# Non-Archimedean analytic continuation, Weirstrass elliptic functions and definability

*Posted on August 15, 2016 by Dima*

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 that 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*. The importance of rational domains lies in the following theorem:

**Theorem** (Gerritzen-Grauert). Every affinoid domain is a finite union of rational domains.

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 \(L^D_{an}\).

(comments)# Incidence structures on algebraic curves

*Posted on June 7, 2016 by Dima*

## Abstract projective geometry

Let \(k\) be a field. Consider the set of points \(P\) and lines \(L\) on the projective \(n\)-space \({\operatorname{\mathbb{P}}}^n(k)\) over \(k\). One finds that the following properties hold:

- there is exactly one line incident to any two distinct points
- there is exactly one point that is incident to two distinct lines
- every line contains at least three distinct points

A tuple \((P,L,I)\) where \(P\) the set of points, \(L\) is the set of lines and \(I \subset P \times L\) is the incidence relation, is an *abstract projective geometry* if properties above hold. These properties in particular hold if \(P\), \(L\) are sets of points and lines on a projective plane over a (skew) field \(k\). There are examples of abstract projective geometries that do not arise this way, the reason is that all such geometries additionally satisfy *Desargues axiom*:

Given two triangles \(ABC\) and \(A'B'C'\) as above, the lines \(AB\) and \(A'B'\), \(BC\) and \(B'C'\), and \(AC\) and \(A'C'\) intersect in three points that lie on the same line.

There are two affine statements that follow from this axioms: called *small* and *big Desargues axioms*.

**small Desargues**: given \(A\) and \(A'\), \(B\) and \(B'\), \(C\) and \(C'\) lying on three parallel lines, \(BC\) and \(B'C'\) are parallel.

**big Desargues**: given that the lines \(AB\) and \(A'B'\), and \(AC\) and \(A'C'\) are pairwise *parallel*, the lines \(BC\) and \(B'C'\) are parallel.

If \(k\) is commutative, then a *Pappus axiom* is also satisfied:

The axiom states that in the configuration as above the three points \(P_7, P_8\) and \(P_9\) lie on the same line.

**Theorem**. If \(\mathfrak{G} = (P, L, I)\) is an abstract projective geometry which satisfies Desargues axiom then there exists a skew field \(k\) such that \(P = {\operatorname{\mathbb{P}}}^n(k)\), \(n > 1\). If \(\mathfrak{G}\) also satisfies the Pappus axiom, then \(k\) is commutative, i.e. is a field.

The proof of this theorem is nicely exposed in Hartshorne’s notes called “Foundations of projective geometry” and in the classical Emile Artin’s book “Geometric algebra”.

I will now sketch the main steps of the proof.

Let \(A = (P,L, I)\) be an affine geometry. A *dilation* is an automorphism of \(A\) such that whenever it sends a point \(P\) to a point \(P'\) and a point \(Q\) to a point \(Q'\), the lines \(PQ\) and \(P'Q'\) are parallel.

On an affine plane over a field \(k\) the set of affine transformations form the group \({\operatorname{\mathbb{G}}}_m(k) \rtimes {\operatorname{\mathbb{G}}}_a(k)\).

Theorem: a dilation is determined by its action on two points.

A *translation* is a dilation with no fixed points. Translations form a normal subgroup of the group of dilations of an affine plane. For an affine plane over a field this subgroup is \({\operatorname{\mathbb{G}}}_a(k)\).

If small Desargues axiom holds then for any two points \(P, P'\) there exists a *unique* translation that maps \(P\) to \(P'\). It follows that translations form an *Abelian* group.

Assuming big Desargues axiom, and given three points \(O, P, P'\) such that \(P, P' \neq O\), there exists a unique dilation that fixes \(O\) and maps \(P\) to \(P'\).

The additive group of the field is reconstructed as the group of translations, with the group of dilations (isomorphic to the multiplicative group) acting on it. More concretely, ix a line \(l\), and fix two points, call them \(0\) and \(1\). The set of points of \(l\) will be our field. In order to add two points \(x\) and \(y\), take two translations \(\tau_x, \tau_y\) that send 0 to \(x\) and \(y\) respectively, and put \(x+y\) to be the result of the application of \(\tau_x \circ \tau_y\) to \(0\). To multiply, let \(\mu_x, \mu_y\) be dilations that fix \(0\) and send \(1\) to \(x\) and \(y\). Then \(x \cdot y\) is defined as \(\mu_x \circ \mu_y (1)\).

Desargues axiom is used to show the distributivity, and Pappus axiom is used in order to show commutativity of the group of dilations.

Recently, the Main theorem of Abstract Projective Geometry has been used by Bogomolov and Tschinkel in one of the key steps of the proof of the function field reconstruction theorem. The birational anabelian program seeks to reconstruct function fields of varieties of dimension \(> 2\) over the algebraic closure of \(\mathbb{F}_p\) from their absolute Galois groups. If \(K\) is the function field of such variety then one notices that \(K^\times / k^\times\) is an infinite-dimensional projective space with an Abelian group structure. The absolute Galois group of \(K\) induces enough structure on the projective space \(K^\times/k^\times\) so that an isomorphism on absolute Galois groups induces a map that preserves the incidence relation. The main theorem of Abstract Projective Gemotry then allows to conclude that the map is induced by a field isomorphism (up to inseparable extensions).

## Zariski geometries

What if we want to look at the incidence systems of curves of higher degree? A generalisation of main theorem of Abstract Projective geometry is possible if we agree to work over an algebraically closed field.

