the functor of points of $E_q$ which has no explicit description.

#### The Picard group

By Theorem 28 we know that we have an isomorphism

$\mathrm{Pic}(\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z})\cong\mathrm{Pic}(E_q)$

and thus it suffices to understand the Picard group of $E_q$. But,

$\mathrm{Pic}(E_q)\cong \mathrm{Pic}^0(E_q)\times \mathbb{Z}$

where $\mathrm{Pic}^0(E_q)$ are the degree $0$ line bundles in on $E_q$. Moreover, we know that

$\mathrm{Pic}^0(E_q)\cong E_q(k)\cong k^\times/q^\mathbb{Z}$

where the first isomorphism has inverse

$p\mapsto \mathcal{O}(p-0)$

where $0\in E_q(k)$ is the identity element, and the second isomorphism is from Theorem 40. To understand the composition explicitly, if one fixes $\alpha$ in $k^\times$ one gets a line bundle $\mathcal{O}(\alpha)$ on $\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ by gluing the trivial line bundles on $q^n(U_1\times U_2)$ via the transition map where $q$ acts as multiplication by $\alpha$ on the trivial bundle. This does descend to a line bundle on $\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$. Moreover, this line bundle only depends on $\alpha$ up to its class in $k^\times/q^\mathbb{Z}$.

#### Dimension

The dimension of $\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ is obviously one. Indeed, $U_1\cup U_2$ has dimension $1$, and since $\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ is obtained by gluing $U_1\cup U_2$ to itself along an open subset, the conclusion follows.

#### Arithmetic applications

Since one of our main motivations for defining rigid analytic geometry is to study arithmetic geometry we would be remiss to not give some number theoretic applications of the theory of Tate elliptic curves.

##### Reduction type and Tate’s theorem

To explain these results, it is first helpful refine the results  of Theorem 40 and Theorem 42 slightly. To do this, let us first recall some basic theory of elliptic curves over a $p$-adic field $F$. We shall denote the ring of integers of $F$ by $\mathcal{O}$ and the residue field of $F$ by $k$ (don’t confuse this with our analytic field from above!). We shall denote the valuation on $F$ (the unique one extending the $p$-adic valuation on $\mathbb{Q}_p$) by $v$ and we shal ldenote the associated absolute value on $F$ by $|\cdot|$.

For the sake of simplicity, we assume that $p>3$. This will allow us to work with simplified Weierstrass models.

We then recall, following [Con4], that a Weierstrass model of $E$ is a relative cubic curve over $\mathcal{O}$

$\mathcal{E}:=V(y^2z+x^3-Ax^2z-Bz^3)\subseteq \mathbb{P}^2_{\mathcal{O}}$

together with an isomorphism $i:E\to \mathcal{E}_F$. Again, note that [Con4] requires that $\mathcal{E}$ is actually an essentially general cubic, but since $p>3$ we have that $6$ is invertible in $\mathcal{O}$ and so we can always put our elliptic curve in this form (e.g. see [KM, (2.2.6)]). We define the discriminant of $\mathcal{E}$, denoted $\Delta(\mathcal{E})$, as follows:

$\Delta(\mathcal{E}):=-16(4A^3+27B^2)$

One can check that $\Delta(\mathcal{E})$ is indpendent of the isomorphism class of $\mathcal{E}$. One can give a more highfalutin description of the discriminant as in [Con4] where this becomes more obvious.

We then note that $\mathcal{E}$ is smooth, and thus an elliptic scheme (in the sense of [KM, (2.1.1)]) if and only if $v(\Delta(\mathcal{E}))=0$. In general there won’t exist a Weierstrass model of $E$ which is smooth, and the best thing we can ask is that $\mathcal{E}$ is a minimal Weierstrass model which means that $v(\Delta(\mathcal{E}))$ is minimal amongst Weierstrass models for $E$. A priori if one has two minimal Weierstrass models $\mathcal{E}_1$ and $\mathcal{E}_2$ one only knows that $v(\Delta(\mathcal{E}_1))=v(\Delta(\mathcal{E}_2))$ and, in particular, there is no reason for $\mathcal{E}_1$ to be isomorphic to $\mathcal{E}_2$. This is the case though. This can be seen by explicit calculation (as in [Sil, 1.3 in VII]) or by abstract theory (as in [Con4, Corollary 4.6]).

