# Zilber’s notion of logically perfect structure: Universal Covers

John T. Baldwin  Department of Mathematics, Statistics and Computer Science,
The University of Illinois at Chicago  Andrés Villaveces  Departamento de
Matemáticas, Universidad Nacional de Colombia, Bogotá

###### Abstract

We sketch the mathematical back ground and the main ideas in the proofs of
categoricity of theories of several examples of universal covers – reducing an
analytic to a model theoretic (discrete) description. We hope this discussion
will be useful to a wide spectrum of mathematicians ranging from those working
in geometry to those working in logic; specifically, model theory.

###### Contents

  1. 1 Introduction
    1. 1.1 Mathematical Encounters
      1. 1.1.1 Some ancient history: In and out of the Zilber world
      2. 1.1.2 An unlikely encounter of two areas: MAMLS at Rutgers, 2001
    2. 1.2 A word of thanks from the second author
  2. 2 Model theory in Mathematics
    1. 2.1 Model theoretic background
    2. 2.2 Various Viewpoints
  3. 3 Categoricity, quasiminimality and excellence
    1. 3.1 The Classical Categoricity Theorems
    2. 3.2 Pregeometries (matroids) and quasiminimality
  4. 4 Modular and Shimura Curves
    1. 4.1 The great confluence
    2. 4.2 Moduli Spaces
    3. 4.3 Quantifier Elimination in Modular and Shimura Curves
    4. 4.4 Galois Representations and finite index conditions
      1. 4.4.1 Two views: domain and field sort
      2. 4.4.2 Galois Representation
  5. 5 First order Excellence
    1. 5.1 The two-sorted structure and fmr groups
    2. 5.2 First order Excellence and fmr groups
    3. 5.3 Abelian Varieties
  6. 6 Higher Dimensional Shimura Varieties
  7. 7 Model Theory and Analysis
  8. 8 Families of covers of algebraic curves
    1. 8.1 Pseudo-analytic covers of modular curves
    2. 8.2 Locally o-minimal covers of algebraic varieties

##  1 Introduction

The goal of this paper is to sketch (hopefully for a wide spectrum of
mathematicians ranging from those working in geometry to those working in
logic; specifically, model theory) some recent interactions between model
theory and a roughly 150-year old study of analytic functions involving
complex analysis, algebraic topology, and number theory that explore the
canonicity of universal covers. Towards this goal we discuss and present
several examples indicating the main ideas of the proofs and the necessary
changes in method for different situations.

Here is Zilber’s description of his own project (from his 2000 Logic
Colloquium talk in Paris [Zil05a]):

> _The initial hope of this author in[Zil84] that any uncountably categorical
> structure comes from a classical context (the trichotomy conjecture), was
> based on the belief that logically perfect structures could not be
> overlooked in the natural progression of mathematics. Allowing some
> philosophical license here, this was also a belief in a strong logical
> predetermination of basic mathematical structures. As a matter of fact, it
> turned out to be true in many cases. …Another situation where this principle
> works is the context of o-minimal structures [PS98]. _

A rather ambitious project aimed at finding _categorical_ axiomatizations
(Definition 3.0.1) of various kinds of _universal covers_ has been unfolding
in the 21st century. The simplest example of such universal covers is given by
the short exact sequence:

$\displaystyle 0\to\rm ker(\exp)\to(\mbox{\msbm C},+,0)\
{\stackrel{{\scriptstyle exp}}{{\to}}}(\mbox{\msbm C},+,\cdot,0,1)\to 1.$ (1)

Zilber’s original project really aimed to understand the sequence

$\displaystyle 0\to\rm ker(\exp)\rightarrowtail(\mbox{\msbm C},+,\cdot,\exp)\
{\stackrel{{\scriptstyle exp}}{{\twoheadrightarrow}}}(\mbox{\msbm
C},+,\cdot,\exp)\to 1.$ (2)

The first diagram describes a two-sorted _cover_ of the multiplicative group
by the additive group. The full field structure is studied on the range space
although the kernel is of the homomorphism from $(\mbox{\msbm C},+,0)$ to
$(\mbox{\msbm C},\cdot,1)$.

The second [Zil04] corresponds to the _theory_ of the complex exponential
field. The domain and range of the map are the same exponential field but the
kernel is again computed with respect to the homomorphism $\exp$ from
$(\mbox{\msbm C},+)\rightarrow(\mbox{\msbm C}^{*},\times)$.

In both cases, first order axioms are supplemented by an
$L_{\omega_{1},\omega}$-sentence asserting the kernel is isomorphic to ℤ,
i.e., is standard. Here, we focus on three main _families_ of generalizations
(described in the chart below) of the first diagram. As this question was
extended to more general algebraic contexts, the fundamental cover diagram
from equation (1) changed to this more general situation:

$\displaystyle C\ {\stackrel{{\scriptstyle
p}}{{\twoheadrightarrow}}}S(\mbox{\msbm C}).$ (3)

Notice two things:

  * •

The map $p$ remains a projection, but it will significantly change as the
family of examples unfolds. Also,

  * •

there is no longer a kernel when $S(\mbox{\msbm C})$ is not a group.

Therefore, in a rather Protean way, the infinitary description that in the
particular case described a ‘standard kernel’ assumes various guises for
different examples. Usually, the descriptions are of ‘standard fibres’ rather
than having a ‘standard kernel’.

Crucially, in all cases except part of § 5 the target will be some kind of
definable set in an algebraically closed field. The necessary vocabulary for
the domain will vary among the situations considered. Shimura varieties
require a more general domain:

######  Notation 1.0.1.

(The general situation)

$\displaystyle X^{+}\ {\stackrel{{\scriptstyle p}}{{\to}}}S(\mbox{\msbm C})\to
1.$ (4)

Here, $S(\mbox{\msbm C})$ is a variety arising as the quotient of the action
of a discrete group on ℍ (hyperbolic space) or more generally (Shimura
varieties) on a hermitian symmetric domain $X^{+}$. The target is described by
a first order theory $T:=\mathop{\rm Th}(S(\mbox{\msbm C}))$ in a large enough
(field) countable vocabulary with quantifier elimination (possible, as $S$ is
definable in $(\mbox{\msbm C},+,\times)$). Notation 1.0.1 thus instantiates
the general schema, with appropriate notations for specific cases to be given
as we discuss them. Zilber describes the value of his project in terms of ‘a
complete formal invariant’ (Remark 5.3.2).

> _The geometric value of the project is perhaps in the fact that the
> formulation of the categorical theory of the universal cover of a variety
> $X$ …is essentially a formulation of a complete formal invariant of $X$. _
>
> [DZ22b, 1]

The following chart organizes the papers which are the major source for this
study. It also provides a keyword describing the main method or context used,
and the section of this paper where issues around the specific variant are
explained.

| topic | paper | method/context | section
---|---|---|---|---
1 | Complex exponentiation | [Zil05b] | quasiminimality |  §1
2 | cov mult group | [Zil06] | quasiminimality |  §1
3 |  | [BZ11] | quasiminimality |
4 |  $j$-function | [Har14] | background |  §4.1
5 | Modular/Shimura Curves | [DH17] | quasiminimality |  §4
6 | Modular/Shimura Curves | [DZ22b] | quasiminimality |
7 | finite Morley rank groups | [BGH14] | fmr & notop |  §5.1
8 | Abelian Varieties | [BHP20] | fmr & notop /quasiminimality |  §5.3
9 |  Shimura varieties | [Ete22] | notop |  §6
10 | Smooth varieties | [Zil22] | o-quasiminimality |  §8

In this chart, the first line [Zil04] (an axiomatization of the exponential
map from the complex field to itself) differs from the others in the role of
the quantifier ‘there exists uncountably many’. In that case it is essential
to directly control the cardinality of the algebraic closure of a countable
set. Moreover, in line 1 the domain has a field structure that disappears in
the two-sorted approach of the rest. In the other lines of the chart, the
infinitary logic $L_{\omega_{1},\omega}$ is used to control the size of fibers
of the cover or when the structure is a group the size of the kernel. This
requirement suffices to also control the cardinality of the algebraic closure.
Lines 2-6 deal with curves ($1$-dimensional objects) where categoricity is
obtained by quasiminimality. The third big horizontal block deals with higher
dimensional varieties. Lines 7 and 9 stray from formal categoricity towards
more traditional descriptions of models; quasiminimality is replaced by a
different version of excellence arising in Shelah’s study of notop theories
(an important notion in Classification Theory). Both quasiminimality and
‘notop’ apply to line 8. The last line considers families of covers of
arbitrary smooth algebraic varieties with an infinitary logic construction
defined over o-minimal expansions of the reals. There, the focus is on
categoricity in $\aleph_{1}$.

It is worth noting that we could have organized our chart under a totally
different scheme. The Abelian varieties and $(\mbox{\msbm C},+)$ are specific
varieties. The $j$-function and the Shimura varieties may be regarded as
moduli spaces for (generalized) families of varieties111Various types of
Shimura varieties include Siegel, PEL-type, and Hodge-type; only some
parameterize algebraic avarieties.. After preliminary discussions on the model
theoretic framework, in Section 4 we sketch in some detail categoricity of
universal covers of modular curves. In the later sections we describe the
modifications to this program necessary for higher dimensions.

###  1.1 Mathematical Encounters

####  1.1.1 Some ancient history: In and out of the Zilber world

The first author turns to the first person singular for some memories:

Zilber and I both received our Ph.D.’s in the early 1970’s. An important
result appeared in both theses: the solution to Morley’s conjecture that an
$\aleph_{1}$-categorical theory has finite Morley rank. Such an overlap was
not an issue during the Cold War. (On the other hand, Baldwin’s advisor,
Lachlan, had to write an entirely new thesis when the result of the proposed
one appeared in the west as he was about to submit.)

I first (given my zero knowledge of Russian) learned in any detail of Zilber’s
work during the 1980-81 model theory year in Jerusalem. Greg Cherlin had no
such deficiency and gave with Harrington and Lachlan an alternate proof of
Zilber’s theorem that there were no finitely axiomatizable totally categorical
theories. They relied on the classification of finite simple groups. A few
years later Boris completed his model theoretic proof of the key combinatorial
lemma avoiding that reliance.

I first knew Boris in any depth during the model theory semester in Chicago
91-92. Unfortunately, I had partially financed a semester by agreeing to be
acting head the Fall semester, thereby restricting my mathematical activity.
In that busy fall, Boris and Angus Macintyre lectured on Tuesday’s on Zariski
geometries and o-minimality, respectively. The lively group include Macintyre,
Zilber, Laskowski, Marker, Otero, D’Aquino and myself, with Pillay driving in
weekly from Notre Dame. Lunch was at a deli that Boris insisted on because of
the soup followed by coffee at Jamoch’s, the first modern coffee house in the
UIC area.

About that time, I began work on the Hrushovski construction, but in a quite
different direction from Boris: predimension with irrational $\alpha$. This
led to my work with Shelah giving the first full proof of the $0-1$ law with
edge probability $n^{-\alpha}$ and that the theory of the Shelah-Spencer graph
was stable, building on the 1992 Ph.D. thesis of my student Shi. And this led
to work with Kitty Holland on fusions, giving the first construction of a rank
$2$ field with a definable infinite predicate. And then back to Boris and his
work on complex exponentiation. Understanding his notion of quasiminimal
excellence inspired the desire to understand Shelah’s more general notion of
excellence. Thence came my monograph on abstract elementary classes and
subsequent work on infinitary logic. In any case, visits several times a
decade to Oxford always were exciting sources of ideas and pleasant times.

####  1.1.2 An unlikely encounter of two areas: MAMLS at Rutgers, 2001

The second author of this paper witnessed and participated in one of those
momentous encounters of two areas that only seldom happen: during the MAMLS
Meeting at Rutgers in February 2001, a group of people working in Abstract
Elementary Classes (including Rami Grossberg, Monica VanDieren, Olivier
Lessmann and the second author of this paper) was very busy discussing
Shelah’s notion of excellence, originally linked to his work in the model
theory of $L_{\omega_{1},\omega}$. The _$n$ -amalgamation diagram_ was very
much part of that discussion. There was a lecture by Boris Zilber at the end
of the day, and we all attended, not expecting to understand much, but eager
to see him speak. To our great surprise, at the end of Zilber’s lecture
(dealing with exponential covers, mentioning many analytic number theoretic
methods that were arcane to us, and mixing in areas such as “Nevanlinna
Theory”, etc.), he asked a final question and drew a picture underscoring his
question. Boris’s picture was _exactly_ the $n$-amalgamation diagram we had
been discussing thoroughly with the AEC people those very same days; Boris’s
question was exactly about the behaviour of types in the amalgam and how it
could be controlled by small pieces in the components. We jumped to talk to
him at the end of his lecture, with the excitement of seeing a potential
connection. Boris said he didn’t know the model theory of
$L_{\omega_{1}\omega}$ but he would look into excellence…

The rest is history: after a few weeks, a first draft of a proof of properties
of pseudoexponentiation drawing on a version of excellence and quasiminimality
in $L_{\omega_{1}\omega}$ was circulated, and Zilber started using many
methods from excellent classes and infinitary logic. The richness of this
approach has provided many interesting connections; we explore some of them in
our paper.

