Non-$\aleph_0$-categorical geometrically trivial strongly minimal sets in $DCF_0$ beyond arithmetic Fuchsian groups (Casale–Freitag–Nagloo, Question 2.15)

In the model theory of differentially closed fields of characteristic zero ($DCF_0$), the Zilber trichotomy classifies strongly minimal sets up to non-orthogonality into three classes, of which the geometrically trivial class is the only one still lacking a general structure theory. Casale, Freitag and Nagloo (Annals of Mathematics 192 (2020), 721–765) proved that the Schwarzian differential equation of the Hauptmodul of any genus-zero Fuchsian group of the first kind defines a geometrically trivial strongly minimal set, and that this set is $\aleph_0$-categorical exactly when the group is non-arithmetic; arithmetic Fuchsian groups thus became the only known source of non-$\aleph_0$-categorical geometrically trivial strongly minimal sets. Their Question 2.15 asks whether this source is exhaustive: does $DCF_0$ contain a non-$\aleph_0$-categorical geometrically trivial strongly minimal set that is not non-orthogonal to the Schwarzian equation (or any fiber thereof) of an arithmetic Fuchsian group?

Background

A differential field is a field equipped with a derivation $\delta$; its elements killed by $\delta$ form the constant field. The theory $DCF_0$ of differentially closed fields of characteristic zero (with one derivation) is the model-theoretic completion of the theory of differential fields, and one works inside a sufficiently saturated model $\mathbb{U}$ playing the role of a universal differential field with constant field $C$. A definable set $X$ is strongly minimal if every definable (with parameters) subset of $X$ is finite or cofinite; strongly minimal sets are the atoms from which all finite-rank definable objects are built, and their internal geometry governs the structure of all definable sets of finite order. The order of $X$ is $\sup\{\operatorname{tr.deg}_K K\langle y\rangle : y \in X\}$ over a base field $K$ of definition, where $K\langle y\rangle$ is the field generated by $y$ and all its derivatives. $X$ is geometrically trivial if whenever finitely many solutions $y_1,\dots,y_m \in X$, together with all their derivatives, are algebraically dependent over a differential extension of the base field, then already some pair among them (with derivatives) is algebraically dependent — that is, all algebraic relations among solutions are pairwise.

The Hrushovski–Zilber trichotomy for $DCF_0$ (see the source paper, Theorem 5.2) states that every strongly minimal set is exactly one of: (1) nonorthogonal to the constants $C$; (2) nonorthogonal to the Manin kernel $A^\sharp$ — the smallest Zariski-dense definable subgroup — of a simple abelian variety $A$ not descending to $C$; or (3) geometrically trivial. Here two strongly minimal sets are nonorthogonal when there is an infinite definable relation between them with finite-to-one projections, an equivalence that preserves order and trichotomy class. Classes (1) and (2) carry a complete further classification (Manin kernels up to isogeny, Fact 5.6 of the source), while class (3) has no general classification; the present problem sits exactly in that gap.

For a geometrically trivial strongly minimal set $X$, $\aleph_0$-categoricity has a concrete criterion (Freitag–Jaoui–Marker–Nagloo, Definition 6.2): $X$ is $\aleph_0$-categorical when for every solution $y$ the intersection $X \cap K\langle y\rangle^{\operatorname{alg}}$ is finite, where $K$ is a differential field over which $X$ is defined; dually, $X$ fails to be $\aleph_0$-categorical when some solution has infinitely many further solutions algebraic over the differential field it generates, linked by definable finite-to-finite correspondences. Finding any non-$\aleph_0$-categorical geometrically trivial strongly minimal set was a long-standing open problem, resolved by Freitag and Scanlon (2018) through the classical modular $j$-function; before that, all known geometrically trivial strongly minimal sets (e.g. the second Painlevé equations) were $\aleph_0$-categorical, and the nonexistence of such sets had been part of a proposed strategy for certain diophantine problems, as recounted in the source paper.

