Generic solutions of equations involving the modular $j$ function

In order to reduce Strong EC to EC under MSCD and MZP, there are two main technical steps.

1. (a)

   MSCD gives a lower bound for transcendence degree, but this inequality is only measured over $\mathbb{Q}$, whereas strong EC requires one to measure transcendence degree over an arbitrary finitely generated field, that is, we need a modular Schanuel-inequality with parameters. This is resolved by using the results on the existence of “convenient generators” of [AEK23, §5], which in turn is based upon the results of [AEK21] on differential existential closedness. See §4.2 for details.

2. (b)

   Under MSCD, a point in $V$ of the form $(z_{1},\ldots,z_{n},j(z_{1}),\ldots,j(z_{n}))$ which is not generic in $V$ will produce what is known as an atypical intersection. MZP speaks precisely about such atypical components, giving us sufficient control over their behaviour.

The motivation for the EC and strong EC problems for $j$ originates in the study of analogous problems stated for the complex exponential function $\exp:\mathbb{C}\to\mathbb{C}^{\times}$, which where first considered in Zilber’s work on pseudo-exponentiation [Zil05], with further details and results in [KZ14] and [BK18]. Zilber’s work gives a model-theoretic approach to the study of the algebraic properties of the complex exponential function, and his ideas have since been expanded to many other settings.

With this in mind, the motivation for Theorem 1.1 is not simply a restating of Zilber’s conjectures on $\exp$ for the case of $j$, but our aim is also to give a general strategy for reducing strong EC problems to EC problems, even though there is no nice model-theoretic “pseudo-$j$” structure like pseudo-exponentiation (yet). The methods presented here are expected to work in more general situations, such as in the case of Shimura varieties for which some cases of (EC) have been shown in [EZ21].

The original name of the EC problem in the case of $\exp$ was Exponential Algebraic Closedness (EAC). Instead of requiring the variety $V$ to be broad and free, EAC requires the varieties $V\subseteq\mathbb{C}^{n}\times\left(\mathbb{C}^{\times}\right)^{n}$ to be rotund, and additively and multiplicatively free, see [KZ14, §3.7] for definitions. The original name for the Zilber–Pink conjecture in the context of the exponential function was the conjecture on the intersection with tori (CIT).

The result below is a direct analogue of Theorem 1.1 for $\exp$, and we reference [KZ14, Theorem 1.5] as a source. We point out however that the original formulation of [KZ14, Theorem 1.5] does not require the variety $V$ to be additively and multiplicatively free, although these conditions are necessary for the theorem to hold. Without them, we cannot even expect the variety to intersect the graph of $\exp$. For example, if we define a variety $V\subset\mathbb{C}^{2}\times\left(\mathbb{C}^{\times}\right)^{2}$ by the following two equations: $X_{1}-X_{2}=0$ on $\mathbb{C}^{2}$ (which prevents $V$ from being additively free) and $Y_{1}-Y_{2}=1$ on $\left(\mathbb{C}^{\times}\right)^{2}$, then $V$ cannot have point in the graph of any function. Not only that, if we slightly modify $V$ to be defined by the equations $X_{1}-X_{2}=0$ and $X_{1}=Y_{1}$, it can be checked that $V$ has an infinite intersection with the graph of $\exp$ (one for every fixed point of $\exp$), but every point in this intersection has transcendence degree at most 1, so they are not generic in $V$.

Since our methods for proving Theorem 1.1 are expected to easily generalise to give a proof of Theorem 1.4, and many aspects of the such a proof can already be found in [BK18] and [KZ14], we will not present a proof of this result here. We remark that Theorem 1.4 only considers complex algebraic varieties, not a general model of the first-order theory of pseudo-exponentiation, and as such it falls short of the full ambition of [KZ14, Theorem 1.5].

On the other hand, we expect that by imposing conditions on the base field of $V$ (analogous to our notion of “having no $C_{j}$-factors”) one can proceed like in the proof of Theorem 1.2 to remove the dependence on Schanuel’s conjecture and CIT from Theorem 1.4. Although not phrased in this way, many aspects of such a result can be recovered from [BK18, Proposition 11.5], with some extra details provided by [AEK23, Theorem 5.6].

Naturally, given the similarities between the results for $\exp$ and $j$ one should ask: when can one expect to obtain analogous results with other functions? The key ingredient in obtaining the existence of the “convenient generators” mentioned in §1.1 is the Ax–Schanuel theorem. As we will see, this theorem is also essential for obtaining uniform weak forms of Zilber–Pink, which are then used in proving Theorem 1.2. We therefore expect that the methods used here can be extended to other situations where one has such a result. Ax–Schanuel theorems have been been obtained in many situations: the exponential map of a (semi-) abelian variety in [Ax72] and [Kir09], uniformizers of any Fuchsian group of the first kind in [BSCFN21] (and more), the uniformisation map of a Shimura variety in [MPT19], variations of mixed Hodge structures in [GK23, Chi21], among many others.

1. §2:

   We set up the basic notation and give some background on the algebraic aspects of the modular $j$ function.

2. §3:

   We go over the technical details of broadness, freeness, the (EC) condition, and the uniform and weak versions of Zilber–Pink.

3. §4:

   We first lay down the necessary groundwork for handling transcendence inequalities over different fields (using the convenient generators alluded to earlier), we then deal with the possible presence of special solutions (using uniform André–Oort), and we prove Theorem 1.1. After that, we give a few cases of Theorem 1.1 were we can remove the dependence on MZP, without imposing conditions on the field of definition of $V$. We also prove Theorem 4.15 in §4.4, which is a version of Theorem 1.1 for a blurring of the $j$ function.