###  1.2 A word of thanks from the second author

Here, the second author turns to the first person singular, for this excerpt:

> _I would like to thank Boris Zilber, at a very personal level, for a life-
> changing conversation we had in 2007 in Utrecht, during a meeting organized
> by Juliette Kennedy, on connections between Mathematics, Philosophy and Art.
> One evening, after dinner, Boris said “let’s go for a walk and speak a bit
> about mathematics.” In the cold night along the canals, he described, for
> about an hour, some of what he had been doing—I kept asking and asking
> questions. At some point, on a bridge, he turned to me and said: “But you,
> in what have you been working?” I tried to gather my thoughts on the spot
> while walking, and started describing a project we had back then, with
> Berenstein and Hyttinen[BHV18], of understanding independence notions in
> continuous logic, trying to extend the work of Chatzidakis and Hrushovski to
> the continuous case, and encountering difficulties. Boris asked me to
> describe briefly continuous model theory and continuous abstract elementary
> classes. At some point, he said I obviously had tools for dealing with model
> theoretical approaches to quantum mechanics. I asked how so. He said “look
> at Gelfand triples, …”. I returned to Helsinki where I was spending a
> sabbatical, and Boris’s remarks made a deep change in my own approach to
> model theory, in the possibilities I started slowly unfolding. I am deeply
> grateful for that momentous conversation, and for all the lines of work that
> have derived from that evening!_
>
> Andrés Villaveces

The authors want to thank many people who helped this project go through.
Among them, hoping not to forget important people, are, most notably Sebastián
Eterović, Jim Freitag, Jonathan Kirby, Anatoli Libgober, Ronnie Nagloo, and
Boris Zilber. Without their attention to our discussions, online, at
conferences, and on campus, this project would have been much harder to
complete. The first author especially wants to thank Ronnie and Sebastián for
hours of conversation. The second author especially thanks Alex Cruz and
Leonardo Cano for many helpful discussions related to these subjects in the
Bogotá seminar before this project started. Finally, discussions with Thomas
Kucera and Martin Bays were very important at earlier stages of the
construction of this paper. Finally, the referee reports were invaluable.

##  2 Model theory in Mathematics

We first deal with some variations in model theoretic and geometric
terminology.

###  2.1 Model theoretic background

Mathematical logic makes a central distinction between a vocabulary and a
collection of sentences in a logic. For this reason, we use ‘language’ only
for the second and reserve ‘vocabulary’ for what is sometimes called
similarity type.

######  Definition 2.1.1 (Vocabulary and Structure).

  1. 1.

A vocabulary $\tau$ is a collection of constant, relation, and function
symbols (with finitely many arguments).

  2. 2.

A $\tau$ -structure is a set in which each $\tau$-symbol is interpreted, e.g.,
an $n$-ary relation symbol as an $n$-ary relation.

######  Definition 2.1.2.

Full formalization involves the following components.

  1. 1.

A vocabulary with associated notion of structure as in Definition 2.1.1.

  2. 2.

A logic ${\cal L}$ has:

a

A class ${\cal L}(\tau)$ of ‘well formed’ formulas.

b

A notion of ‘truth of a formula’ from the class $\mathcal{L}\left(\tau\right)$
in a $\tau$-structure, usually denoted $\mathfrak{A}\models\varphi$.

c

A notion of a “formal deduction” for this logic.

  3. 3.

Axioms: Specific sentences of the logic that specify the basic properties of
the situation in question.

######  Example 2.1.3.

(Three important logics.)

  1. 1.

The first order language ${\cal L}_{\omega,\omega}(\tau)$ associated with
$\tau$ is the least set of formulas containing the atomic $\tau$ -formulas and
closed under _finite_ Boolean operations and quantification over finitely many
individuals.

  2. 2.

The ${\cal L}_{\omega_{1},\omega}(\tau)$ language associated with $\tau$ is
the least set of formulas containing the atomic $\tau$ -formulas and closed
under _countable_ Boolean operations and quantification over finitely many
individuals.

  3. 3.

The second order language associated with $\tau$, denoted ${\cal
L}^{2}(\tau)$, is the least set of formulas extending ${\cal
L}_{\omega,\omega}(\tau)$ by allowing quantification over sets and relations.
${\cal L}^{2}(\\{=\\})$ is symbiotic (‘morally equivalent’, roughly speaking)
with set theory.

Morley rank (corresponding to the Krull/Weil dimension in the particular case
of fields) was introduced in [Mor65] to study theories categorical in
uncountable power. Section 5 explores the role of finite Morley rank groups in
studying covers. Three good sources for the more advanced model theory used
here are [Mar02, TZ12, Poi85].

###  2.2 Various Viewpoints

We now discuss two quite different uses of the three words _automorphism_ ,
_model_ and definable, coming from areas of mathematics relevant to this
paper. (The difference in use depending on the area of mathematics has been at
times a source of confusion.)

######  Remark 2.2.1.

(Automorphism: two notions)

In Model Theory:

An automorphism of a $\tau$-structure ${\mathcal{A}}$ is a permutation of its
universe $A$ that preserves (in both directions) each relation or function
symbol for $\tau$. For instance, the automorphisms of a geometry (when given
in terms of lines and points together with an incidence relation) are the
_collineations_.

In Algebraic Geometry:

An automorphism of a variety is an invertible morphism222This begs the
question of defining morphism. A good approximation is ‘definable map’. In
algebraic geometry a morphism is (cf [Poi87, p 79: section 4.4]) a
constructible (generically quasi-rational) bijection. Biregular and birational
are more specific syntactic restrictions on an isomorphism..

######  Remark 2.2.2.

(Model: two notions)

In Model Theory:

The word model also sees different uses depending on the area. In logic, a
model is sometimes just a $\tau$-structure but often signifies that the
structure satisfies a theory (as in ‘$\left(\mathbb{C},+,\cdot,0,1\right)$ is
a model of the theory $ACF_{0}$’). Minimal model might mean ‘no proper
elementary submodel’ or, very differently, ‘every definable subset is finite
or cofinite’.

In Algebraic Geometry:

A model is a specific biregularity class within a birational equivalence
class. In Weil/Zariski style, a variety is determined by a coordinate ring,
but only up to isomorphism of this coordinate ring. A ‘model’ of the variety
might be a specific affine variety with that coordinate ring, but any
biregularly isomorphic variety would also be a model.

Thus, unlike model theory, algebraic geometry does not identify ‘models’ up to
isomorphism. Rather, it looks for a specific ‘canonical representation’ among
‘isomorphic solution sets’. A minimal model is a smooth variety $X$ with
function field $K$ such that if $Y$ is another smooth variety with function
field $K$ and $f\colon X\mapsto Y$ is birational, then $f$ is an isomorphism.

######  Remark 2.2.3.

(Definable/defined: two notions)

In Model Theory:

A subset $X$ of a model $M^{n}$ is defined over a set $A$ if there is a
formula $\phi({\bf x},{\bf a})$ with solution set $X$.

In usual mathematics

the word ‘defined’ is often short for ‘well-defined’ saying that the value of
a function defined on a quotient space does not depend on the choice of a
representative.

In model theory, we add the adjective ‘definable’ when there is a formula of
the language that captures the notion. Thus, the algebraic geometric
‘automorphism’ becomes ‘definable bijection’. It is worth noting that many
important automorphisms in algebraic geometry do not necessarily preserve
structure.

######  Remark 2.2.4 (Why infinitary logic?).

A natural question at this point is: Why is axiomatizability in
$L_{\omega_{1},\omega}$ relevant to geometric questions? The answer to this
question is not univocal, and strongly reflects different historical issues
arising in different areas of mathematics. We discuss four responses, two from
ordinary mathematics, two from logic.

  1. 1.

In ordinary mathematics:

    1. (a)

The constraints of expressibility offered by a particular logic force a
detailed analysis of the hypotheses of a result. This analysis in similar
earlier cases has led to, for example, the Zilber-Pink conjecture and the
Conjecture on the Intersection of Tori (see e.g. [BMPTW20]).

    2. (b)

Of course, each of the ‘canonical structures’ is explicitly definable in set
theory. But this definition in most cases is useless for studying the object.
Useful succinct second order axioms are available for the real and complex
numbers but are only partially known for universal covers. First order logic
is stymied a priori by the intractability of arithmetic. Thus, categoricity in
infinitary logic is essential for giving an ‘algebraic’ account of an
‘analytic object’. This use of model theory can be seen as part of the larger
scale _GAGA_ mathematical program of bridging analytical concepts and
algebraic ones.

  2. 2.

In logic (in particular, in model theory):

    1. (a)

A natural question is: are there important mathematical notions expressible in
infinitary logic which are not expressible in first order? The study of
complex exponentiation yielded a superb initial example: the categoricity of
the covering map of $\mbox{\msbm C}^{*}$ in [BaysZil].

    2. (b)

This raises the question of what are the new axioms in this paper that require
an infinitary description. The infinite dimension axioms are well known and
the switch from ‘standard kernel’ to ‘standard fiber over $z$’ (i.e.
$q^{-1}(z)$) is unremarkable. It seems the finite index conditions (Section
4.4) are not first order expressible.

##  3 Categoricity, quasiminimality and excellence

We give a quick sketch of notions around categoricity333More specifically,
when in model theory we use the word _categoricity,_ we mean categoricity in a
specific cardinality or ‘in power’. See a thorough discussion of categoricity
in various logics in [Bal18, §3.1] and an exposition of the philosophical
import of the notion in [CMVZ21]. and the history of their logical
development.

######  Definition 3.0.1 (Categoricity).

  1. 1

A _theory_ $T$ in a logic ${\cal L}$ is a collection of ${\cal L}$-sentences
in a vocabulary $\tau$.

  2. 2

$T$ is _categorical in cardinality $\kappa$_ ($\kappa$-categorical) if all
models $M$ of $T$ with $|M|=\kappa$ are isomorphic.

Although certain canonical mathematical structures are fruitfully axiomatized
in second order logic, rather than second order categoricity, we usually
consider these characterizations as defining these structures _in set theory._
Such definitions are exactly what it means to be a structure. Second order
categoricity per se gives no useful mathematical information. In contrast,
$\kappa$-categoricity in first order logic or in $L_{\omega_{1},\omega}$
provides very significant (combinatorial geometric) information; it assigns a
dimension to each model.

###  3.1 The Classical Categoricity Theorems

The following results survey the spectrum of cardinals in which certain types
of theory can be categorical. These theorems are of the form _if a theory (or
a sentence) is categorical in some high enough cardinal(s), then it must be
categorical on a tail of cardinals._

######  Theorem 3.1.1 (Morley’s Categoricity Theorem).

A countable first order theory is categorical in one uncountable cardinal if
and only if it is categorical in all uncountable cardinals. [Mor65].

######  Theorem 3.1.2 (Shelah’s Categoricity under the weak continuum
hypothesis below $\aleph_{\omega}$).

Assuming $2^{\aleph_{n}}<2^{\aleph_{n+1}}$ a sentence in
$L_{\omega_{1},\omega}$ that is categorical in $\aleph_{n}$ (for every
$n<\omega$) is categorical in all uncountable cardinals [She83a], [She83b].

######  Theorem 3.1.3 (Shelah’s Categoricity theorem for excellent sentences).

An excellent sentence in $L_{\omega_{1},\omega}$ is categorical in one
uncountable cardinal if and only if it is categorical in all uncountable
cardinals [She83a], [She83b].

######  Theorem 3.1.4 (Zilber’s Categoricity for quasi-minimal excellent
classes).

A quasi-minimal excellent class is categorical in all uncountable cardinals
[Zil04].

###  3.2 Pregeometries (matroids) and quasiminimality

The presence of quasiminimal pregeometries provides an extremely fruitful and
natural control of models in a class (and of their interactions).

######  Definition 3.2.1 (Combinatorial Geometry).

A _closure system_ is a set $G$ together with a ‘closure’ relation on subsets
of $G$

$cl:{\cal P}(G)\rightarrow{\cal P}(G)$

satisfying the following axioms.

A1. $cl(X)=\bigcup\\{cl(X^{\prime}):\ X^{\prime}\subseteq_{fin}X\\}$

A2. $X\subseteq cl(X)$

A3. $cl(cl(X))=cl(X)$

$(G,{\rm cl})$ is a pregeometry if, in addition, we have:

A4. If $a\in cl(Xb)$ and $a\not\in cl(X)$, then $b\in cl(Xa)$.

If points are closed ($cl(\\{a\\})=\\{a\\}$, for each $a$) the structure is
called a _geometry_.

Pregeometries are virtually the same mathematical objects as matroids.

######  Definition 3.2.2.

  1. 1.

A subset $D$ of a $\tau$-structure M is first order-definable in $M$ if there
is ${\bf a}\in M$ and an ${\cal L}_{\omega,\omega}(\tau)$-formula
$\varphi(x,{\bf y})$ such that $D=\\{m\in M:M\models\varphi(m,{\bf a})\\}$. If
${\bf a}\in A\subseteq M$, $D$ is definable with parameters from $A$.

  2. 2.

${\rm acl}_{M}(A)$ (the algebraic closure of $A$ in $M$) is $\\{m\in
M\colon\phi(m,\overline{a}),\overline{a}\in A\\}$, where
$\phi(x,\overline{a})$ has only finitely many solutions in $M$.

  3. 3.

${\rm dcl}_{M}(A)$ (the definable closure of $A$ in $M$) is defined as was the
algebraic closure, but replacing ‘finitely many’ by ‘one’.

  4. 4.

An infinite definable subset $D$ (or its defining formula $\varphi(x)$) is
strongly minimal if every definable subset of $D$ in every elementary
extension of $M$ is finite or cofinite.

  5. 5.

A theory is _strongly minimal_ if the formula $x=x$ is strong minimal.

The notion of type is a crucial tool in model theory.

######  Definition 3.2.3.

  1. 1.

The first order type of $a$ over $B$ (in $M$), denoted ${\rm tp}_{M}(a/B)$, is
the set of ${\cal L}_{\omega,\omega}$-formulas with parameters from $B$ that
are satisfied in $M$ (for $a,B\subseteq M$).

  2. 2.

The quantifier-free type of $a$ over $B$ (in $M$), denoted ${\rm tp}_{\rm
qf}(a/B\colon M)$, is the set of _quantifier-free_ first order formulas
$\varphi(x,{\mathbf{b}})$ such that $M\models\varphi(a,{\mathbf{b}})$ (as
before, $\mathbf{b}$ ranges over tuples of $B$).

In most contexts, when we just say ‘the type of $a$ over $B$,’ we mean the
first order type. Note also that if a property is defined without parameters
in $M$, then it is uniformly defined in all models of $\mathop{\rm Th}(M)$
(the theory of $M$, i.e., the set of all $\tau$ sentences that are true in
$M$).

Here are three fundamental observations on strongly minimal sets.

  * •

A strongly minimal set admits a combinatorial geometry when the closure is
taken as ${\rm acl}$ (Definition 3.2.2).