A Fuchsian group is a discrete subgroup $\Gamma$ of $PSL_2(\mathbb{R})$ acting on the upper half plane $\mathbb{H}$; it is of the first kind and genus zero when the compactified quotient $X(\Gamma)=\Gamma\backslash(\mathbb{H}\cup\text{cusps})$ is the Riemann sphere. A Hauptmodul $j_\Gamma$ is a generator of the field of automorphic functions of $\Gamma$, i.e. a uniformizing map $X(\Gamma)\to\mathbb{P}^1(\mathbb{C})$. The Schwarzian derivative of $y$ is $S(y)=y'''/y'-\tfrac{3}{2}(y''/y')^2$; every branch of $j_\Gamma$ composed with a Möbius transformation satisfies a third-order algebraic ODE of Schwarzian type $S(y)+(y')^2\,R_{j_\Gamma}(y)=0$ with $R_{j_\Gamma}$ an explicit rational function determined by the fundamental domain of $\Gamma$; replacing the right-hand side $0$ by a parameter $a$ gives the fibers $\chi_\Gamma^{-1}(a)$. The group $\Gamma$ is arithmetic (in the sense of the source paper, Definition 2.7) when it is commensurable with the norm-one group of an order in a quaternion algebra over a totally real field splitting at exactly one real place, $PSL_2(\mathbb{Z})$ being the basic example; by a theorem of Margulis, $\Gamma$ is arithmetic exactly when it has infinite index in its commensurator $\operatorname{Comm}(\Gamma)$, which yields infinitely many Hecke-type algebraic correspondences on $\mathbb{P}^1$.

Casale–Freitag–Nagloo proved (Theorem 2.12) that the Schwarzian equation above defines a geometrically trivial strongly minimal set for every genus-zero Fuchsian group $\Gamma$ of the first kind, and (Theorems 2.13, 2.14, 6.2) that this set — and each of its fibers — is $\aleph_0$-categorical if and only if $\Gamma$ is non-arithmetic: the infinitely many $\operatorname{Comm}(\Gamma)$-correspondences of an arithmetic group are precisely the binary algebraic relations that force non-$\aleph_0$-categoricity. They also proved that distinct fibers of the same $\Gamma$ are orthogonal (Theorem 6.3) and that the equations of two groups not commensurable in the wide sense are orthogonal (Theorem 6.5). Arithmetic Fuchsian groups therefore constitute one uniform source of non-$\aleph_0$-categorical geometrically trivial strongly minimal sets in $DCF_0$, and the authors pose, as 'the next major challenge in the classification of geometrically trivial strongly minimal sets in differentially closed fields', the question of whether this source is exhaustive. The question as printed omits the words 'geometrically trivial', but the authors' own footnote to it fixes the reading: 'arise from' means 'non-orthogonal to the differential equation $(\star)$ or one of its other fibers', a notion that only applies within the trivial class — read without the triviality restriction the literal question would be trivially affirmative through the nontrivial trichotomy classes (1) and (2).

Problem Statement

Question (Casale–Freitag–Nagloo, Question 2.15, with the authors' footnote 4 reading). Does there exist, in a differentially closed field of characteristic zero ($DCF_0$), a strongly minimal set $X$ such that:

(i) $X$ is geometrically trivial;

(ii) $X$ is not $\aleph_0$-categorical — by the criterion used throughout this literature for geometrically trivial strongly minimal sets (Freitag–Jaoui–Marker–Nagloo, Definition 6.2), $X$ is $\aleph_0$-categorical when $X \cap K\langle y\rangle^{\operatorname{alg}}$ is finite for every solution $y \in X$, where $K$ is a differential field over which $X$ is defined, so this clause requires the existence of a solution $y \in X$ with $X \cap K\langle y\rangle^{\operatorname{alg}}$ infinite; and

(iii) $X$ is orthogonal to the solution set of the Schwarzian equation $S(y)+(y')^{2}\,R_{j_\Gamma}(y)=0$ of every arithmetic genus-zero Fuchsian group $\Gamma$ of the first kind, and to every fiber $\chi_\Gamma^{-1}(a)$ of these equations?

Equivalently: is arithmetic Fuchsian uniformization the only source of non-$\aleph_0$-categorical geometrically trivial strongly minimal sets in differentially closed fields of characteristic zero? The question is to be answered at the source's full generality — no restriction on the order of the defining equation, on autonomy, on the base differential field, or on the form of the defining formulas is permitted — either by an explicit example with proof of properties (i)–(iii), or by a theorem showing that every non-$\aleph_0$-categorical geometrically trivial strongly minimal set in $DCF_0$ is non-orthogonal to the Schwarzian equation of some arithmetic Fuchsian group (or one of its fibers).

References

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