One uses the language of mathematical logic. Let \(M\) be a first-order structure such that a Noetherian (proper descending chains of closed irreducible sets must be finite) topology is defined on every Cartesian power \(M^n\). Assume that every closed set is named by a predicate, so that every constructible set is definable. Further assume that every definable set is constructible.

An example of such structure is the so-called full Zariski structure on an algebraic variety \(V\) over an algebraically closed field \(k\): name every Zariski closed subset of \(V^k\), \(k \geq 1\) by a predicate.

Question: given a full Zariski structure on an algebraic variety \(V\), is it possible to recover the field \(k\)?

Here is more precise definition of “recover” which is a standard notion in model theory. A first-order structure \(M\) *interprets* a field, if there exists a definable subset \(F \subset M^n\) and a definable equivalence relation \(R \subset F \times F\) such that \(R/F\) can be endowed with definable addition and multiplication maps which make it into a field.

**Theorem** (Hrushovski, Zilber). Let \(M\) be an irreducible algebraic curve over an algebraically closed field \(k\), considered as a full Zariski structure. Then \(M\) interprets a field, definably isomorphic to \(k\).

This is actually a corollary of a more general theorem of Hrushovski and Zilber, which describes axiomatically a class of structures, similar to full Zariski structures which interpret a field.

(comments)*dd*^{c} lemma

^{c}

*Posted on May 4, 2016 by Dima*

In this note I will prove the easy local case of \(dd^c\) lemma.

Let \((M,I)\) be a complex manifold. Extend the action of the complex structure on exterior powers of the complexified tangent bundle forms by \[ \mathbf{I}: \bigwedge{}^* M \to \bigwedge{}^* M, \mathbf{I} := \sum i^{p-q} \cdot \Pi^{p,q} \] where \(\Pi^{p,q}: \bigwedge^{p+q} \to \bigwedge^{p,q}\) is the natural projection. Define a dwisted differential \(d^c = I \circ d \circ I^{-1}\). On the cotangent bundle define it to be \(\mathbf{I}(\alpha)(v_1, \ldots, v_n) := \alpha(\mathbf{I}v_1, \ldots, \mathbf{I}v_n)\).

**Proposition**. \(\partial = \dfrac{d + i \cdot d^c}{2}, \bar \partial = \dfrac{d - i \cdot d^c}{2}\), where \(\partial\) and \(\bar\partial\) are projections of \(d\) on \((\cdot +1, \cdot)\) and \((\cdot, \cdot+1)\) components.

*Proof.* Indeed \[
\partial + \bar\partial + i \mathbf{I}^{-1} (\partial +
\bar\partial) \mathbf{I} = \partial + \bar\partial + i (i^{q-p-1}
\partial + i^{q-p+1}\bar\partial)i^{q-p}=\\
= \partial + \bar\partial +
\partial + i^2\bar\partial=2\partial
\] Similarly for the second equality. \(\square\)

In particular, \(\partial\bar\partial = -\dfrac{i}{2}dd^c\) on a complex manifold.

**Lemma** (Poincaré lemma). If \(\alpha\) is a closed form on a polydisc then it is exact.

**Lemma** (Poincaré-Dolbeault-Grothendieck lemma). If \(\alpha\) is a \(\partial\)-closed, not holomorphic (i.e. \(\alpha \notin A^{n,0} M\)), form on a polydisc then it is \(\bar\partial\)-exact.

(this lemma means in particular that Dolbault resoltions of sheaves of holomorphic forms are acyclic)

Note that since \(\bar(\partial\alpha) = \bar\partial \bar\alpha\), the PDG lemma also holds for \(\partial\).

**Lemma** (local \(dd^c\) lemma). Let \(\eta\) be a \((1,1)\)-form on a polydisc. Assume either

- \(\eta\) is exact, or
- \(\eta\) is \(\partial\)-exact, \(\bar\partial\)-closed.

# The Picard scheme

*Posted on April 24, 2016 by Dima*

## Functors \(Pic\) and \(Div\)

We will denote base change with a subscript: \(X_T = X \times T\).

If \(X\) is a scheme, the *Picard group* of \(X\) is defined to be the group of isomorphism classes of invertible sheaves on \(X\). The relative Picard functor of an \(S\)-scheme \(X\) is defined as \[
{\operatorname{Pic}}_{X/S}(T) := {\operatorname{Pic}}(X_T) / {\operatorname{Pic}}(T)
\] where the embedding \({\operatorname{Pic}}(T) \hookrightarrow {\operatorname{Pic}}(X_T)\) is given by the pullback along the structure maps \(X_T \to T\).

Caveat: all representability results work with a sheafification of this functor in some topology, Zariski, étale, or fppf. Unless \(X \to S\) is proper and has a section, they need not be isomorphic.

An *effective divisor* is a closed subscheme of codimension 1. If \(f: X \to S\) is a morphism of schemes then a relative effective divisor is an effective divisor \(D\) such that \(D\) is flat over \(S\). For a morphism \(X \to S\) define the functor of relative divisors \[
{\operatorname{Div}}_{X/S}(T) := {\{\ \textrm{ relative effective divisors on } X_T/T
\ \}}
\]

We are interested in representability of this functor, so to this end we prove a little lemma.

**Lemma**. Let \(X \to S\) be a flat morphism. Let \(D\) be a closed subscheme of \(X\). Then \(D\) is codimension 1 in a neighbourhood of \(x \in X\) if and only if \(D_s\) is codimension 1 in the fibre \(X_s\) where \(s\) is the image of \(x\).