Regardless, we now know that a minimal Weierstrass model $\mathcal{E}^\text{min}$ for $E$ exists and is unique up to isomorphism. We then define the reduction of $E$ to be the curve $\overline{E}:=\mathcal{E}^\text{min}_k$.

Of course, this reduction is a priori just just a cubic curve embeddable in $\mathbb{P}^2_k$. In particular, there is no reason that $\overline{E}$ should be smooth over $k$. That said, $\overline{E}$ is necessarily very mildly singular (if it’s singular at all):

Proposition 44: Let $k$ be any field and let $C$ be a singular cubic hypersurface in $\mathbb{P}^2_k$. Then, $C$ has exactly one singular point $x_0$ and $x_0$ is in $C(k)$.

Proof: Suppose first that $k$ is algebraically closed. Let $p$ and $q$ be points of $C(k)$ which are both singular. Note then that there exists a line $\ell\subseteq \mathbb{P}^2_k$ such $p$ and $q$ both lie in $\ell(k)$. Note then that by Bézout’s theorem (e.g. see [Liu, Corollary 1.20 in §9.1.2]) we have that the $\ell\cdot C=3$ and in particular we see that one of $i_p(\ell,C)$ or $i_q(\ell,C)$ (using [Lieu, Definition 1.1 §9.1.1]) must be $1$. Without loss of generality we may assume that $i_p(\ell,C)=1$. But, this implies that $C$ is smooth at $p$intuitively this means that $\ell$ and $C$ meet transversally at $p$ so that

$T_p\mathbb{P}^2_k=T_p\ell\oplus T_p C$

(where $T_p$ is the tangent space) which then implies that $T_p C$ has dimension $1$, which means that $C$ is smooth at $p$ (for details see [Liu, Proposition 1.8 in §9.1.1]). This is a contradiction.

Suppose now that $k$ is not algebraically closed and assume that $p$ is a singular point of $C$. Note that if $p$ is not a $k$-point of $C$ then the preimage of $p$ under the map $C_{\overline{k}}\to C$ has more than $1$ point. Each of these points is necessarily singular which contradicts the first paragraph. $\blacksquare$

We thus can try to understand the geometric properties of $\overline{E}$, and thus the properties of $E$, by trying to suss out the properties of the singularity of $\overline{E}$ or, equivalently (perhaps not so obviously) the properties of of the smooth locus $\overline{E}^\text{sm}$ which, by Proposition 44, is just $\overline{E}-\{x_0\}$ for some $k$-point $x_0$ in $\overline{E}(k)$. Let us note that, as can be checked by hand, the point $e=[0:1:0]$ in $\overline{E}(k)$ is smooth and so, in particular, $x_0\ne e$.

Before we state the marquee classification in this direction, we first note the following beautiful fact:

Proposition 45: There exists the unique structure of a group variety on $\overline{E}^\text{sm}$ such that $e$ is the identity element. Moreover, $\overline{E}^\text{sm}$ is smooth, integral, and dimension $1$. If $\overline{E}$ is not smooth then $\overline{E}^\text{sm}$ is affine.

Proof(Sketch): The point is that the chord-tangent construction as in the theory of elliptic curves works here as well.

More explicitly, let $p,q\in \overline{E}^{\text{sm}}(\overline{k})$ be distinct. There exists a unique line $\ell_{p,q}\subseteq \mathbb{P}^2_{\overline{k}}$ such that $\ell_{p,q}$ passes through $p$ and $q$. If $p=q$ we set $\ell_{p,q}$ to be the unique line in $\mathbb{P}^2_{\overline{k}}$ passing through $p$ and which is tangent to $\overline{E}_{\overline{k}}$ at $p$.