  * •

There is a unique type of elements in a strongly minimal set that are not
algebraic. This is called the generic type for $D$.

  * •

In many important examples (e.g. $DCF_{0}$), the structure of the model is
controlled by its strongly minimal sets.

Shelah’s abstract notion of independence (for some first order theories,
crystallized as _non-forking_) weakens the notion of combinatorial geometry by
dropping A3; in some desirable cases this property is recovered on the points
realizing a regular type and in even better cases the dimensions of the
regular types determine the isomorphism type of the model. However, a priori,
the existence of a global dimension is unusual.

We now look at the generalization of strong minimality, introduced by Zilber,
that is central in the connections between model theory and algebraic geometry
described in this paper.

######  Definition 3.2.4 (Quasiminimal structure).

A structure $M$ is quasiminimal if every first order ($L_{\omega_{1},\omega}$)
definable subset of $M$ is countable or cocountable. Algebraic closure is
generalized by saying $b\in{\rm acl}^{\prime}(X)$ if there is a first order
formula with countably many solutions over $X$ which is satisfied by $b$.

######  Definition 3.2.5 (Quasiminimal excellent geometry).

Let K  be a class of $L$-structures such that $M\in\mbox{\boldmath K }$ admits
a closure relation ${\rm cl}_{M}$ mapping $X\subseteq M$ to ${\rm
cl}_{M}(X)\subseteq M$ that satisfies the following properties.

  1. 1.

Basic Conditions

    1. (a)

Each ${\rm cl}_{M}$ defines a pregeometry on $M$.

    2. (b)

For each $X\subseteq M$, ${\rm cl}_{M}(X)\in\mbox{\boldmath K }$.

    3. (c)

countable closure property (ccp): If $|X|\leq\aleph_{0}$ then $|{\rm
cl}(X)|\leq\aleph_{0}$.

  2. 2.

Homogeneity

    1. (a)

A class K  of models has $\aleph_{0}$-homogeneity over $\emptyset$ (Definition
3.2.5) if the models of K  are pairwise qf-back and forth equivalent
(Definition 4.3.7)

    2. (b)

A class K  of models has $\aleph_{0}$-homogeneity over models if for any
$G\in\mbox{\boldmath K }$ with $G$ empty or a countable member of K , any
$H,H^{\prime}$ with $G\leq H,G\leq H^{\prime}$, $H$ is qf-back and forth
equivalent with $H^{\prime}$ over $G$.

  3. 3.

K  is an almost quasiminimal excellent geometry if the universe of any model
$H\in\mbox{\boldmath K }$ is in ${\rm cl}(X)$ for any maximal ${\rm
cl}$-independent set $X\subseteq H$.

  4. 4.

We call a class which satisfies these conditions an almost quasiminimal
excellent geometry  [BHH+14].

An almost quasiminimal excellent geometry with strong submodel taken as $A\leq
M$, if ${\rm acl}_{M}(A)=A$, gives an _abstract elementary class_ (AEC)444See
[GL02] for the early history of the model theory of AECs.. But the distinct
notion of a quasiminimal AEC (defined in terms of $\leq$ rather than any
axioms) is due to [Vas18].

To obtain that the class is complete for $L_{\omega_{1},\omega}$, [Kir10,
BHH+14] add the requirement of $\aleph_{0}$-categoricity.

######  Remark 3.2.6.

This definition differs only superficially from those in e.g. [Kir10], where
the connections with the combinatorial geometry was emphasized by
distinguishing the treatment of elements depending on whether they were in
${\rm cl}(H)$. However, [BHH+14] required a quasiminimal structure to have a
unique generic type. This requirement fails in the two-sorted treatment we
deal with here; there may be ${\rm acl}$-bases in each sort. So we replace
quasiminimality with _almost quasiminimality_ (less explicit in [BHP20]) and
we thus restore Zilber’s first intuition (Definition 3.2.4) that
quasiminimality means that all definable sets are countable or co-countable.

######  Remark 3.2.7 (Excellence).

From Zilber’s introduction of the notion in [Zil04], it has been known that
the axioms 3.2.5 imply $\aleph_{1}$-categoricity. See the exposition in
[Bal09]. But, without further ‘excellence’ hypotheses, it was unknown whether
the class had larger models. Two formulations of excellence are 1) [She83a,
She83b]: $n$-amalgamation of independent systems of models, for all
$n<\omega$, and 2) A local condition on the properties of a ‘crown’ [Kir10].
Either of these implies the existence of arbitrarily large models for theories
in $L_{\omega_{1},\omega}$. As we discuss in Section 5.2, influenced by work
Hart and Shelah on first order classification theory, the next result (here
modified by ‘almost’) clarified the relationship.

######  Remark 3.2.8.

Crucial Fact [Theorem: Bays, Hart, Hyttinen, Kesälä, Kirby]. Every almost-
quasiminimal class (Definition 3.2.5) is excellent as described in Remark
3.2.7. Thus, it is categorical in all uncountable cardinalities.

##  4 Modular and Shimura Curves

We begin with an astronaut’s view of the $j$-function and then turn to the
model theoretic treatment of some generalizations.

###  4.1 The great confluence

The general form (over a field of characteristic $0$) of an elliptic curve is

$y^{2}=x^{3}+ax+b.$

At least since Diophantus (3rd century AD), the search for integer solutions
for such equations has been a central question. The cataloguing of such
equations was a major achievement of the 19th century. One key step toward
this classification is to generalize the original problem and look first for
complex solutions. The solution set of an elliptic curve is then a smooth,
projective, algebraic curve of genus one. It can be thought of as a ‘classical
torus’ $\mbox{\msbm T}_{\tau}:=\mbox{\msbm C}/\Lambda_{\tau}$, where
$\tau\in\mbox{\msbm C}$ and $\Lambda_{\tau}$ is the lattice in ℂ (the subgroup
of $(\mbox{\msbm C},+)$ generated by $\langle 1,\tau\rangle$.

Klein studied modular and automorphic functions, which provide surprising and
deep links between geometry, complex analysis and number theory. The most
famous example is the $j$-function, analytic on $\mathbb{H}=\\{z:{\rm
im}(z)>0\\}$, the upper half plane, and maps onto ℂ and meromorphic with some
poles on the real axis and the following remarkable properties.

######  Theorem 4.1.1 (Classification of tori by the $j$-function).

The following are equivalent:

  1. 1.

There exists $s=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in
SL_{2}(\mbox{\msbm Z})$ such that
$s(\tau)=\frac{a\tau+b}{c\tau+d}=\tau^{\prime}$,

  2. 2.

$\mbox{\msbm T}_{\tau}\approx\mbox{\msbm T}_{\tau^{\prime}}$ in the algebraic
geometry sense of Definition 2.2.1.

  3. 3.

$j(\tau)=j(\tau^{\prime})$

This rather astonishing classical fact paves the way toward modern day
classifications. It provides equivalences between analytic and number-
theoretic notions. Strikingly, $j$ is defined as a rational function of two
analytic functions $g_{2}$ and $g_{3}$ (each of them coding so-called
‘modularity’ properties):

$j(\tau)=12^{3}\cdot\frac{g_{2}(\tau)^{3}}{g_{2}(\tau)^{3}-27g_{3}(\tau)^{3}}.$

But where does the word ‘elliptic’ come from? A meromorphic function is called
an elliptic function, if it is doubly periodic: there are two $\mathbb{R}$
-linear independent complex numbers $\omega_{1}$ and $\omega_{2}$ such that
$\forall z\in\mbox{\msbm C}$, ${\displaystyle f(z+\omega_{1})=f(z)}$ and
${\displaystyle f(z+\omega_{2})=f(z)}$. Abel discovered such doubly periodic
functions arose from the solutions of elliptic integrals – originally defined
to find the arc length of an ellipse. Weierstraß used the symbol $\wp$ to
denote a family of functions $\wp(z,\Lambda_{\tau})$ where the defining double
sum runs over the elements of the lattice $\Lambda_{\tau}$, generated by $1$
and $\tau$. The crucial property of the function is that every meromorphic
function that is periodic on $\Lambda_{\tau}$ is a rational combination of
$\wp(z,\Lambda_{\tau})$ and $\wp^{\prime}(z,\Lambda_{\tau})$. This field of
functions is precisely Abel’s field of elliptic functions.

Klein’s discovery of the $j$ function unified the results of Weierstraß. In
his famous investigation of the psychology of mathematical investigation,
Hadamard devotes several pages to Poincaré’s generalization of the
$j$-function to the family of functions derived from Fuchsian group actions.
The crucial phrase for us is _‘the transformations I had used to define the
Fuchsian functions were identical with those of non-Euclidean geometry’_
[Had54, p 33].

This completes a very quick summary of the 19th century predecessors of the
theory of moduli spaces, developed in the next section. This study involves
complex analysis, actions by a discrete group, number theory, and non-
Euclidean geometry. The crucial model theoretic step is to formalize in a
vocabulary for two-sorted structures of the form

${\mathfrak{A}}=\langle\langle H;\\{g_{i}\\}_{i\in\mbox{\msbm
N}}\rangle,\langle F,+,\cdot,0,1\rangle,j:H\to F\rangle$

where $\langle F,+,\cdot,0,1\rangle$ is an algebraically closed field of
characteristic $0$, $\langle H;\\{g_{i}\\}_{i<\omega}\rangle$ is a set
together with countably many unary function symbols, and $j\colon H\to F$.

In the next section we provide some of the mathematical background for a
formal analysis of these two-sorted structures.

###  4.2 Moduli Spaces

Moduli spaces in geometry are parametrized collections of objects, together
with equivalences that allow us to see when two objects are in some sense ‘the
same’, and with families that articulate the variation between the objects in
the collection. Paraphrasing the important survey [BZ08], ‘moduli spaces are a
geometric solution to a geometric classification problem.’ They parametrize
collections of geometric objects, they define equivalences to say when two
objects are the ‘same’, and establish families that determine how we allow our
objects to vary or modulate.

In model theory, the notion of a uniform family of definable sets has been
thoroughly studied. Such a family is given by a formula of the form $\phi({\bf
x},{\bf y})$. Each set in the family is the solution set of $\phi({\bf a},{\bf
y})$ (for some ${\bf a}$), and the set $\\{{\bf a}:(\exists{\bf y})\phi({\bf
a},{\bf y})\\}$ is an indexing set of the family. In the algebraic geometry
setting, one can require that the ${\bf x}$ fall into a variety $V$ and the
${\bf y}$ into a variety $W_{{\bf a}}$. $V$ is a step toward the notion of a
moduli space.

Except in § 5, we consider moduli spaces arising from a pair $(G,X)$
consisting of a group $G$ acting on a space $X$. The algebraic varieties we
study arise as quotients $\Gamma\setminus X$ (for $\Gamma$ a subgroup of $G$,
see Definition 4.2.2). A modular curve arises as a connected component of
quotient of ℍ by congruence subgroups (Definition 4.2.9) of $\mathop{\rm
GL}_{2}({\mathbb{R}})$. Shimura generalized the topic to groups acting on
wider classes of domains. Shimura curves are rather more complicated yet
generally share similar categoricity properties. Shimura varieties of higher
dimension raise many new issues that we sketch in Section 6. In this section,
we consider only covers of modular curves by ℍ.

Here, $\mathbb{H}$ refers, as in the rest of this paper, to the upper half
complex plane (as the set of points: $\mathbb{H}=\\{z\in\mathbb{C}:{\rm
Im}(z)>0\\}$). $\mathbb{H}$ is also called the hyperbolic plane (when endowed
with a metric and topology that make it hyperbolic rather than Euclidean). See
[Miy89] for a detailed description. In all our examples, the function $p$ maps
the hyperbolic plane into a complex variety. We consider the action of
$\mathop{\rm PSL}_{2}({\mathbb{R}})$ on ℍ as fractional linear
transformations: for
$A=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in\mathop{\rm
SL}_{2}(\mbox{\msbm Z})$ and $\tau\in\mbox{\msbm H}$,
$A(\tau)=\left(\frac{a\tau+b}{c\tau+d}\right)$.

The group of bijections (isometries, ${\rm isom}(\mbox{\msbm H})$) that
preserve the hyperbolic metric of ℍ is generated by $\mathop{\rm
PSL}_{2}({\mathbb{R}})$ and the map $z\mapsto-\overline{z}$; $\mathop{\rm
PSL}_{2}({\mathbb{R}})$ consists precisely of all those isometries that
preserve orientation (e.g. [Kat92]). After outlining here the classical theory
of such actions and moduli spaces, in section 4.3 we describe a model
theoretic approach.

######  Definition 4.2.1 (Fuchsian group).

  1. 1.

A subgroup $G\leq{\rm isom}(\mbox{\msbm H})\approx\mathop{\rm
PSL}_{2}({\mathbb{R}})$ is _discrete_ if it is discrete in the induced
topology.

  2. 2.

A Fuchsian group is a discrete subgroup of $\mathop{\rm
PSL}_{2}({\mathbb{R}})$.

The most important example of a Fuchsian group is $\mathop{\rm
PSL}_{2}(\mbox{\msbm Z})$. Underlying this entire study and almost one and a
half centuries of interactions between number theory and complex analysis is
the remarkable fact that the quotient of ℍ by certain discrete subgroups has
the structure of a Riemann surface [Miy89, §1.8] and even an algebraic variety
which, in important cases, is a moduli space [Mil12].

######  Definition 4.2.2 (Quotient of ℍ by a group).

If a group $G$ acts on a set $X$, $G\setminus X$ has universe the collection
of $G$-orbits of the action. $\pi$ is the canonical map taking $x$ to its
orbit $Gx$. The prototypical example corresponds to $X=\mathbb{H}$.

######  Definition 4.2.3.

The quotients $V=S(\mbox{\msbm C})$ of ℍ by a discrete group $\Gamma$ that we
consider are examples of moduli spaces. $V=\bigcup_{a\in C}V_{a}$ is the image
of a map $p$ from ℍ that acts as a uniformizer for a family of varieties
$V_{a}$. Namely for each $a,b\in\mbox{\msbm H}$, $V_{a}\cong V_{b}$ iff for
some $\gamma\in\Gamma$, $\gamma(a)=b$ iff $p(a)=p(b)$.

