tags: non-Archimedean geometry

Let \(K\) be an algebraically closed complete non-Archimedean valued field. Let \(q \in \mathbb{G}_m(K)\) be an element of strictly positive valuation. Then the quotient \(\mathbb{G}_m(K) / q^{\mathbb Z}\) is a rigid analytic manifold which can be embedded into \(\mathbb{P}^2(K)\) by means of analytic functions \(x, y: \mathbb{G}_m(K) \to \mathbb{A}^2(K)\) that are periodic under multiplication by \(q\) (one takes the image in \(\mathbb{A}^2\) and takes the closure in \(\mathbb{P}^2\)). Here is the series that give \(x\) (taken from Tate’s article ``A review of non-Archimedean elliptic functions’’): \[ x(w) = \sum_{q = -\infty}^{\infty} \dfrac{q^m w}{(1 - q^m w)^2} - 2 \sum_{m=1}^\infty \dfrac{q^m}{(1 - q^m)^2} \]

Recall that algebras of the form \[ K\{x_1, \ldots, x_n\} := \{\ \sum a_\delta x^\delta \ \mid\ v(a_\delta) \to \infty \textrm{ as } |\delta| \to \infty\ \} \] where \(\delta\) is a multiindex, are called Tate algebras. They are normed algebras with the norm \[ || \sum a_\delta x^\delta || = \sum e^{-v(a_\delta)} \]

The power series that constitute these algebras converge on unit polydiscs \[ \{\ x_i \in K \ \mid\ ||x_i|| \leq 1 \ \} = \{\ x_i \in K \ \mid\ v(x_i) \geq 0 \ \} \] Variants include (isomorphic) algebras \[ K\{e^{-r_1}x_1, \ldots, e^{-r_n}x_n\} := \{\ \sum a_\delta x^\delta \ \mid\ v(a_\delta) \to \infty \textrm{ as } |\delta| \to \infty\ \} \] of functions analytic on polydiscs of varying radii \[ \{\ x_i \in K \ \mid\ v(x_i) \geq r_i\ \} \] They are however not isometric, since they are normed in a different way \[ || \sum a_\delta x^\delta || = \sum e^{-v(a_\delta)} e^{-r_{\delta_i}} \]

An annulus with inner valuative radius \(a\) and outer valuatie radius \(0\) \[ \{\ x \in K \ \mid\ 0 \leq v(x) \leq a\ \} \] is an affinoid domain with the corresponding algebra of analytic functions \(K\{x, y\}/(xy = \gamma)\); here, \(\gamma \in K\) is such that \(v(\gamma) = a\).

This is an example of a rational domain in \(\mathbb{A}^1\). The importance of rational domains lies in the following theorem:

Theorem (Gerritzen-Grauert). Let \(U\) be an affinoid domain. Every affinoid subdomain of \(U\) is a finite union of rational subdomains of \(U\).

An affinoid algebra is a quotient of a Tate algebra by a closed ideal. Global analytic functions on affinoid domains are called affinoid functions. Recall that a function is called meromorphic on a domain if it can be locally represented as a quotient of analytic functions. We will see that on affinoid domains meromorphic functions are quotients of affinoid functions globally.

Intersections of affinoid domains are affinoid. Affinoids play the role of opens in rigid geometry. In fact, one can define the ``rigid \(G\)-topology’’: it is a Grothendieck topology where admissible covers are finite covers of \(\mathrm{Specm} A\) by affinoid domains.

Both elliptic functions introduced above, \(x\) and \(y\), are meromorphic on \(\mathbb{G}_m(K)\). Indeed, restricting to annuli \[ \{\ x \in K \ \mid\ n \leq v(x) \leq m\ \} \] for \(n,m \in \mathbb{Z}\), functions \(x\) and \(y\) have finitely many poles, therefore, after multiplication by a suitable polynomial, they become analytic.

Note that \(x\) and \(y\) cannot be extended to a function meromorphic on \(\mathbb{A}^1\) (or unit disc), since their poles converge to 0, and meromorphic functions can only have isolated poles (the proof is similar to the Archimedean case).

Quite expectedly though, they can be uniquely extended to the punctured unit disc. It took me some effort to find out how this can be proved: analytic continuation works quite differently in non-Archimedean situation than in the complex-analytic setting.

Theorem (Tate acyclicity theorem) Let \(\{ A_i \}\) be finitely many affinoid domains such that \(\cup A_i = A\). Then the following sequence (with Čech differentials: taking restrictions with alternating signs) is exact \[ 0 \to \mathcal{O}_A(A) \to \mathcal{O}_A ( \coprod A_i) \to \mathcal{O}_A ( \coprod A_i \cap A_j) \] where \(\mathcal{O}_A\) is the sheaf of affinoid functions. In particular, affinoid domains are acyclic and cohomology of coherent sheaves in \(G\)-topology can be computed using Čech complex of an admissible covering.