Note then that by Bézout’s theorem that for any $p,q\in \overline{E}(\overline{k})$ (possibly with $p=q$) we have that $\ell_{p,q}\cap \overline{E}_{\overline{k}}$ is a finite $\overline{k}$-scheme with $3$-dimensional global sections. If $p\ne q$ then it’s easy to see that $\ell_{p,q}\cap \overline{E}_{\overline{k}}$ is the disjoint union of three $\overline{k}$-points of $\overline{E}(\overline{k})$ which are $p,q$ and a third point $r$. Moreover, note that necessarily $r$ is another point of $\overline{E}^\text{sm}(\overline{k})$ since the intersection of $\ell_{p,q}$ and $\overline{E}_{\overline{k}}$ neccessarily has multiplicity (cf. see the proof of Proposition 44). If $p=q$ then $\ell_{p,p}\cap E$ is the disjoint union of $\mathrm{Spec}(\mathcal{O}_{\mathbb{P}^2_{\overline{k}},p}/(I_p^{\overline{E}},I_p^{\ell_{p,p}}))$ (where for a closed subscheme $Z$ of $\mathbb{P}^2_{\overline{k}}$ we denote by $I^Z_p$ the ideal of $\mathcal{O}_{\mathbb{P}^2_{\overline{k}},p}$ induced by $Z$) and an $\overline{k}$-point $r$. Again, since the multiplicity of $\ell_{p,p}\cap \overline{E}_{\overline{k}}$ at $p$ is $2$, we see that $r$ is in $\overline{E}^\text{sm}(\overline{k})$. In either case we see that we obtain a third point $r\in \overline{E}^\text{sm}(\overline{k})$ from $p,q$.

Note then the exact same ideas yield associated to the pair $r,e\in \overline{E}^\text{sm}(\overline{k})$ a line $\ell_{r,e}\subseteq \mathbb{P}^2_{\overline{k}}$ and a third point of intersection of the line $\ell_{r,e}$ and $E_{\overline{k}}$ which is in $\overline{E}^\text{sm}(\overline{k})$. We denote this third $\overline{k}$-point of $\overline{E}(\overline{k})$ by $p+q$.

The proof that this process gives a group structure on $\overline{E}^\text{sm}(\overline{k})$ is the same as in [Sil, Proposition 2.2 in §III.2] and the fact that this is algebraic and defined over $k$ is the same as in [Sil, Group Law Algorithm 2.3 in §III.2].

Let us now verify the claimed geometric properties of $\overline{E}^\text{sm}$. The fact that it’s smooth and dimension $1$ are obvious, and thus it suffices to prove that it’s integral. Since it’s smooth it suffices to prove it’s connected (e.g. by [Stacks, Tag033M]). But, to do this it suffices to show that $\overline{E}$ is irreducible. But, this is clear since

$\overline{E}=V(y^2z-x^3-\overline{A}xz^2-\overline{B}z^3)\subseteq \mathbb{P}^2_k$

where $\overline{A}$ and $\overline{B}$ are the images of $A$ and $B$ in $k$. In particular, note that the equation

$y^2z-x^3-\overline{A}xz^2-\overline{B}z^3$

is evidently irreducible by setting $z=1$ and arguing that any factorization of $y^2-x^3-\overline{A}x-\overline{B}$ would have a non-zero $xy$ term. From this, the claim that $\overline{E}$ is irreducible is clear from basic theory.

Finally we verify the affineness claim. But, this follows from totally general theory (e.g. see Lemma 13 from this post). $\blacksquare$

Thus, a natural question is what group $\overline{E}^\text{sm}$ is. As it turns out there are not many options:

Proposition 46: Let $k=\mathbb{F}_q$ be a finite field and $G$ a smooth integral connected group variety of dimension $1$. Then, either $G$ is an elliptic curve or it’s isomorphic to one of the following three groups:

1. The additive group $\mathbb{G}_{a,\mathbb{F}_q}$.
2. The split multiplicative group $\mathbb{G}_{m,\mathbb{F}_q}$.
3. The non-split multiplicative group $\mathsf{Res}^1_{\mathbb{F}_{q^2}/\mathbb{F}_q}\mathbb{G}_{m,\mathbb{F}_{q^2}}$.