We explored in Section 4.1 the ur-example of a moduli space, elliptic curves
as uniformized by the $j$-function. The next definition relies on the fact
that, while elements of $\mathop{\rm PSL}_{2}({\mathbb{R}})$ fix ℍ setwise,
they also act on all of ℂ.

######  Definition 4.2.4 (Cusp).

For a discrete subgroup $\Gamma$ of $\mathop{\rm PSL}_{2}({\mathbb{R}})$:

  1. 1.

We say $c\in{\mathbb{R}}\cup\\{\infty\\}$ is a cusp of $\Gamma$ if $c$ is the
unique fixed point of some $\gamma\in\Gamma$.

  2. 2.

$P_{\Gamma}$ is the set of cusps of $\Gamma$ and $\mbox{\msbm
H}^{*}=\mbox{\msbm H}^{*}_{\Gamma}=\mbox{\msbm H}\cup P_{\Gamma}$.

We relate some standard facts (see [Har14, p 15]). The first relies on the
fact that while some of the quotients we study are not compact, they can be
compactified by adding finitely many cusps from
${\mathbb{R}}\cup\\{\infty\\}$.

######  Fact 4.2.5.

For any discrete subgroup $\Gamma\subseteq\mathop{\rm PSL}_{2}({\mathbb{R}})$,
the quotient $\Gamma\setminus\mbox{\msbm H}^{*}_{\Gamma}$ is a compact
Hausdorff space that can be given the structure of a Riemann surface.
Therefore if $\Gamma^{\prime}$ is of finite index in $\Gamma$, the quotient
$\Gamma^{\prime}\setminus\mbox{\msbm H}^{*}_{\Gamma}$ is a compact Riemann
surface, and is therefore algebraic by the Riemann existence theorem.
$\mbox{\msbm H}^{*}_{\Gamma}$ is the compactification of the quasi-projective
algebraic variety (so first order definable) $\mbox{\msbm H}_{\Gamma}$, .

For the purposes of this paper since the quasiprojective variety $\mbox{\msbm
H}_{\Gamma}=\Gamma\setminus\mbox{\msbm H}$ determines the (classical)
algebraic variety (set of solutions of a system of polynomial equations),
$\mbox{\msbm H}^{*}_{\Gamma}$ we work hereafter with $\mbox{\msbm
H}_{\Gamma}$. This is natural from a model-theoretic standpoint since (in this
situation) there are only finitely many cusps and so the sets differ by only
finitely many points.

Notation 4.2.6 fixes the group $G$ for the rest of § 4. Setting the
determinant as $1$ and modding out the center guarantees the group action
preserves both distance and orientation.

######  Notation 4.2.6.

Let $G=\mathop{\rm GL}_{2}^{ad}(\mbox{\msbm Q})^{+}=_{\rm
def}PSL_{2}({\mathbb{Q}})/Z(PSL_{2}({\mathbb{Q}}))\approx PSL_{2}(\mbox{\msbm
Q})$ modulo its center. $\Gamma$ varies over subgroups of $G$

We now distinguish two kinds of points in $\mathbb{H}$: ‘special’ points and
‘Hodge-generic’ points. The equivalence of the following definition with the
usual notion [DH17, Definition 2.2] for Shimura varieties is in [DH17, Theorem
2.3].

######  Definition 4.2.7 (Special points).

Fix $\langle\mbox{\msbm H},S(\mbox{\msbm C}),p\rangle$ with $S(\mbox{\msbm
C})$ biholomophic to $\Gamma\setminus\mbox{\msbm H}$. A point $x\in\mbox{\msbm
H}$ is special if there is a $g\in G$ whose unique fixed point is $x$.

We omit the definition of a Hodge generic point arising in algebra, as it does
not enter our discussion; we use only the equivalent characterization [DH17,
Prop 2.5] given in Fact 4.2.8.1) and the dichotomy in 2) noted just after that
proposition. It is worth mentioning that for a point the fact of being
“special” or “Hodge generic” does not depend on the choice of the group
$\Gamma$; furthermore, these two notions are preserved by the action of
$G=GL^{\rm ad}_{2}(\mathbb{Q})^{+}$.

######  Fact 4.2.8.

Special and Hodge generic points [DH17, Proposition 2.5]

  1. (1)

If $x$ is Hodge generic the only $g\in G$ that fixes $x$ is the identity.

  2. (2)

Every point in ℍ is either Hodge generic or special.

Although we are studying the categoricity of the universal cover of a specific
modular curve (e.g. the image of the $j$-function, $\Gamma\setminus\mbox{\msbm
H}$), other modular curves naturally arise in the analysis. The study of
families of such curves is expounded in [Shi71, §6, 7]. A key tool to give a
uniform treatment to a family is the existence of a common commensurator of
the generating Fuchsian groups. In fact, the members of the family are
interalgebraic and the entire family (indexed by the $\Gamma_{N}$) is studied
in [DZ22a].

######  Definition 4.2.9.

  1. 1.

The groups $\Gamma_{N}$ ($N$ a fixed integer) are given by

$\Gamma_{N}=\left\\{\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]\in\Gamma:\ b\equiv c\equiv 0,a\equiv d\equiv 1\ {\rm
mod}\ N\right\\}.$

Note that each $\Gamma_{N}$ has finite index in $\Gamma$ and if $N|M$ then
$\Gamma_{M}\subseteq\Gamma_{N}$.

  2. 2.

Two subgroups $\Gamma$ and $\Gamma^{\prime}$ of a group $H$ are said to be
_commensurable_ if $\Gamma\cap\Gamma^{\prime}$ is of finite index in both of
them.

  3. 3.

A congruence subgroup is a subgroup $\Gamma^{\prime}$ of $\Gamma$ such that
some $\Gamma_{N}$ is a finite index subgroup of $\Gamma^{\prime}$.

  4. 4.

The _commensurator_ ${\rm comm}(\Gamma)$ of a subgroup $\Gamma$ of
${\mathop{\rm PSL}}_{2}(\mbox{\msbm R})$ is

$\\{\delta\in{\mathop{\rm PSL}}_{2}(\mbox{\msbm
R})\colon\delta\Gamma\delta^{-1}\ \text{\ is commensurable with}\ \Gamma\\}.$

We rely on the following standard fact.

######  Lemma 4.2.10.

The group $G=\mathop{\rm GL}_{2}^{ad}(\mbox{\msbm Q})^{+}$ (Notation 4.2.6) is
the commensurator of any congruence subgroup $\Gamma$ of $SL_{2}(Z)$.

Because the functions $g\in G$ are in the formal vocabulary, we employ
congruence subgroups $\Gamma_{\bf g}$ from Notation 4.2.11 rather than the
$\Gamma_{N}$. The $Z_{{\bf g}}$ defined in Notation 4.2.11 play a central role
both in the quantifier elimination and via an inverse limit in Section 4.4.

######  Notation 4.2.11.

With $G$ as fixed in Notation 4.2.6, as each of the congruence subgroups of
$\mathop{\rm PSL}_{2}(\mbox{\msbm Z})$ act on ℍ we can define for any finite
sequence of the form ${\bf g}=\langle e,g_{2},\ldots,g_{n}\rangle$ from $G$
(by convention, $g_{1}=e$),

  1. 1.

$\Gamma_{{\bf g}}=\Gamma\cap g_{2}^{-1}\Gamma g_{2}\ldots\cap g_{n}^{-1}\Gamma
g_{n}$.

  2. 2.

Let $p:\mbox{\msbm H}\rightarrow S(\mbox{\msbm C})$.

    1. (a)

$Z_{{\bf g}}$ is defined as
$\\{\left(p(x),p(g_{2}x),\ldots,p(g_{n}x)\right)\in S(\mbox{\msbm
C})^{n}:x\in\mathbb{H}\\}$.

    2. (b)

Let $p_{{\bf g}}:\mbox{\msbm H}\rightarrow Z_{{\bf g}}\subseteq S(\mbox{\msbm
C})^{n}$ be defined by

$x\mapsto p({\bf g}x)=\langle p(x),p(g_{1}(x)),\ldots,p(g_{n}(x))\rangle.$

    3. (c)

Let $[\phi_{{\bf g}}]$ be the map from $\mbox{\msbm H}_{{\bf g}}$ onto
$Z_{{\bf g}}$ (Lemma 4.2.12) given by $[\phi_{{\bf g}}]{x}_{\Gamma_{\bf
g}}=p_{\bf g}(x)$.

  3. 3.

Let $\mbox{\msbm H}_{\bf g}$ denote $\Gamma_{{\bf g}}\setminus\mbox{\msbm H}$.

All the previous are well-defined by our choice of $p$ and $\Gamma$.

The following lemma [Ete22, 3.31] is central to Section 4.4.2. Its proof uses
Shimura theory very heavily.

######  Lemma 4.2.12.

The map $[\phi_{\bf g}]$ is bijective on the Hodge generic points and the
image $Z_{\bf g}$ is a variety contained in ${\cal S}^{n}(\mbox{\msbm C})$,
$n=\lg({\bf g})$. Moreover, [Ete22, p 17], for all ${\bf g}$, $Z_{\bf g}$ is
defined over the maximal Abelian extension $L$ of the field of definition,
$E$, of $S$.

######  Remark 4.2.13.

From the model theoretic standpoint, it makes no sense to say the $[\phi_{\bf
g}]$ are definable since their domains $\mbox{\msbm H}_{\bf g}$ are not. While
the maps $[\phi_{\bf g}]$ are bijective on Hodge generic points, they may
identify special points.

###  4.3 Quantifier Elimination in Modular and Shimura Curves

We now lay out the vocabulary and first order theory for studying modular
curves. The mathematical input is a Fuchsian group $\Gamma$ acting on
hyperbolic space ℍ and the image curve $S(\mbox{\msbm
C})=\Gamma\setminus\mbox{\msbm H}^{*}_{\Gamma}$ (Definition 4.2.4) with a
standard model ${\bf p}=\langle\mbox{\msbm H},S,p\rangle$. The structure of a
discrete group is unwieldy from a traditional model theoretic standpoint
because its first order theory is unstable and undecidable. Just as modules
are usually studied in model theory by adding unary function symbols $f_{r}$
for the elements of the ring, in order to represent the action of $G$ on ℍ, we
add symbols $f_{g}$ for $g\in G$ as unary functions that act on ℍ. We thus use
a two-sorted presentation of our structures: a sort for the domain, a sort for
the target, and a map $p$ connecting them.

######  Remark 4.3.1 (Sorts).

A two-sorted structure interprets two sort symbols and additional relation and
function symbols with the understanding that each such relation/function
either is restricted to one of the predicates or explicitly connects them.

######  Notation 4.3.2 (The formal vocabulary $\tau$).

The two-sorted vocabulary $\tau$ consists of the sorts (unary predicate
symbols) $D$ (the covering sort), $S$ the target sort, and a function $q$
mapping $D$ onto the sort $S$.

We write $\tau_{G}$ for the vocabulary of the first sort with
$G=G^{ad}(\mbox{\msbm Q}^{+})$. The second $\tau_{F}={\cal R}$ where ${\cal
R}$ is the set of formulas in $\\{+,-,0,1,\times\\}$ specified in Definition
4.3.3. $\tau$ is $\tau_{G}\cup\tau_{F}\cup\\{p\\}$. There are constant symbols
for each element of the field $E^{ab}(\Sigma)$ defined in Notation 4.3.3. We
use $f_{g}$ to name the functions acting on $D$, but often write the shorter
$g(x)$ or $gx$ instead of $f_{g}(x)$.

The following notation is essential to understand the Axioms 4.3.5. Note in
the prototype $q$ is replaced by the known covering map $p$.

######  Notation 4.3.3.

The standard model for a modular curve determined by a Fuchsian group
$\Gamma\subseteq G=G^{ad}(\mbox{\msbm Q}^{+})$ will consist of a
$\tau$-structure ${\bf p}=\langle\mbox{\msbm H},S,p\rangle$ with the domain ℍ,
the variety $S(F)$ over the algebraically closed field $F$ defined by
$\Gamma\setminus\mbox{\msbm H}$, and ${\cal R}$ the set of all Zariski closed
relations on $S(F)^{n}$ (for all $n$) with constants from a field
$E^{ab}(\Sigma)$ that are true in $F$. $E^{ab}$ is the maximal abelian
extension of the defining (reflex) field $E$ of $S$. $E^{ab}(\Sigma)$ is the
extension of $E^{ab}$ ($F_{0}$ in [Ete22, p 19]) obtained by adding the
coordinates of the ($\leq\aleph_{0}$) special points, and closing to a field.

######  Notation 4.3.4.

For a structure ${\bf p}$, we write $\mathop{\rm Th}({\bf p})$ for the
complete first order theory of all sentences true in ${\bf p}$ and $T({\bf
p})$ for the specified set of axioms true of ${\bf p}$. Clearly, $T({\bf
p})\subseteq Th({\bf p})$.

We must distinguish $\mathop{\rm Th}({\bf p})$ from its subset $T({\bf p})$
until we prove $T({\bf p})$ is a complete axiomatization of $\mathop{\rm
Th}({\bf p})$.

######  Definition 4.3.5 (First Order Axioms).

$T({\bf p})$ is the following collection of first order sentences that are to
hold in a structure $\langle D,S(F),q\rangle$.

  1. 1.

Each sentence in $\mathop{\rm Th}(\langle\mbox{\msbm H},\\{f_{g}:g\in
G\rangle)$. These include ‘Special Point axioms’ $SP_{g}$: For each $g\in G$
that fixes a unique point in $D$

$\forall x,y\in D[(g(x)=x\wedge g(y)=y)\Rightarrow x=y]$

  2. 2.

$\mathop{\rm Th}(S(\mbox{\msbm C}),{\cal R})$ (${\cal R}$ from Definition
4.3.2.)

  3. 3.