Theorem (Kiehl’s Theorem A) Any coherent sheaf \(\mathcal F\) over an affinoid domain is generated by global sections.

Proof. Let \(a \in A\) and let \(\mathcal{I}_{a}\) be the skyscraper sheaf. Consider the exact sequence \[ 0 \to \mathcal{I}_a \mathcal F \to \mathcal F \to \mathcal F / \mathcal{I}_a \mathcal F \to 0 \] Then the associated cohomology long exact sequence is \[ \ldots \to H^0(A,\mathcal F) \to H^0(A,\mathcal F / \mathcal{I}_a \mathcal F) \to H^1(A,\mathcal{I}_a \mathcal F) \to \ldots \] where the last term vanishes by Tate’s acyclicity. Therefore, there exist (finitely many, by coherence) non-vanishing at \(a\) global sections of \(\mathcal F\). The second term is a finitely generated \(\mathcal{O}_{A,a}\)-module, and by Nakayama’s lemma, these sections generate the germs of \(\mathcal F\) at \(a\).

Proposition (Poincaré’s problem) Meromorphic function on an affinoid domain is a quotient of affinoid functions.

Proof. This is a consequence of Tate acyclicity theorem. Let \(h \in H^0(A, \mathcal{K}_A)\) be a meromorphic function. Consider the map of sheaves \(\varphi: \mathcal{O}_A \to h \cdot \mathcal{O}_A \hookrightarrow \mathcal{K}_A\) and let \(\mathcal{I}\) be the pre-image sheaf \(\varphi^{-1}(\mathcal{O}_A \cap h \cdot \mathcal{O}_A\). By Kiehl’s theorem A there exists a non-zero global section \(g \in H^0(A,\varphi^{-1}(\mathcal{O}_A \cap h \cdot \mathcal{O}_A))\). Then \(g\cdot h\) is regular by the definition of the sheaf in question.

Proposition Denote \(\bar x\) the restriction of the function \(x\) to the domain \[ A_1 := \{\ x \ \mid\ 0 \leq v(x) \leq 1 \ \} \] Then \(x\) is the unique meromorphic function on \[ A_n := \{\ x \ \mid\ 0 \leq v(x) \leq n \ \} \] (for any \(n \in \Gamma\)) that restricts to \(\bar x\) on \(A_1\).

Proof. Let \(f_1, f_2\) be two meromorphic functions on \(A_2\) that coincide on \(A_1\), they are then quotients of pairs of affinoid on \(A_n\) functions, \(f_1 = \dfrac{g_1}{h_1}, f_2 = \dfrac{g_2}{h_2}\). Then \(g_1 h_2 = g_2 h_2\) on \(A_1\), but then by Tate acyclicity theorem \(g_1 h_2 = g_2 h_1\) on \(A_2\), and therefore \(\dfrac{g_1}{h_1} = \dfrac{g_2}{h_2}\).

In their 1987 paper van den Dries and Denef have introduced subanalytic domains and functions over the field of p-adic numbers, and have proved a quantifier elimination result for \(\mathbb{Q}_p\) equipped with restricted subanalytic functions. The language of the theory they define consists of the Denef-Pas language, extended with graphs of \(D\)-functions, defined inductively as follows.

Let \(D\) be the function \[ \begin{array}{l} D: \mathbb{Z}_p^2 \to \mathbb{Z}_p, \\ D(x,y) = \left\{ \begin{array}{l} x/y, \qquad v(x) \geq v(y) \\ 0 \end{array} \right. \end{array} \]

• \(D\) is a \(D\)-function
• affinoid functions are \(D\)-functions
• the composition \(f(g_1(\ldots), \ldots, g_n(\ldots))\) of a \(D\)-function \(f(x_1, \ldots, x_n)\) with \(D\)-functions \(g_1, \ldots, g_n\) is a \(D\)-function.

The language of valued rings expanded with symbols of \(D\)-functions, naturally interpredet, is called \(L^D_{an}\), and Denef and van den Dries have shown that in this language, definable subsets of \(\mathbb{Z}_p^n\) are quantifier free definable.

Here is an illustration of expressiveness of this expansion: restrictions of Weirstrass ellptic functions to their fundamental domain are definable.

Firstly, observe that any affinoid domain is definable. Indeed, by Gerritzen-Grauert theorem an affinoid domain is a union of rational domains, and these can be represented as projections of a vanishing set of a family of affinoid functions on a polydisc. Secondly, it immediately follows that affinoid functions are also definable (restrict functions on the polydisc to the closed subset).

Now Weirstrass elliptic functions are meromorphic on the annulus \(0 \leq v(x) \leq v(q)\), so they are quotients of an analytic — hence affinoid — functions by polynomials, therefore, they are definable in \(L^D_{an}\).

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