Proof: This follows from Theorem 29 of this post. $\blacksquare$

Here we using the notation $\mathsf{Res}^1_{\mathbb{F}_{q^2}/\mathbb{F}_q}\mathbb{G}_{m,\mathbb{F}_{q^2}}$ to denote the norm $1$ elements of the Weil restriction (e.g. see [Mil1, §2.i]).

Abstractly this is the group scheme which associates to an $\mathbb{F}_q$-algebra $R$ the group

$\ker\left(N:(\mathbb{F}_{q^2}\otimes_{\mathbb{F}_q} R)^\times\to R^\times\right)$

where $N(a)$ is defined to be the determinant of the $R$-linear map

$[a]:\mathbb{F}_{q^2}\otimes_{\mathbb{F}_q} R\to \mathbb{F}_{q^2}\otimes_{\mathbb{F}_q}R:x\mapsto ax$

which makes sense since $\mathbb{F}_{q^2}\otimes_{\mathbb{F}_q}R$ is a free $R$-module of rank $2$.

Concretely if $\mathbb{F}_{q^2}=\mathbb{F}_q[\sqrt{\alpha}]$ for $\alpha\in\mathbb{F}_q$ then

$\mathsf{Res}^1_{\mathbb{F}_{q^2}/\mathbb{F}_q}\mathbb{G}_{m,\mathbb{F}_{q^2}}$

can be thought about as the subgroup of $\mathsf{Res}_{\mathbb{F}_{q^2}/\mathbb{F}_q}\mathrm{GL}_{2,\mathbb{F}_{q^2}}$ given as the set of matrices of the form

$\left\{\begin{pmatrix}a & \alpha b\\ b& a\end{pmatrix}:a^2+\alpha b^2=1\right\}$

with the usual matrix multiplication.

So, with this setup we can divide up the elliptic curves over $F$ into categories depending on which group $\overline{E}^\text{sm}$ is which constitutes four cases:

1. If $\overline{E}^\text{sm}$ is an elliptic curve we say that $E$ has good reduction.
2. If $\overline{E}^\text{sm}$ is the additive group we say that $E$ has additive reduction.
3. If $\overline{E}^\text{sm}$ is the split multiplicative group we say that $E$ has split multiplicative reduction.
4. If $\overline{E}^\text{sm}$ is the non-split multiplicative group we say that $E$ has non-split multiplicative reduction.

There is further terminology concerning reduction type that is useful. Namely, if $E$ has either split or non-split multiplicative reduction, we say that $E$ has multiplicative reduction. If $E$ has either good reduction or multiplicative reduction then $E$ is said to have semistable reduction. If $E$ has either good or additive reduction it is said to have potentially good reduction. The reason for this naming scheme will soon be clear.

As we said before, we can also understand the reduction type of $E$ in terms of the singularity of $E$ at its singular point $x_0$ (if it has one) as follows:

Proposition 47: Let $E$ be an elliptic curve over $F$ and assume that $k=\mathbb{F}_q$. Then, the following are equivalent:

1. $E$ has good reduction.
2. $v(\Delta(\mathcal{E}^\text{min}))=0$.
3. If one writes $\mathcal{E}^\text{min}$ as the projective closure of $y^2=x^3+Ax+B$, then $x^3+\overline{A}x+\overline{B}$ is separable.

So are equivalent the following:

1. $E$ has additive reduction.
2. The completed local ring $\widehat{\mathcal{O}}_{\overline{E},x_0}$ is isomorphic as $k$-algebra to $k[[x,y]]/(y^2-x^3)$.
3. $v(\Delta(\mathcal{E}^text{min}))>0$ and $\overline{B}=0$.
4. If one writes $\mathcal{E}^\text{min}$ as the projective closure of $y^2=x^3+Ax+B$, then $x^3+\overline{A}x+\overline{B}$ has a repeated root of multiplicity $3$.