The covering map; for each ${\bf g}\in G^{m}$ and all $m<\omega$:

    1. (a)

$Mod^{1}_{{\bf g}}$,

$\forall x\in D\ (q(g_{1}(x),\ldots q(g_{m}(x))\in Z_{{\bf g}})$

    2. (b)

$Mod^{2}_{{\bf g}}$:

$\forall z\in Z_{{\bf g}}\exists x\in D(q(g_{1}(x)),\ldots q(g_{m}(x))=z)$

    3. (c)

${\bf MOD}=\\{Mod^{1}_{\overline{g}}\wedge
Mod^{2}_{\overline{g}}:\overline{g}\in G^{m},m<\omega\\}$

Note that $\bf{MOD}$ is a countable collection of first order sentences.

######  Notation 4.3.6.

By the choice of $E^{ab}(\Sigma)$, special points belong to ${\rm
dcl}\left(\emptyset\right)$. Therefore, we can name each one of them by
$d_{g}$, where $g\in G$ fixes $d_{g}$. Any $g$ that fixes a point is in
$G-\mathop{\rm SL}_{2}(\mbox{\msbm Z})$ [Ete22, Lemma 3.18]. There will be
distinct $g_{1},g_{2}$ that fix the same point (e.g. if $g_{2}=g_{1}^{2}$). If
so, $T({\bf p})\vdash d_{g_{1}}=d_{g_{2}}$ The theory of $(D,G)$ contains the
uniqueness axiom (Definition 4.3.5.1) that entails $g(d_{g})=d_{g}$.

The cover sort is a set with unary functions. Both its theory (since the
universe is a union of orbits) and that of the field sort (since algebraically
closed) are strongly minimal and quantifier eliminable.

######  Definition 4.3.7.

We say two structures $M$ and $N$ are qf-back and forth equivalent if the
system $I$ of partial isomorphisms of $M$ and $N$ between isomorphic finitely
generated substructures satisfies the back and forth condition: For each $f\in
I$ and each $m\in M-\mathop{\rm dom}f$, there exists an $n\in N$ such that
$f\cup\\{\langle m,n\rangle\\}\in I$, and symmetrically, for each $n\in
N-\mathrm{im}\,f$, there exists $m\in M$ such that $f\cup\\{\langle
m,n\rangle\\}\in I$. In this situation $\mathop{\rm dom}f$ is definably close.

######  Notation 4.3.8.

We write ${\bf g}(x)$ for $(g_{1}(x),\ldots g_{n}(x))$ where ${\bf g}$ has
length $n$ and begins with $e$. And then ${\bf g}({\bf x})$ denotes the
sequence of length $nm$ obtained when ${\bf g}$ is applied to each element of
a sequence ${\bf x}\in(D)^{m}$. When convenient we write $gx$ or ${\bf g}x$
for the action, omitting the parentheses.

We now sketch the proof of Theorem 4.3.13 that $T({\bf p})$ axiomatizes a
complete, quantifier eliminable $\tau$-theory.

######  Definition 4.3.9 (The back and forth).

Fix two models ${\bf q}=\langle D,S(F),q\rangle$ and ${\bf q}^{\prime}=\langle
D^{\prime},S(F^{\prime}),q^{\prime}\rangle$ of $T({\bf p})$. We define the qf-
back-and-forth system $I$ of substructures of ${\bf q}$ and ${\bf q}^{\prime}$
For each $f\in I$, $\mathop{\rm dom}f$ and ${\mathop{\rm rg}\ f}$ are each
finitely generated over $E^{ab}(\Sigma)$ . A typical member $f$ of the system
for ${\bf q}$ has $\mathop{\rm dom}f=U=U_{D}\cup U_{S}$. Since $U$ is finitely
generated, $U_{D}$ consists of the $G$-orbits of a finite number of $x\in D$;
$U_{S}$ is $S(L_{U})$ where $L_{U}$ is the field generated by $E^{ab}(\Sigma)$
(since the elements of $E^{ab}(\Sigma)$ elements are named), the coordinates
of the $q(x)$ for $x\in U_{D}$ and finitely many additional points of $F\cap
U$. Note that the additional points determine finitely many new field elements
since $q$ is constant on each orbit, so the field remains finitely generated.
Define a similar subsystem for ${\bf q}^{\prime}$, labeling by putting primes
on corresponding objects. By Lemma 4.2.8 every point of $D$ is either special
and so named in the vocabulary (Remark 4.3.6), or Hodge generic. Thus we can
ignore the special points in building the back and forth system.

Suppose $f$ is an isomorphism between $U\subseteq{\bf q}$ and
$U^{\prime}\subseteq{\bf q}^{\prime}$. Then $f$ restricts to a $G$-equivariant
(elements in the same orbit have the same image) injection of $U_{D}$ into
$U_{D^{\prime}}$ and an embedding of $S(L_{U})$ into $S(F^{\prime})$ induced
by an embedding $\sigma$ of $L$ into $S(F^{\prime})$, that fixes
$E^{ab}(\Sigma)$.

Note that the following claim is for arbitrary finite sequences ${\bf g}$, but
only singleton $x$. The type $r_{d}$ of an infinite sequence (here represented
by an infinite tuple of variables ${\bf v}$) includes the types of ${\bf g}x$
for any finite ${\bf g}$.

The main consequence of the following claim is that we may reduce types of
points in the domains sort to quantifier-free types of their images in the
field sort.

######  Claim 4.3.10.

[DH17, Prop 3.3] If $d\in D-U_{D}$ is Hodge generic:

$r_{d}({\bf v})\models{\rm tp}_{qf}(d/U),$

where $r_{d}({\bf v})=\bigcup_{{\bf g}\in G}{\rm tp}_{qf}(q({\bf
g}(d))/U)={\rm tp}_{qf}(\langle q(g(d)):g\in G\rangle/U)$.

###### Proof.

We show that there is a unique quantifier-free type over $U$ of an element of
$D$ that restricts to $r_{d}$. The consistent non-trivial types in $\tau_{G}$
are i) $\\{x\neq f:f\in U_{D}\\}$ and ii) $\\{x\neq gx\\}$ for any non-
identity $g\in G$. The first is captured by $(q(x),q(f))\not\in Z_{e,e}$ for
each $f\in U_{D}$ and the second by $(q(x),q(x))\not\in Z_{e,g}$ if
$g\not\in\Gamma$ and these are both in $r({\bf v})$.

Suppose ${\bf h}\in S(M)^{\omega}$ (for a saturated $M\models T({\mathbf{p}})$
containing $U$) realizes $r_{d}({\bf v})$ and ${\bf h}$ with $d^{\prime}\in
D(M)$ satisfy ${\bf h}=\langle q(g(d^{\prime})):g\in G\rangle$. By the
previous paragraph $d^{\prime}\not\in U_{D}$. So $d^{\prime}$ realizes ${\rm
tp}_{qf}(d/U)$ as required. ∎

######  Notation 4.3.11.

For a type $r(v)$ over a set $A$ and an isomorphism $f$ from $A$ to $B$,
$f(r)$ is the set of $B$-formulas $\phi(v,f({\bf a}))$ with $\phi(v,{\bf
a})\in r$.

######  Claim 4.3.12.

[DH17, Prop 3.4] Fix ${\bf g}$. If $x\in U_{D}$, there is an $x^{\prime}\in
U_{D^{\prime}}$ such that $q({\bf g}(x^{\prime}))\in S(F^{\prime})^{m}$
realizes $f({\rm tp}_{qf}(q({\bf g}(x))/L_{U}))$.

###### Proof.

We write $Z_{{\bf g}}^{{\bf q}}$ for the points in $S(F)$ satisfying (the
formula defining) $Z_{{\bf g}}$. Using Notation 4.3.11, Claim 4.3.10 implies
that the smallest algebraic subvariety $W_{{\bf g}}^{{\bf q}}$ of $S(F)^{n}$
that is defined over $L_{U}$ and contains $q({\bf g}({\bf x}))\in S(F)^{n}$
determines ${\rm tp}_{qf}({\bf g}({\bf x}))/L_{U})$. Since $Mod^{1}_{{\bf g}}$
is true in ${\bf q}$, $W_{{\bf g}}^{{\bf q}}\subseteq Z_{{\bf g}}^{{\bf q}}$.
But since (by Lemma 4.2.12) $Z_{{\bf g}}^{{\bf q}}$ is fixed setwise by
$\sigma$ (the map described after Definition 4.3.9) - being defined over
$E^{ab}(\Sigma)$, we have that $Z_{{\bf g}}^{{\bf q}^{\prime}}=Z_{{\bf
g}}^{{\bf q}}$, and therefore $W_{{\bf g}}^{{\bf q}^{\prime}}\subseteq Z_{{\bf
g}}^{{\bf q}^{\prime}}$. Now applying $Mod^{2}_{{\bf g}}$ in ${\bf
q}^{\prime}$, we find the required $x^{\prime}$. ∎

Having proved Claim 4.3.12, we can finish the argument. We need one more
crucial piece for the ‘forth’. What if $x\in D-U_{D}$? For this, we need ${\bf
q}^{\prime}$ to be $\omega$-saturated (realize all types over finite sets).

######  Theorem 4.3.13.

Suppose that ${\bf q}$ and ${\bf q}^{\prime}$ are $\omega$-saturated. Then the
$qf$-system described in Remark 4.3.9 is a back and forth; hence, $T({\bf p})$
admits elimination of quantifiers and is complete.

###### Proof.

Suppose $f$ is an isomorphism between $U\subseteq{\bf q}$ and
$U^{\prime}\subseteq{\bf q}^{\prime}$. Then $f$ restricts to a $G$-equivariant
injection of $U_{D}$ into $U_{D^{\prime}}$ and an embedding of $S(L_{U})$ into
$S(F^{\prime})$ induced by an embedding $\sigma$ of $L_{u}$ into
$S(F^{\prime})$, that fixes $E^{ab}(\Sigma)$.

For $x\in{\bf q}-U$, we must find $x^{\prime}\in U^{\prime}$ so that
$f\cup(x,x^{\prime})$ generates an isomorphism between the structures
generated by $U\cup\\{x\\}$ and $U^{\prime}\cup\\{x^{\prime}\\}$. If $x\in S$,
$x=q(\check{x})$ for some $\check{x}\in D$ so we restrict to that case. If
$x\in U_{D}$, $x^{\prime}$ exists as $U^{\prime}_{D}$ is closed under action
by $G$. Since the coordinates of special points are in $E^{ab}(\Sigma)$, whose
points are all named, for a special point $x$, $x^{\prime}$ must equal $x$.

The difficult case is when $x\in(D-U_{D})$ is Hodge generic. But we noted in
Claim 4.3.10 that it suffices to simultaneously realize all types ${\rm
tp}_{qf}((q(g_{1}x),\ldots q(g_{n}x))/U)$ for all ${\bf g}$ (of arbitrary
length). A slight variant on the argument for Claim 4.3.12 still holds if for
fixed $x$, we replace a single ${\bf g}$ by an arbitrary finite set of ${\bf
g}$. By compactness, the entire type is consistent and so satisfied in the
$\omega$-saturated ${\bf q}^{\prime}$. There is one final step. By induction
we have to choose $x^{\prime}$ for a sequence ${\bf x},{\bf y},x$ where ${\bf
x}\in U_{D}$ and ${\bf y}\in U_{S}^{k}$ for some $k$. But what if $x\in
U_{S}$? By Claim 4.3.10, ${\rm tp}_{qf}({\bf x},{\bf y})$ is determined by
${\rm tp}_{qf}({\bf g}({\bf x}),{\bf y})$ (in the field sort). That we can
choose of $x^{\prime}\in U^{\prime}_{S}$ to satisfy $f({\rm tp}_{qf}({\bf
g}({\bf x}),{\bf y}))$ is now immediate by $\omega$-saturation and quantifier
elimination in the field-sort.

By Karp’s theorem [Bar73, Theorem 3], the existence of the back and forth
implies all $\omega$-saturated models of $T({\bf p})$ are
$L_{\omega_{1},\omega}$ (indeed, $L_{\infty,\omega}$) elementarily equivalent.
Every model has an $\omega$-saturated elementary extension, so $T({\bf p})$ is
complete. ∎

###  4.4 Galois Representations and finite index conditions

In this section we begin by considering the action of discrete and Galois
groups on the domain and field sorts. Then we unite these approaches by
defining a Galois representation. We then state the key to establishing
categoricity, a consequence of Serre’s open mapping theorem.

####  4.4.1 Two views: domain and field sort

We explore the following diagram which links the domain sort (via the
quotient) with the field sort.

${\mathbb{H}_{\overline{h}}\approx\Gamma_{\overline{h}}\setminus\mathbb{H}}$${Z_{\overline{h}}}$${\mathbb{H}_{\overline{g}}\approx\Gamma_{\overline{g}}\setminus\mathbb{H}}$${Z_{\overline{g}}}$$\scriptstyle{[\phi_{\overline{h}}]}$$\scriptstyle{{\rm
id}_{\mathbb{H}_{\overline{g}}}}$$\scriptstyle{\psi_{\overline{h},\overline{g}}}$$\scriptstyle{[\phi_{\overline{g}}]}$

######  Convention 4.4.1.

${\bf g}=\langle e,g_{1}\ldots g_{n-1}\rangle$ has length $n$. We restrict to
${\bf g}$ with $\Gamma_{{\bf g}}\unlhd\Gamma$ (normal subgroup). Recall
$Z_{{\bf g}}\subseteq S(\mbox{\msbm C})^{\lg({\bf g})}$.

We have two views of ‘essentially’ the same map. The first moves to a quotient
on the domain side which is not $\tau$-definable; the second ‘names’ the range
of the first in the target side. We begin with quotient data but with
manifestations in both the domain and target.

Domain/Quotient data: The first view motivates $id$ for identity.

######  Definition 4.4.2.

Let ${\bf g}\subseteq{\bf h}$. Define $\rm id_{{\bf h}{\bf g}}:\mbox{\msbm
H}_{{\bf h}}\rightarrow\mbox{\msbm H}_{{\bf g}}$ by $[x]_{\Gamma_{\bf
h}}\mapsto[x]_{\Gamma_{\bf g}}$.