So are equivalent the following:

1. $E$ has split multiplicative reduction.
2. The completed local ring $\widehat{\mathcal{O}}_{\overline{E},x_0}$ is as a $k$-algebra isomorphic to $k[[x,y]]/(xy)$.
3. $v(\Delta(\mathcal{E}^\text{min}))=0$, $\overline{B}\ne 0$, and $\overline{A}$ is a square in $k$.
4. If one writes $\mathcal{E}^\text{min}$ as the projective closure of $y^2=x^3+Ax+B$, then $x^3+\overline{A}x+\overline{B}$ can be factorized in $k[x]$ in the form $(x-\alpha)(x-\beta)^2$ for some $\alpha,\beta\in k$.

So are equivalent the following:

1. $E$ has non-split multiplicative reduction.
2. The completed local ring $\widehat{\mathcal{O}}_{\overline{E},x_0}$ is isomorphic as a $k$-algebra to $k[[x,y]]/(q(x,y))$ where $q(x,y)$ is an irreducible homogenous polynomial of degree $2$.
3. $v(\Delta(\mathcal{E}^\text{min}))=0$, $\overline{B}\ne 0$, and $\overline{A}$ is not a square in $k$.
4. If one writes $\mathcal{E}^\text{min}$ as the projective closure of $y^2=x^3+Ax+B$, then $x^3+\overline{A}x+\overline{B}$ is irreducible in $k[x]$.

Proof:  All of these are fairly straightforward and are covered in [Sil, Proposition 5.1 in §VII.5] except the statements concerning the completed local rings, but these can be checked elementarily given the explicit description of the polynomials cutting out $\overline{E}$ in $\mathbb{P}^2_k$ as explained in each part. $\blacksquare$

Using these characterizations, especially those involving the valuation of $\Delta(\mathcal{E}^\text{min})$, it’s not super hard to deduce the following:

Corollary 48 (Semistable reduction theorem): Let $E$ be an elliptic curve over $F$. Then, if $E$ has multiplicative reduction then so does $E_{F'}$ for every finite extension $F'/F$. If $E$ has good reduction, then so does $E_{F'}$ for every finite extension $F'/F$. If $E$ has additive reduction, then there exists a finite extension $F'/F$ such that $E_{F'}$ has good reduction.

Proof: See [Sil, Proposition 5.4 in §VII.5]. $\blacksquare$.

Remark 49: Be careful to not make the following mistake. It seems tempting to think the above theorem is wrong, for examples that additive reduction becomes good reduction over an extension, given Proposition 47. Namely, if one assumes that the minimal Weierstrass model of $E_{F'}$ is $(\mathcal{E}^\text{min})_{\mathcal{O}_{F'}}$. That said, this claim only necessarily holds true if $F'/F$ is unramified (i.e. that $\mathcal{O}_F\to \mathcal{O}_{F'}$ is étale). So, the extension which changes an additive reduction curve to good reduction is necessarily ramified.

Another useful thing that follows from Proposition 47 is the following:

Corollary 50: An elliptic curve $E$ over $F$ has multiplicative reduction if and only if $|j(E)|>1$. Consequently, $E$ has potentially good reduction if and only if $|j(E)|\leqslant 1$.

Proof: For example, see [Sil, Proposition 5.5 in §VII.5]. $\blacksquare$

We can then understand Tate’s theorem as follows:

Theorem (Tate) 51: The map

$\{q\in F^\times : |q|<1\}\to \left\{\begin{matrix}\text{Elliptic curves over }F\\ \text{with split multiplicative}\\ \text{reduction}\end{matrix}\right\}/\text{iso.}$

given by $q\mapsto E_q$ is a bijection.

Proof: See, for example, [Sil1, Theorem 5.3 in §V.5]. $\blacksquare$

This makes intuitive sense since one might imagine that $E_q=\mathbb{G}_{m,k}^\mathrm{an}/q^\mathbb{Z}$ should have $\overline{E_q}$ the result of $\mathbb{G}_{m,k}$ since $q$ projects to $0$ in $k$, and this looks quite a bit like $\overline{E_q}$ should have split multiplicative reduction. Of course, this is clearly missing part of the picture since, for example, we don’t see where the singularity of $\overline{E_q}$ is entering into the picture. We will revisit this discussion when we discuss Raynaud’s perspective of rigid geometry.

##### Arithmetic applications

Now that we have a more refined sense of which elliptic curves are Tate elliptic curves, they are precisely the ones with split multiplicative reduction, we can now more properly state the arithmetic applications of the theory.

###### Tate modules

The first application is to the study of the $p$-adic Tate module of an elliptic curve over $F$. Namely, recall that if $E/F$ is an elliptic curve then its $p$-adic Tate module is the $\mathrm{Gal}(\overline{F}/F)$ representation given by

$T_pE(E):=\varprojlim E[p^n](\overline{F})$

where $E[p^n]$ is the closed subgroup scheme of $E$ given by the $p^n$-torsion and the transition maps are the multiplication-by-$p$ maps (e.g. see [Sai, §1.3] or [Sil, §III.7]). We also have the rational $p$-adic Tate module which is the $\mathrm{Gal}(\overline{F}/F)$-representation given by

$V_p(E):=T_p(E)\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$

Note that as an abstract $\mathbb{Z}_p$-module one has that $T_p(E)$ is isomorphic to $\mathbb{Z}_p^2$ and thus $V_p(E)$ is isomorphic abstractly to $\mathbb{Q}_p^2$ as a $\mathbb{Q}_p$-vector space (e.g. see [Sai, Proposition 1.19] or [Sil, Proposition 7.1 in §III.7]).

Remark 52: More abstract one has identifications of Galois representations

$T_p(E)=\left(H^1_\text{et}(E_{\overline{F}}, \mathbb{Z}_p)\right)^\vee,\qquad V_p(E)=\left(H^1_\text{et}(E_{\overline{F}},\mathbb{Q}_p\right)6\vee$

where these objects are the étale cohomology groups of $E$, but we won’t really use or need this fact. See this post for more details.

Using the theory of Tate elliptic curves we deduce the following:

Proposition 53: Let $E$ be an elliptic curve over $F$ with split multiplicative reduction. Then, for every $n\geqslant 1$ one has an extension of Galois representations

$0\to (\mathbb{Z}/p^n\mathbb{Z})(1)\to E[p^n](\overline{F})\to \mathbb{Z}/p^n\mathbb{Z}\to 0$

which is non-split for $n$ sufficiently large and therefore one has a non-split extensions

$0\to \mathbb{Z}_p(1)\to T_p(E)\to \mathbb{Z}_p\to 0$

and

$0\to \mathbb{Q}_p(1)\to V_p(E)\to \mathbb{Q}_p\to 0$

Here we are using the notation $(\mathbb{Z}/p^n\mathbb{Z})(1)$ to denote the $\mathrm{Gal}(\overline{F}/F)$-representation given by $\mu_{p^n}(\overline{F})$ and by $\mathbb{Z}_p(1)$ the representation $\varprojlim \mu_{p^n}(\overline{F})$ with the transition maps being the $p^\text{th}$-power maps. We are also using $\mathbb{Z}/p^n\mathbb{Z}$ and $\mathbb{Z}_p$ to denote the trivial Galois representation on these groups.

Proof (of Proposition 53): Let us make an identification $E\cong E_q$ for some $|q|<1$ . Let us also fix for each $n\geqslant 1$ a primitive $p^\text{th}$ root of unity $\zeta_{p^n}$ in $\overline{F}$ and a $p^n$-th root $q^{\frac{1}{p^n}}$ of $q$. Let’s assume that we’ve made this choice compatabily so that

$\zeta_{p^{n+1}}^p=\zeta_{p^n},\qquad \left(q^{\frac{1}{p^{n+1}}}\right)^p=q^{\frac{1}{p^n}}$

for all $n\geqslant 1$.