The normality hypothesis implies that $\Gamma_{\bf g}/\Gamma_{{\bf h}}$ acts
on $\mbox{\msbm H}_{{\bf g}}$: for $\lambda\in\Gamma_{\bf g}$,
$\lambda[x]_{\Gamma_{\bf g}}:=[\lambda x]_{\Gamma_{\bf g}}$, so the
representatives $\lambda_{i}$ of the cosets of $\Gamma_{{\bf g}}/\Gamma_{{\bf
h}}$ index the equivalence classes; thus the action is transitive.

Field data: We define the right hand column of the diagram.

######  Definition 4.4.3.

  1. 1.

For ${\bf g}\subseteq{\bf h}$, $\lg({\bf g})=n$, $\lg({\bf h})=m$, $\psi_{{\bf
h},{\bf g}}$, denotes the restriction of the natural projection from
$S(\mbox{\msbm C})^{m}$ onto $S(\mbox{\msbm C})^{m}$ to a map from $Z_{{\bf
h}}\subseteq S(\mbox{\msbm C})^{m}$ onto $Z_{{\bf g}}\subseteq S(\mbox{\msbm
C})^{n}$.

  2. 2.

Choose $z\in Z_{\bf g}$ and let $L=L_{z}$ be a finitely generated extension of
the defining field for $S$ such that $z$ is defined over $L$. Write
$\overline{L}$ for ${\rm acl}(L)$.

  3. 3.

Now, $\rm Aut(\mbox{\msbm C}/L)$ acts on the fiber of $\psi_{{\bf h},{\bf g}}$
over $z$, by its action on the coordinates of $z$; as it would for any
definable finite-to-one map from $Z^{m}_{{\bf h}}\rightarrow Z^{n}_{{\bf g}}$.

To connect the two sides, conjugating by $[\phi_{\bf h}]$, $\rm
Aut(\overline{L}/L)$ acts on $\rm id_{{\bf h}{\bf g}}^{-1}(z)$.

######  Lemma 4.4.4.

[Ete22, §3.5 p. 18] $\rm Aut(\mbox{\msbm C}/L)$ acts on the fiber of
$\psi_{{\bf h},{\bf g}}$ over $z$, (and so via $[\phi_{\bf h}]$ on $\rm
id_{{\bf h}{\bf g}}^{-1}(z)$). This action commutes with the action of the
free and transitive (simply transitive) action of $\Gamma_{\bf g}/\Gamma_{\bf
h}$ on the fibers of $\rm id_{{\bf h},{\bf g}}$. Thus we have a homomorphism
(Galois representation) $\rho^{z}_{{\bf g},{\bf h}}$ from $\rm
Aut(\overline{L}/L)$ into $\Gamma_{\bf g}/\Gamma_{\bf h}$.

####  4.4.2 Galois Representation

While the notion of a representation of a group $A$ frequently refers to
linear representations, a homomorphism of $A$ into a matrix group $B$, here we
will discuss specific examples of a more general notion: a representation of
$A$ is a homomorphism of $A$ into a group $B$. This is a Galois representation
if $A$ is the Galois group of one field over another. In Section 4.4.1, we
gave Galois representations of $\rm Aut(\overline{L}/L)$ into $\Gamma_{\bf
g}/\Gamma_{\bf h}$. In order to understand how to combine the actions of the
$\Gamma_{\bf g}/\Gamma_{\bf h}$ as ${\bf g},{\bf h}$ vary, we need the notion
of inverse limit.

######  Definition 4.4.5 (Inverse Limit).

Given a directed set $(I,\leq)$ an inverse system on $I$ is a family of
structures $\langle A_{i}:i\in I\rangle$, and for $i<j$, maps $f_{ij}$ from
$A_{j}$ to $A_{i}$ such that $i<j<k$ implies $f_{ij}\circ f_{jk}=f_{ik}$.

An inverse limit of this inverse system is an object
$\hat{A}=\underleftarrow{lim}A_{i}$ and a family of morphisms
$g_{i}:\hat{A}\rightarrow A_{i}$ such that (1) for all $i<j$ in $I$,
$f_{ij}\circ g_{j}=g_{i}$ and (2) given any $A^{\prime}$ and family
${g^{\prime}}_{i}$ satisfying (1) there is a unique morphism
$h:\hat{A}\rightarrow A^{\prime}$ such that for all $i\in
I,{g^{\prime}}_{i}=g_{i}\circ h$.

######  Definition 4.4.6.

Galois Representations of Inverse Limits We work with a modular curve
$S(\mbox{\msbm C})=\Gamma\setminus\mbox{\msbm H}$ which is defined over
$E^{ab}(\Sigma)$. (Notation 4.3.3). Since each $\Gamma{\bf g}\subseteq\Gamma$,
$\rho^{z}_{{\bf g},{\bf h}}:\rm Aut(\overline{L}/L)\rightarrow\Gamma$ and by
taking an inverse limit of the representations $\rho^{z}_{{\bf g},{\bf h}}$,
we obtain:

$\rho^{z}\colon{\rm Gal}(\overline{L}/L)\rightarrow\overline{\Gamma}$

where $\overline{\Gamma}=\underleftarrow{\lim}_{\bf h}\ \Gamma/\Gamma_{{\bf
h}}$. The ${\bf h}$ range over all finite sequences as Convention 4.4.1. See
Definition 4.4.5 and [Ete22, §3.6 p 17].

For any groups $H_{1}\leq H_{2}$ that act on a set $X$ the $H_{1}$-orbits of
$X$ partition the $H_{2}$-orbits. So if $[H_{2}:H_{1}]$ is finite and $H_{2}$
is infinite, the obits will have the same cardinality and the smaller
$[H_{2}:H_{1}]$ is, the closer we are to an isomorphism.

Now, we can state the first of two crucial sufficient conditions for
categoricity.

######  Definition 4.4.7.

First Finite Index Condition (FIC1) The first finite index condition is
satisfied by a modular curve $p\colon\mbox{\msbm H}\rightarrow S(\mbox{\msbm
C})$ if:

For any non-special points $x_{1},\ldots x_{m}\in\mbox{\msbm H}$ in distinct
$G$-orbits (Definitions 4.4.2, 4.4.3) and for any field $L$ containing the
field over $E^{ab}(\Sigma)$ along with the coordinates of the $p(x_{i})$, the
image of the induced homomorphism $\rho:{\rm
Gal}(\overline{L}/L)\rightarrow\overline{\Gamma}^{m}$ has finite index in
$\overline{\Gamma}^{m}$.

Recall from Lemma 4.3.10 that

$r_{d}({\bf v})\models{\rm tp}_{qf}(d/U).$

where $r_{d}({\bf v})=\bigcup_{{\bf g}\in G}{\rm tp}_{qf}(q({\bf
g}(d))/U)={\rm tp}_{qf}(\langle q(gd):g\in G\rangle/U)$. The argument for
Lemma 4.3.10 began with the observation that $r_{d}({\bf v})$ implied, in
particular, that $d\not\in D_{U}$, so $d$ is an independent Hodge generic. We
will deduce from Lemma 4.4.8 that (under FIC1) only finitely many tuples ${\bf
g}$ from $r_{d}$ are really needed.

######  Lemma 4.4.8.

Assume FIC1. Then, for each $z$, for some $\hat{\bf g}$, the map

$\rho_{z}:\rm Aut(\overline{L}/L_{\hat{\bf
g}})\mapsto\overline{\Gamma}_{\hat{\bf g}}^{m}=\underleftarrow{lim}_{{\bf
h}\supseteq{\hat{\bf g}}}({\Gamma_{\hat{\bf g}}/\Gamma_{\bf h}})^{m}$

is surjective.

###### Proof.

Let $I=\mathrm{im}(\rho_{z})$ and let $k=[\overline{\Gamma}:I]$. Suppose not.
Choose $\hat{{\bf g}}$ with ${\bf g}\subseteq{\hat{{\bf g}}}$ such that
$[\Gamma_{{\bf g}}:\Gamma_{\hat{\bf g}}]=k$. Thus, for any ${\bf
h}\supseteq{\hat{\bf g}}$, $\rho_{z}$ must be onto $\Gamma_{\hat{\bf
g}}/\Gamma_{{\bf h}}$. For, if not, there is an $\eta\in\Gamma_{\hat{\bf
g}}/\Gamma_{{\bf h}}$ and that is not in $I$; it must be in a new coset of $I$
in $\overline{\Gamma}$, contrary to the choice of $\hat{{\bf g}}$. ∎

######  Corollary 4.4.9.

(Under FIC1) For $d\in D-U$,

${\rm tp}_{qf}(q(\hat{\bf g}(d))/U)\models r_{d}({\bf v})\models{\rm
tp}_{qf}(d/U).$

###### Proof.

The second implication is Lemma 4.3.10. For the first, choose any ${\bf
h}\supseteq\hat{\bf g}(d)$ and let $m=\lg(\hat{\bf g})$, $r=\lg({\bf h})$. Let
$\mathcal{F}\subseteq Z^{r}_{\bf h}$ be the fiber over $\hat{\bf
g}(d^{\prime})\in Z^{m}_{\hat{\bf g}}$ of the finite-to-one map $\psi_{{\bf
h}\hat{\bf g}}:Z^{r}_{\bf h}\rightarrow Z^{m}_{\hat{\bf g}}$. Similarly, ${\rm
tp}_{qf}({\bf h}(d)/L_{U})$ is determined by the $\rm Aut(\mbox{\msbm
C}/L_{U})$-orbit $\mathcal{G}\subseteq\mathcal{F}$ containing ${\bf h}(d)$.
Then, ${\rm tp}_{qf}({\bf h}({\bf x})/L_{U})$ is determined by the $\rm
Aut(\mbox{\msbm C}/L_{U})$-orbit $\mathcal{G}\subseteq\mathcal{F}$ containing
${\bf h}({\bf x})$. But $\mathcal{G}=\mathcal{F}$, since $\rho_{z}$ induces a
homomorphism from $\rm Aut(\mbox{\msbm C}/L_{U})$ onto $\Gamma_{\hat{\bf
g}}/\Gamma_{{\bf h}}$ and $\Gamma_{\hat{\bf g}}/\Gamma_{{\bf h}}$ acts
transitively on the fiber. Since this holds for any such ${\bf h}$, we finish.
∎

We turn now to the infinitary axioms that are needed to obtain categoricity.

######  Notation 4.4.10 (Infinitary Axioms).

  1. 1.

$\Phi_{\infty}$ is the $L_{\omega_{1},\omega}$ sentence asserting that for
$(D,S,q)$ both the dimension of the field bi-interpretable with $S$ and of the
strongly minimal structure $\langle D,\\{f_{g}:g\in\Gamma\\}\rangle$ are
infinite.

  2. 2.

$SF$ (standard fibers) denotes the $L_{\omega_{1},\omega}$-axiom:

$(\forall x\forall y\in
D)(q(x)=q(y)\rightarrow\bigvee_{g\in\Gamma}x=f_{g}(y)).$

  3. 3.

$T^{\infty}({\bf p})$ denotes $\mathop{\rm Th}({\bf
p})\cup\\{\Phi_{\infty}\\}$ and

  4. 4.

$T^{\infty}_{SF}({\bf p})$ denotes $\mathop{\rm Th}({\bf
p})\cup\\{SF\\}\cup\\{\Phi_{\infty}\\}$.

######  Definition 4.4.11.

For $\langle D,S(F),q\rangle\models T^{\infty}_{SF}({\bf p})$ and $X\subset
D\cup S(F)$,

${\rm cl}(X)=q^{-1}({\rm acl}(q(X)))$

where ${\rm acl}$ is the field algebraic closure in $F$.

An essential consequence of the standard fibers axiom is that Definition
4.4.11 defines an almost quasiminimal closure relation satisfying the
countable closure condition from Definition 3.2.4. This closure dimension
restricts on the separate sorts to the dimension of the constituent strongly
minimal sets that is expressed in $\Phi_{\infty}$. This accomplishes the aim
of an ($L_{\omega_{1},\omega}$-complete so $\aleph_{0}$-categorical)
$L_{\omega_{1},\omega}$ theory with arbitrarily large models.

A class K  of models has $\aleph_{0}$-homogeneity over $\emptyset$ (Definition
3.2.5) (the precise statement is from [Ete22, p 4]) if the models of K  are
pairwise qf-back and forth equivalent (Definition 4.3.7).

######  Theorem 4.4.12.

[DH17, Theorem 4.11] If the standard model ${\bf p}$ of a modular curve
satisfies FIC1, then the class of models of $T^{\infty}_{SF}({\bf p})$ is
$\aleph_{0}$-homogenous over $\emptyset$. In particular, by Karp [Kar64,
Bar73], all models of $T^{\infty}_{SF}({\bf p})$ are back and forth equivalent
and so satisfy the same sentences of $L_{\omega_{1},\omega}$.

###### Proof.

Our task is to replace the $\omega$-saturation hypothesis from Lemma 4.3.13 by
adding the infinitary axioms and the condition FIC1. As in the proof of
theorem 4.3.13 we need only worry about Hodge generic points. Suppose we have
a partial function $f$ from ${\bf q}$ to ${\bf q}^{\prime}$ with domain and
range $U$ and $U^{\prime}$ as in Lemma 4.3.13 between models ${\bf q}$ and
${\bf q}^{\prime}$ of $T^{\infty}_{SF}({\bf p})$. Proceed as in the proof of
the second paragraph of Lemma 4.3.13. We vary the argument for the ‘difficult
case’ from the 3rd paragraph. Choose $\hat{\bf g}$ by Lemma 4.4.8. Taking
$\hat{\bf g}$ for the ${\bf g}$ in Lemma 4.3.12, for $x\in U_{D}$, there is an
$x^{\prime}\in U_{D^{\prime}}$ such that (*) $q(\hat{\bf g}(x^{\prime}))\in
S(F^{\prime})^{m}$ realizes $f({\rm tp}_{qf}(q(\hat{\bf g}(x))/L_{U}))$. We
want to show that the same choice $x^{\prime}$ satisfies (*) for every ${\bf
h}\supseteq\hat{\bf g}$. This is immediate from Lemma 4.4.9. The argument is
completed by induction as in the ‘final step’ of the proof of Lemma 4.3.13.