Let us then note that from the isomorphism of $\mathrm{Gal}(\overline{F}/F)$

$E_q(\overline{\mathbb{Q}_p})\cong \overline{\mathbb{Q}_p}^\times/q^\mathbb{Z}$

we easily see that

$E_q[p^n](\overline{F})=\left\{\left(q^{\frac{1}{p^n}}\right)^a \zeta_{p^n}^b:a,b\in\{0,\ldots,p^n-1\}\right\}$

as Galois representations. But, evidently we then have the subrepresentation given by $\mu_{p^n}(\overline{F})$ and generated by $\zeta_{p^n}$. Note moreover that for each $\sigma$ in $\mathrm{Gal}(\overline{F}/F)$ one has that we can write $\sigma\left(q^{\frac{1}{p^n}}\right)=q^{\frac{1}{p^n}}\zeta_{p^n}^b$ for a unique $b\in\{0,\ldots,p^n-1\}$. Thus, we see that the quotient $E[p^n]/\mu_{p^n}(\overline{F})$ is isomorphic to $\mathbb{Z}/p^n\mathbb{Z}$ with the trivial action as desired.

The fact that this sequence doesn’t split for $n$ sufficiently large is clear. Indeed, such a section for the projection map must send $1$ to $q^{\frac{1}{p^n}}\zeta_n^b$ for some $b\geqslant 0$ and for this section to be Galois equivariant we need that this element is Galois fixed. This implies that $q^{\frac{1}{p^n}}\zeta_{p^n}^b$ is in $F$, and evidently if $n$ is sufficiently large this is impossible.

The $p$-adic statement follows by passing to the limit. Checking non-splitting at the rational level requires a small argument, but we leave this to the reader. $\blacksquare$

In fact, we can elaborate on this result even further. Namely, note that one has

$\mathrm{Ext}^1(\mathbb{Z}/p^n\mathbb{Z}_p,(\mathbb{Z}/p^n\mathbb{Z})(1))=H^1_\text{cont.}(\mathrm{Gal}(\overline{F}/F),\mu_{p^n}(\overline{F}))= F^\times/(F^\times)^{p^n}$

where the first object is the set of isomorphism classes of extensions in the category of Galois $\mathbb{Z}/p^n\mathbb{Z}$-representations, and the second equality follows from Kummer theory (e.g. see this post). We can then understand the above as saying that the association of an elliptic curve with split multiplicative reduction to the extension class of $E_q[p^n](\overline{F})$ in $\text{Ext}^1(\mathbb{Z}/p^n\mathbb{Z},(\mathbb{Z}/p^n\mathbb{Z})(1))$ is the map sending $E_q$ to the class of $q$ in $F^\times/(F^\times)^{p^n}$. The non-splitness is then clear since $q$ is not in $(F^\times)^{p^n}$ for sufficiently large $n$. The non-splitting of the rational Tate module is then clear since the natural map

$\{q\in F^\times : |q|<1\} \to \left(\varprojlim F^\times/(F^\times)^{p^n}\right)\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$

is injective.

Of course, one might be interested in understanding what happens in the case of non-split multiplicative reduction. But, this is quite simple as the following shows:

Proposition 54: Let $E_1$ and $E_2$ over any field $F$ and suppose that $E_1$ becomes isomorphic to $E_2$ upon base extension to the quadratic extension $L/F$. Then, there is an isomorphism of Galois representations

$T_p(E_2)\cong T_p(E_1)(\chi)$

where $\chi$ is the quadratic character associated to $L/F$.

Proof: For example this follows from [Sil1, Lemma 5.2(c) in §V.5]. $\blacksquare$

Here the quadratic character associated to $L/F$ is the unique non-trivial homomorphism $\mathrm{Gal}(\overline{F}/F)\to \{\pm 1\}$ factoring through $\mathrm{Gal}(L/F)$. By $T_p(E_1)(\chi)$ we mean the twist of $T_p(E_1)$ by $\chi$ which means the same underlying module of $T_p(E_1)$ but with the twisted Galois action. Namely, for $x\in T_p(E_1)$ and $\sigma\in\mathrm{Gal}(\overline{F}/F)$ if $\sigma(x)$ denotes the action of $\sigma$ on $x$ in $T_p(E_1)$, then the action of $\sigma$ on $x$ in $T_p(E_1)(\chi)$ is $\chi(\sigma)\sigma(x)$.