∎

######  Remark 4.4.13 (FIC2).

Like FIC1, FIC2 is a finite index condition on Galois representations into
inverse limits. Now, however there are independence conditions over the ground
field. [DH17, Condition 4.8] provides sufficient conditions so that a minor
modification of the proof of Theorem 4.4.12, shows FIC2 implies homogeneity
over models; pairs of models are back and forth equivalent over a countable
submodel. This is the first place in the argument where types over countable
algebraically closed fields rather than the empty set (i.e. a fixed countable
field) are encountered. Combining this result with Theorem 4.4.12, the
homogeneity conditions are now stronger than those defining quasiminimal
excellence in [BHH+14]. Thus, we apply that paper and obtain:

######  Theorem 4.4.14.

For any modular curve interpreted as a standard model ${\bf p}$ (Definition
4.3.3) for $T^{\infty}({\bf p})$, $T^{\infty}({\bf p})$ is almost quasiminimal
excellent and so categorical in every infinite power.

###### Proof.

We need only that FIC1 and FIC2 hold for all modular curves. This is proved in
[DH17, §5], where the proof for FIC1 relies heavily on [Ser72, §6] and FIC2 on
[Rib75]. ∎

With further effort they extend this result to Shimura curves.

######  Remark 4.4.15.

Keisler’s theorem [Kei70, Corollary 5.10] and work of Shelah [Bal09, §7] show
that an $\aleph_{1}$-categorical sentence $\phi$ of $L_{\omega_{1},\omega}$
not only has only countably many types in any countable fragment of
$L_{\omega_{1},\omega}$ containing $\phi$ (Keisler) but has a
completion555That is, a sentence $\phi^{*}$ that implies $\phi$ and decides
every $L_{\omega_{1},\omega}$-sentence. (Shelah). Equivalently, the completion
must specify the isomorphism type of the countable model. The only such
completion consistent with having an uncountable model is adding
$\Phi_{\infty}$.

We have used FIC1 to prove categoricity in all powers. In fact,
$\aleph_{1}$-categoricity implies FIC1. For this, [DH17, Ete22] argue that the
weaker hypothesis of having just countably many types over the empty set in
the theory $T^{\infty}_{SF}$ implies FIC1. If FIC1 holds, for some $z$, by
Lemma 4.4.8, for every ${\bf g}$, there is ${\bf h}\supseteq{\bf g}$ with a
$\Gamma_{\bf g}/\Gamma_{\bf h}$-orbit contained in $\psi_{{\bf h}{\bf
g}}^{-1}(z)$ that projects to that $\Gamma_{\bf g}$ orbit. So under the
assumption that FIC1 fails, there is a ${\bf g}$, such that for every ${\bf
h}\supseteq{\bf g}$ there are distinct $\Gamma_{\bf g}/\Gamma_{\bf h}$-orbits
$O_{1},O_{2}$ contained in $\psi_{{\bf h}{\bf g}}^{-1}(z)$ that project to the
same $\Gamma_{\bf g}$-orbit.

By Lemma 4.3.10, if two points are Galois equivalent they realize the same
quantifier free $\tau$-type; so $O_{1},O_{2}$ realize distinct Galois orbits
(and so any two orbits that project to them must realize distinct
$\tau$-types). But since $\overline{\Gamma}$ acts transitively on each $Z_{\bf
g}$, there is a complete tree of splittings of $\rm Aut(\mbox{\msbm C}/L)$
orbits that all project to $z$. This contradicts Keisler’s theorem. So
$\aleph_{1}$-categoricity of $T^{\infty}_{SF}$ implies FIC1.

######  Remark 4.4.16.

[DH17, §5], using both Serre’s open mapping theorem [Ser72, §6] for the finite
index condition and work by [Rib75] on Shimura curves show FIC1 and FIC2 hold
for all modular and Shimura curves. So our remaining sections concern higher
dimensional varieties. FIC1 is known for some higher dimensional Shimura
varieties and conjecturally for others, while FIC2 is true for all [Ete22].

[DH17] use both to prove categoricity. Since the Galois group is not
accessible in our formal language, FIC1 cannot be directly expressed in the
two-sorted theory. So the goal of a ‘fully formal invariant’ cannot be
achieved unless explicit reliance on the finite index conditions as an
hypothesis is avoided.

##  5 First order Excellence

Here is the opening paragraph of [BHP20].

> _Let $\mbox{\msbm G}=\mbox{\msbm G}^{n}$ be a complex algebraic torus, or
> let 𝔾 be a complex abelian variety. Considering $\mbox{\msbm G}(\mbox{\msbm
> C})$ as a complex Lie group, with $\bf L\mbox{\msbm G}=T_{0}(\mbox{\msbm
> G}(\mbox{\msbm C}))$ its (abelian) Lie algebra, the exponential map provides
> a surjective analytic homomorphism_
>
> $\exp:\bf L\mbox{\msbm G}\twoheadrightarrow\mbox{\msbm G}(\mbox{\msbm C}).$

In the spirit of Zilber, their paper aims at finding ‘algebraic descriptions’
of the cover $\exp$ which characterize the standard structure (at least up to
categoricity in power). They solve a more general problem by providing a first
order theory $\hat{T}$ for the situation and showing each model $\tilde{M}$
($\hat{M}$ here) of $\hat{T}$ is determined by relations among two designated
substructures and a certain transcendence degree. In this generality, the
result is proved for any abelian group of finite Morley rank (henceforth fmr
groups). Then, under slightly stronger hypotheses, the result becomes a true
categoricity result for, in particular, an abelian variety defined over a
number field.

We address in this section four new ingredients: formalized non-standard
covers, ‘first order excellence’, Kummer theory, and a distinction between
classification and categoricity. First order excellence appears to be both
necessary and applicable for higher order Shimura varieties.

As noted in [BHP20], the quasiminimal approach studied earlier in this paper
suffices to prove the $L_{\omega_{1},\omega}$-categoricity in power for
Abelian varieties. The goal of this section is to identify the distinctive
elements of the [BHP20] proof that later reappear in [Ete22].

###  5.1 The two-sorted structure and fmr groups

A first order theory $T$ is stable in $\kappa$ if any $M\models T$, with
$|M|=\kappa$, $|S(M)|=\kappa$. ($S(M)$ denotes the set of $1$-types over $M$.)
Morley showed that $\omega$-stability (more properly, $\aleph_{0}$-stability)
of a theory $T$ is equivalent to stability in all powers (and also to the
Morley rank having an ordinal value for each type). We need here a slightly
weaker condition called superstability: $T$ is stable in $\kappa$ if
$\kappa\geq 2^{\aleph_{0}}$.

The theory of $(\mathbb{Z},+)$ is one of the prototypical strictly superstable
theories666The other one is the theory of countably many equivalence relations
$E_{n}$ such that for each $n$, each $E_{n}$-class is split into infinitely
many $E_{n+1}$-classes (and $E_{n+1}\subseteq E_{n}$). (that is, superstable,
but not $\aleph_{0}$-stable). One can fix arbitrarily the congruence class of
an element $x$ for each $n$. This gives $2^{\aleph_{0}}$ distinct types
realized by non-standard integers.

There is an extensive theory of fmr groups (see [BN94, ABC08]). We need here
only the basics. In particular, Macintyre’s result [Mac70] that an
$\omega$-stable group is divisible by finite. We now introduce the two-sorted
theory; with that notation we are able at the end of this section to outline
the main steps of the proof.

Unlike [DH17] where $\underleftarrow{\lim}\ Z_{\bf g}$ is in the background of
the proof of (our) Theorem 4.4.12 but not the statement, [BHP20] build the
structure of non-standard covers into the vocabulary of the two sorted
structure by the $\rho_{n}$ below.

[BHP20, §2.2] use the inverse limit of Definition 5.1.1 for divisible abelian
groups; although it is not profinite, they refer to it as a profinite
universal cover denoted $\hat{G}$ of $G$ and $G$ is renamed as $M$. Although
the hat has only one meaning in [BHP20], it becomes overloaded here so we
denote the inverse limit defined below as $\tilde{M}$. While in [BHP20] a
typical 2-sorted (3-sorted in §8) structure $\hat{\tau}$ is represented as
either $(\tilde{M},M)$ or $\tilde{M}$, we write $\hat{M}=(\tilde{M},M)$ and
$\tilde{M}$ for the or (profinite cover) inverselimit from [BHP20, 1.2, 2.1]
as that is the actual usage in most of the cited paper.

######  Definition 5.1.1 ($\tilde{M}$).

Given a commutative, divisible, abelian group $(M,+)$, consider the inverse
limit $\tilde{M}=\underleftarrow{\lim}\ M_{n}$ of isomorphic copies $M_{m}$ of
$M$ with the index set partially ordered by $m\leq n$ if and only $m|n$ and
with maps $\eta_{nm}$ (multiplication by $\frac{m}{n}$) taking $M_{n}\mapsto
M_{m}$. Concretely, $(\tilde{M},+)$ is the subgroup of the direct product of
$\omega$ copies of $M$, containing those sequences ($\langle g_{k}:1\leq
k<\omega\rangle$) such that if $k=nm$, $g_{m}=n\times g_{k}$ and
$g_{n}=m\times g_{k}$.

######  Notation 5.1.2 (The vocabulary $\hat{\tau}$).

Let 𝔾 be the given abelian group and $T:=\mathop{\rm Th}(\mbox{\msbm G})$ in a
large enough countable language that $T$ has quantifier elimination. Further,
let $\hat{T}$ be the theory of $(\hat{}\mbox{\msbm G},\mbox{\msbm G})$ in the
two-sorted language $\hat{\tau}$ consisting of the maps
$\rho_{n}:\hat{}\mbox{\msbm G}\rightarrow\mbox{\msbm G}$ for each $n$, the
theory $T$ and, for each ${\rm acl}^{eq}(\emptyset)$-definable subgroup $H$ of
$G$, a predicate $H$ for $H$ and a predicate $\hat{H}$ for
$\\{x\in\hat{G}:\rho_{n}(x)\in H,n\in\mbox{\msbm N}\\}$.

Although the kernel of $\rho=\rho_{1}$ is definable in the vocabulary given, a
further predicate $\rm ker^{0}$ is included denoting the divisible part of the
kernel (otherwise, it is only type-definable).

The axioms [BHP20, 2.5] of $\hat{T}$ are chosen so that

######  Theorem 5.1.3.

[BHP20, 2.7, 2.8, 2.21] For an fmr group 𝔾, $(\underleftarrow{\mbox{\msbm
G}},\mbox{\msbm G},\rho_{0})\models\hat{T}$ and therefore $\hat{T}$ admits
quantifier elimination and is superstable of finite $U$-rank.

Although the $T$ in Notation 5.1.2 is $\omega$-stable, $\hat{T}$ is only
superstable; also, many elements of $\rm ker(\rho)$ are not divisible in $\rm
ker(\rho)$.

######  Remark 5.1.4 (Quasiminimality, unidimensionality, notop).

Abelian varieties as opposed to fmr groups, can be handled either by the
quasiminimality methods of Section 4 or by the methods described in this
section. A crucial distinction from Section 4 is that the former considered
only the theory of unary functions from a group acting on the domain, while
here we have the full group structure.

To explain the fmr proof we need some further model theoretic background. In
general two types $p,q$ over $M$ are orthogonal when in different models $N$
extending $M$ the number of realizations of $p$ and $q$ can be varied
arbitrarily. Non-orthogonality for strongly minimal sets has a particularly
clear meaning. The strongly minimal sets $D_{1}$ and $D_{2}$ are non-
orthogonal if there is a definable finite to finite binary relation on
$D_{1}\times D_{2}$. A theory is unidimensional if all types are non-
orthogonal.

The three features that underlie the [BHP20] proof are.

  1. 1.

A fmr abelian group has finite width [Bal88, XV.1] (aka almost
$\aleph_{1}$-categorical [Las85]): Any model is the algebraic closure of the
union of the bases of a collection of strongly minimal $D_{i}$ for
$i<n<\omega$. The $D_{i}$ are defined over the prime model (the unique up to
isomorphism model elementarily embedded in every model of the theory).

  2. 2.

In models of $\hat{T}$ with $M_{0}$ the prime model of $T$ and where 𝔾 is
defined over a number field $k_{0}$, Kummer theory allows the control of
$\rho^{-1}(M_{0})$ by the kernel $\rho^{-1}(0)$.

  3. 3.

In studying Abelian varieties the $n$ in 1) can be taken as $1$ because the
variety is interalgebraic with an algebraically closed field and so almost
strongly minimal ($M={\rm acl}(D)$ for strongly minimal $D$).

Since Kummer theory doesn’t apply to arbitrary Shimura varieties, both 2) and
3) fail for more general higher dimensional Shimura varieties (see Section 6).

###  5.2 First order Excellence and fmr groups

Shelah’s main gap program defines a sequence of properties $X$ of countable
first order theories forming a sequence of dichotomies [Bal18, §5.5] such
that: if $T$ satisfies $X$, $T$ has the maximal number of models in every
uncountable cardinal. If $T$ fails $X$, the models of $T$ satisfy conditions
useful for classification. (e.g. stability implies the existence of the ‘non-
forking’ independence relation). The positive side of the final dichotomy in
the sequence is superstable without the omitting types order property (denoted
notop). Under this hypothesis, Shelah ([She90] and earlier papers) showed that
an appropriate class of models of $T$ had a notion of independence among
structures with $n$-amalgamation for all $n$ that yields the classification of
models. Hart [Har87] reduced the amalgamation requirement to $2$-amalgamation
and this reduction was extended to the quasiminimal excellent case in
[BHH+14]. In Section 6, we note this ‘notop’ approach is used to study higher
dimensional Shimura varieties.

In Section 3 of [BHP20] the techniques of [Har87] are adapted to the specific
framework here to establish a decomposition of models of $\hat{T}$ analogous
to that in Remark 5.1.4 for models of $T$. This yields

######  Theorem 5.2.1.

[BHP20, Theorem 3.31] Each model $\hat{\mbox{\msbm M}}$ of $\hat{T}$ is
determined up to isomorphism by the transcendence degree of the algebraically
closed field $K$ such that $M\cong\mbox{\msbm G}(K)$, the isomorphism type of
the inverse image, $\hat{M_{0}}$, of the prime model $M_{0}$ of $T$, and the
isomorphism type of $M$ over $M_{0}$.

###  5.3 Abelian Varieties

From the model theoretic standpoint, an Abelian variety is a complete
algebraic variety whose points form a group such that the group operations are
definable in the ambient field. For Abelian varieties, Kummer theory
eliminates (as in [Gav08, BGH14]) the reliance in Theorem 5.2.1 on knowing the
isomorphism type of $\check{M}_{0}$ over the kernel. The situation described
in the opening paragraph of § 5 is a special case. Namely, let 𝔾 be (the
formula defining) an abelian variety $\mbox{\msbm G}(K)$ over a field $K$ as
in the introduction to Section 5. Assume $\mbox{\msbm G}(\mbox{\msbm C})$ and
its ring of endomorphisms are definable over a number field $k_{0}$. With this
notation:

######  Theorem 5.3.1.

[BHP20, Theorem 4.6] a model $\hat{M}=\langle\tilde{M},M,q\rangle$ of
$\hat{T}$ is determined up to isomorphism by the transcendence degree of the
algebraically closed field $K$ such that $M\cong\mbox{\msbm G}(K)$, and the
$\hat{\tau}$ isomorphism type of $\rm ker\rho$.

######  Remark 5.3.2 (Complete formal invariant).

Theorem 5.3.1 gives categoricity in all uncountable cardinalities by adding
the $L_{\omega_{1},\omega}$ sentence characterizing the standard kernel. But
Theorem 5.3.1 is more general than categoricity; it shows that models with
non-standard (possibly uncountable) kernel are characterized by the
$\hat{\tau}$-diagram of the kernel. Of course, this statement cannot be
formalized in languages with bounded length of conjunctions since the kernels
can be arbitrarily large. But Zilber’s goal (just after Notation 1.0.1) only
aimed at complete formal characterization for prototypical mathematical
structures.

##  6 Higher Dimensional Shimura Varieties

A Shimura variety is a higher-dimensional generalization of a modular curve
that arises as a quotient variety of a Hermitian symmetric space $X^{+}$ by a
congruence subgroup of a reductive algebraic group defined over ℚ. We consider
Shimura varieties that are moduli spaces for generalized algebraic varieties.
Rather than discussing further technical details on the definition of a
Shimura datum $(G,X)$, we survey the differences that arise in generalizing
the results in Remark 4.4.16 about Shimura curves to higher dimensional
Shimura varieties: $S(\mbox{\msbm C})=\Gamma\setminus X^{+}$.

Central difficulties arise directly from the higher dimension in two ways.
First, in the curve case the $2$-sorted structure is (almost)-quasiminimal
because the variety in field sort is a curve and so strongly minimal and the
geometric closure on the cover sort is given by $a\in{\rm cl}(X)$ if $a\in
q^{-1}({\rm acl}((q(X))$. Quasiminimality can fail in the higher dimensions.
Second, rather than special points which are fixed points of some $g$, one
must treat special subvarieties [Ete22, §3.4] and finite unions thereof,
special domains. The fact that these are not merely points leads to several
difficulties.

  1. 1.

The structure of the covering sort is no longer strongly minimal. Even after
naming the elements of the group the special subvarieties give a complicated
structure on the covering sort.

  2. 2.

In the curve case the intersection of special domains was a point; that may
fail in higher dimensions.

  3. 3.

The theories of two inverse limit structures $\hat{\bf p}$ and $\tilde{\bf p}$
are considered as the covering space. The first structure is the analog of
$\underleftarrow{\lim}\ Z_{{\bf g}}$ (Definition 4.2.11). The second consists
only of the standard points of this limit. The canonical universal cover ${\bf
p}$ satisfies the first order $\mathop{\rm Th}(\tilde{\bf p})$ but not in
general $\mathop{\rm Th}(\hat{\bf p})$ [Ete22, Example 5.7, Corollary 5.14].

  4. 4.

An $L_{\omega_{1},\omega}$ categorical axiomatization is not claimed. Each
model can be precisely characterized but the characterization is not in
$L_{\omega_{1},\omega}$. See Remark 5.3.2.

  5. 5.

Finally, even this characterization depends on whether the variety under
consideration satisfies finite index conditions as in the modular case.
Although FIC1 and FIC2 are true in the modular curve case, here the truth of
$FIC1$ for ${\bf p}$ is actually equivalent to the characterizability of
models of $T^{\inf}_{SF}({\bf p})$ since [Ete22] shows FIC2 is true.

##  7 Model Theory and Analysis

One can signal three different model theoretic approaches to analysis:

  1. 1.

Axiomatic analysis studies behavior of fields of functions with operators but
without explicit attention in the formalism of continuity but rather to the
algebraic properties of the functions. The function symbols of the vocabulary
act on the functions being studied; the functions are elements of the domain
of the model.

Example: $DCF_{0}$ as discussed below.

  2. 2.

Definable analysis has a lower level of abstraction; the domain of the
functions remains the universe of the model. The functions being studied are
the compositions of the functions named in the vocabulary; one cannot quantify
over them.

Example: $o$-minimality.

  3. 3.

Implicit analysis Attempts to provide ‘algebraic characterizations of
important mathematical structure by axiomatizations in infinitary logic that
are categorical in power. Example: the material in this paper.

The first two are discussed in [Bal18, §6.3]. The work expounded in this paper
has many commonalities with a prime example of axiomatic analysis: the study
of transcendence results for solutions of differential equations by the study
of the $\omega$-stable theory $DCF_{0}$ of differentially closed fields of
characteristic zero. The notion of ‘not integrable by elementary functions
(Painlevé said ‘irreducible’) is formalized by ‘the solution set is strongly
minimal’ [Nag14]. The study of Schwartzian equations provides a general
framework in which the $j$-function and modular curves are explored. The work
includes, variations on the Ax-Lindemann-Weierstrass theorem, proofs that
Generic differential equations are strongly minimal [DF23] and Differential
Chow Varieties are Kolchin-constructible [FLS17], and analysis of strongly
minimal solution sets defined by differential equations in terms of the Zilber
trichotomy and $\aleph_{0}$-categoricity.

But while the mathematical topics are the same, the aims are different: The
covers project tries to assign a categorical description of each cover. The
$DCF_{0}$ approach tries to understand transcendence results for solutions of
the differential equations.

The crucial methodological difference is the two-sorted nature of the cover
program. The axiomatic analysis framework is preserved in that there is no
explicit treatment of convergence or continuity. But connecting the domain and
target by quotients under an explicit group action as well as the use of
infinitary logic provides tools not available in the earlier examples of
axiomatic analysis.

##  8 Families of covers of algebraic curves

In recent work Zilber and Daw [DZ22b, DZ22a] deal with _families_ of covers of
curves. They build on earlier constructions we have discussed in this paper.
Rather than a cover of a _single_ variety, albeit one that parameterized a
family of varieties, an entire family of such covers is studied and the
covering space becomes an analytic Zariski structure [Zil10]. In [Zil22] the
analysis of families is generalized by being placed in a geometric algebraic
setting.

The most salient difference between these works and those discussed earlier in
this paper is that, rather than a cover of a _single_ variety, an _entire
family of covers_ is now the main subject. Our earlier Definition 4.2.9 is now
replaced by a basic vocabulary consisting of _three_ sorts, together with maps
$\Gamma_{N}\setminus\mbox{\msbm H}\mapsto\mbox{\msbm C}$ covering a family of
curves $S_{N}(\mbox{\msbm C})$.

###  8.1 Pseudo-analytic covers of modular curves

Major differences of paper [DZ22b] from the earlier discussion of modular
curves include:

  1. 1.

The basic vocabulary is now $3$-sorted. More specifically, [DZ22b] considers
structures $(D,G,j_{N},\mbox{\msbm C})$ where the $j_{N}:\mbox{\msbm
H}\twoheadrightarrow S_{N}(\mbox{\msbm C})$. The discrete group is now given
as a third sort incorporating _a group operation_ (so its pregeometry is
locally modular, rather than trivial). This sort contains group with
distinguished subsets777$E$ is the elliptic Möbius transformations and the
${\bf d}_{{\bf q}},{\bf d}_{{\bf q}}$ are specific diagonal matrices.
$(\mathop{\rm GL}^{+}_{2}(\mbox{\msbm Q}),\times,\mathop{\rm
SL}_{2}(\mbox{\msbm Z}),E(\mbox{\msbm Q}),\\{{\bf d}_{q},{\bf
d}^{\prime}_{q}:q\in\mbox{\msbm Q})\\}$, where $E$ is the collection of
elliptic elements of the group; those that have unique fixed points. This
structure is specified up to isomorphism by a sentence of
$L_{\omega_{1},\omega}$. But not all group elements are still named in the
formal language.

  2. 2.

The uniformizing functions $j_{N}$ each map into
${\mathbf{P}}^{3}(\mathbb{C})$ rather than into the arbitrarily high
dimensional spaces of the maps $[\phi_{{\bf g}}]$ in [DH17, Ete22].
Furthermore, these are now defined over $\mathbb{Q}$ rather than over
$E^{ab}(\Sigma)$.

  3. 3.

As well as an almost quasiminimal axiomatization of the $3$-sorted structure,
the domain is considered as a Zariski Analytic set with a quasiminimal
geometry. Both of these structures are shown to be uncountably categorical.

  4. 4.

The special points _are not named_. However as in Definition 4.3.5 they are
uniquely associated with elliptic elements of the group.

In many ways, this last distinction is the most important for the general
program, as naming of the special points trivializes some of the arithmetic.
In [DZ22b], the structure of the _family_ is proved to be categorical in all
uncountable cardinalities.

###  8.2 Locally o-minimal covers of algebraic varieties

The paper [Zil22] takes a more general approach. It abstracts away from naming
all elements of the discrete groups as earlier in this paper. The relations
among the universal and finite covers are given more abstractly as properties
of maps from a domain (whose smoothness is defined topologically and
geometrically but not algebraically) onto families of algebraic varieties.
This smoothness as well as the eventual quasiminimality for curves888The set-
up is for arbitrary algebraic varieties, but the categoricity result is only
for curves and we restrict to that case. is controlled by _external_ o-minimal
structures.

######  Remark 8.2.1.

  1. 1.

The formalization is new. For a fixed model $\mathsf{R}$ of the theory $T$ of
a fixed o-minimal expansion of the reals (e.g the restricted analytic
functions) a structure $\mbox{\msbm U}(\mathsf{R})$ is defined. The resulting
structure $\mbox{\msbm U}(\mathsf{R})$ is an abstract Zariski
structure999Actually, $\mbox{\msbm U}(\mathsf{R})=U(K)$ where $\mathsf{K}$ is
taken as an algebraically closed field ${\mathsf{R}}+i{\mathsf{R}}$ and $U(R)$
is constructed analogously to $U(\mbox{\msbm C})$..

  2. 2.

Generalizing the last paragraph of Section 8.1, in the standard model the
domain is a complex manifold $\mbox{\msbm U}(\mbox{\msbm C})$ with holomorphic
maps $f_{i}$ onto algebraic varieties $X_{i}(\mbox{\msbm C})$ with natural
projections ${\rm pr}_{i,j}$ among the $X_{i}$. These analytic properties are
definable using theory of $\mathsf{K}$-analytic sets in o-minimal expansions
of the reals developed in [PS08, PS10]. We fix $k\subseteq\mathbb{C}$ a
subfield over which the varieties $\mathbb{X}_{i}$ are all defined.

  3. 3.

The ostensibly two-sorted structure of 1) becomes one-sorted because the field
can be interpreted in the abstract Zariski structure. And the third sort of
Section 8.1 has disappeared because the group is no longer referenced
directly.

  4. 4.

The $o$-minimal geometry of algebraic closure in $\mbox{\msbm U}(\mathsf{R})$
imposes the desired quasiminimal geometry on $\mbox{\msbm U}(\mathsf{R})$. The
dimension function is denoted ${\rm cdim}$ for ‘combinatorial dimension’. Note
that the ordering is not externally imposed on 𝕌: rather, it is implied by the
predicates described in (1) above and the dimension just mentioned.

  5. 5.

As before, there is an $L_{\omega_{1},\omega}$ sentence that axiomatizes the
quasiminimal (excellent) geometry and whose models form an AEC that is
categorical in all cardinalities.

Zilber provides a proof of the following theorem [Zil22]:

######  Theorem 8.2.2 (Categoricity of families of smooth complex algebraic
varieties [Zil22]).

Let $\mathbb{U}$ be a cover of a family of smooth complex algebraic variety,
formalized as in Remark 8.2.1, and let $\mathfrak{U}(\mathsf{R})$ be its
associated $L_{\omega_{1}\omega}$-definable class. If
$\dim_{\mathbb{C}}(\mathbb{U})=1$, (i.e. if the varieties are curves) and
${\rm cdim}(\mathsf{R}/k)$ is infinite, then $\mathfrak{U}(\mathsf{R})$ is
categorical in all uncountable cardinals.

Zilber remarks that in the case of higher dimensional varieties, categoricity
in $\aleph_{1}$ can still be proved.

######  Example 8.2.3.

Here are some examples from [Zil22]. Fix the o-minimal expansion
$\mathbb{R}_{\rm An}=\mathbb{R}_{\exp,an}$ of the reals with the exponential
function and the restricted (to bounded intervals) analytic functions.

  * •

Let $I=\mathbb{N}$, $\mathbb{U}=\mathbb{C}$, $f_{k}(z)=\exp(\frac{z}{k})$,
$D_{n}=\\{z\in\mathbb{C}:-2\pi n<{\rm Im}(z)<2\pi n\\}$. These are easily seen
to provide a cover system.

  * •

The $j$-function with variants $j_{N}$ as uniformizers for the modular curves
$\Gamma_{N}\setminus\mbox{\msbm H}$ are examples; this study allows one to
formalize their analytic properties in terms of o-minimality. Finally, other
examples include the Siegel half-space and polarized algebraic varieties
(these last examples are claimed but not developed by Zilber).

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