Note that one interesting consequence of the above discussion is that $p$-adic Tate modules (and thus $p$-adic étale cohomology groups by Remark 52) needn’t be semisimple Galois representations.

###### Bounding torsion

One of the first major theorems that one proves about the arithmetic of elliptic curves is the Mordell–Weil theorem which states that if $E$ is an elliptic curve over a number field $K$, then $E(K)$ is finitely generate. This result requires quite a bit of sophistication in the form of the weak Mordell–Weil theorem (e.g. see this post) and techniques using height functions (see [Sil, §VIII.4-§VIII.6] for details).

One consequence of this is that the group $E(K)_\text{tors}$ of torsion points of $E(K)$ is finite. That said, one can get a significantly easier proof of this, at least assuming the Tate theory, of elliptic curves. The point of this is not necessarily to claim that this is a ‘more elementary’ or ‘simpler’ proof than the usual proof of the Mordell–Weil theorem (or, of course, this can be deduced more simply by using Nagell–Lutz and some theory about reductionfor example see [Mil2, §II.4]) but to illustrate that we can get a purely arithmetic-geometric result by analytic methods.

Regardless, we see the following result quite easily:

Proposition 55: Let $E$ be an elliptic curve over a number field $K$. Suppose that $K$ has a prime $\mathfrak{p}$ such that $E_F$ has multiplicative reduction. Then, $E(K)_\text{tors}$ is finite.

Proof: Set $L$ to either be $F$, if $E$ has split multiplicative reduction over $F$, or the quadratic extension of $F$ that splits $E$ if $E$ has non-split multiplicative reduction. Note then that we have an injection

$E(K)_\text{tors}\hookrightarrow E(L)_\text{tors}$

and we know from Tate’s theorem that

$E(L)_\text{tors}=(L^\times/q^\mathbb{Z})_\text{tors}$

for some $q\in L^\times$ with $|q|<1$. That said, $L^\times$ is clasically isomorphic as a group to

$\mathbb{Z}\times \mu(L)\times M$

where $\mu(L)$ is the set of roots of unity in $L$ (which is finite) and $M$ is a finite free $\mathbb{Z}_p$-module (e.g. see [Lor, §25.4]). But, this group has the property that all of its quotients have finite torsion from where the claim follows. $\blacksquare$

Of course, by Corollary 50 we know that $E/K$ has

###### Endomorphism rings

As a last example, we borrow the following result from [Sil1, §V.6] for which a $p$-adic proof using the theory of Tate elliptic curves is due to Serre.

Namely, let us recall that if $K$ is a characteristic $0$ field and $E$ is an elliptic curve over $K$ then we denote by $\text{End}(E)$ the ring of endomorphisms of $K$-group scheme $E$. As it turns out, there are not many options for the ring $\text{End}(E)$. Namely, this ring must either be $\mathbb{Z}$ or an order in an imaginary quadratic extension of $\mathbb{Q}$ (e.g. see [Sil, Corollary 9.4 in §III.9]). We say that $E$ has geometric complex multiplication if $\text{End}(E_{\overline{K}})$ is an order in a quadratic imaginary extension of $\mathbb{Q}$

Despite first appearances, curves with geometric complex multiplication play a profound role not only in the theory of elliptic curves. Namely, they are certain ‘special’ (or ‘exceptional’) types of curves that are often exceptions for many of the deep theorems in the arithmetic of elliptic curves (cf. [Sil, Theorem 4.9 in §V.4] or [Sil, Corollary 6.3 in §IV.6]) but also serve as the conceptual basis for the theory of canonical models of Shimura varieties (e.g. see [Mil3, §10]).

For this reason, any statement which helps understand elliptic curves with geometric complex multiplication are incredibly significant from the point of view of arithmetic. It is then interesting that Serre was able to prove the following theorem using the theory of $p$-adic Tate curves:

Proposition 56: Let $K$ be a number field and $E$ an elliptic curve over $K$. If $E$ has geometric complex multiplication then $j(E)\in\mathcal{O}_K$.

Proof: For a proof using the theory of Tate elliptic curves see [Sil1,  §V.6]. $\blacksquare$

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