4. §5:

   We prove Theorem 1.2. This proof, combined with the technicalities explained in the proof of Theorem 4.15, produce Theorem 1.3 in §5.2.

5. §6:

   We give an analogue of Theorem 1.2 including the derivatives of $j$.

• •

  If $L$ is any subfield of $\mathbb{C}$, then $\overline{L}$ denotes the algebraic closure of $L$ in $\mathbb{C}$.

• •

  Given sets $A,B$ we define $A\setminus B:=\{a\in A:a\not\in B\}$.

• •

  Tuples of elements will be denoted with boldface letters; that is, if $x_{1},\ldots,x_{m}$ are elements of a set $X$, then we write $\mathbf{x}:=(x_{1},\ldots,x_{m})$ for the ordered tuple. We will also sometimes use $\mathbf{x}$ to denote the (unordered) set $\left\{x_{1},\ldots,x_{m}\right\}$, which should not lead to confusion.

• •

  Suppose $X$ is a non-empty subset of $\mathbb{C}^{m}$. If $f$ denotes a function defined on $X$ and $\mathbf{x}$ is an element of $X$, then we write $f(\mathbf{x})$ to mean $(f(x_{1}),\ldots,f(x_{m}))$.

• •

  Given a subfield $L$ of $\mathbb{C}$ and a subset $A\subseteq\mathbb{C}^{n}$, we say that a subset $Z\subseteq\mathbb{C}^{n}$ is the $L$-Zariski closure of $A$ if $Z$ is the smallest Zariski closed set containing $A$ that is defined over $L$.

• •

  The term algebraic variety for us will just mean a Zariski closed set, not necessarily irreducible. We also identify complex algebraic varieties $V$ with the set of their $\mathbb{C}$-points $V(\mathbb{C})$.

We denote by $\mathbb{H}$ the complex upper-half plane $\{z\in\mathbb{C}:\mathrm{Im}(z)>0\}$. The group $\mathrm{GL}_{2}^{+}(\mathbb{R})$ of $2$ by $2$ matrices with coefficients in $\mathbb{R}$ and positive determinant, acts on $\mathbb{H}$ via the formula

This action can be extended to a continuous action of $\mathrm{GL}_{2}^{+}(\mathbb{R})$ on the Riemann sphere $\widehat{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$. Given a subring $R$ of $\mathbb{R}$ we define $M_{2}^{+}(R)$ as the set of $2$ by $2$ matrices with coefficients in $R$ and positive determinant. We put

which is a subgroup of $\mathrm{GL}_{2}^{+}(\mathbb{R})$. The modular group is defined as

The modular $j$ function is defined as the unique holomorphic function $j:\mathbb{H}\to\mathbb{C}$ that satisfies

and has a Fourier expansion of the form

This function allows us to identify $\Gamma\backslash\mathbb{H}\simeq\mathbb{C}$. The quotient space $Y(1):=\Gamma\backslash\mathbb{H}$ is known to be a (coarse) moduli space for one-dimensional complex tori, or equivalently, elliptic curves over $\mathbb{C}$. If $\Gamma z$ is a point in $Y(1)$ and $E_{z}$ denotes an elliptic curve in the corresponding isomorphism class, then $j(z)$ is simply the $j$-invariant of the curve $E_{z}$.

It is well-known ([Mah69]) that $j$ satisfies the following algebraic differential equation (and none of lower order):

Let $\left\{\Phi_{N}(X,Y)\right\}_{N=1}^{\infty}\subseteq\mathbb{Z}[X,Y]$ denote the family of modular polynomials associated with $j$ (see [Lan87, Chapter 5, Section 2] for the definition and main properties of this family). We recall that $\Phi_{N}(X,Y)$ is irreducible in $\mathbb{C}[X,Y]$, $\Phi_{1}(X,Y)=X-Y$, and for $N\geq 2$, $\Phi_{N}(X,Y)$ is symmetric of total degree $\geq 2N$. Also, the action of $G$ on $\mathbb{H}$ can be traced by using modular polynomials in the following way: for every $g$ in $G$ we define $\widetilde{g}$ as the unique matrix of the form $rg$ with $r\in\mathbb{Q}$ and $r>0$, so that the entries of $\widetilde{g}$ are all integers and relatively prime. Then, for every $x$ and $y$ in $\mathbb{H}$ the following statements are equivalent:

• (M1):

  $\Phi_{N}(j(x),j(y))=0$;

• (M2):

  There exists $g$ in $G$ with $gx=y$ and $\det\left(\widetilde{g}\right)=N$.

A point $w$ in $\mathbb{C}$ is said to be special (also known as a singular modulus) if there is $z$ in $\mathbb{H}$ such that $[\mathbb{Q}(z):\mathbb{Q}]=2$ and $j(z)=w$. Under the moduli interpretation of $j$, special points are those that correspond to elliptic curves endowed with complex multiplication. We set

A theorem of Schneider [Sch37] says that $\mathrm{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(z,j(z))=0$ if and only if $z$ is in $\Sigma$. We say that a point $\mathbf{w}$ in $\mathbb{C}^{n}$ is special if every coordinate of $\mathbf{w}$ is special.

Given a subset $A$ of $\mathbb{H}$, we define $\dim_{G}(A)$ as the number of distinct $G$-orbits in

(which may be infinite). Equivalently, $\dim_{G}(A)$ is the cardinality of the quotient set $G\backslash(G\cdot A)$. Given another subset $C$ of $\mathbb{C}$, we define $\dim_{G}(A|C)$ as the number of distinct $G$-orbits in $(G\cdot A)\setminus(G\cdot C)$. In plain words, $\dim_{G}(A|C)$ counts the number of orbits generated by elements of $A$ that do not contain elements of $C$.

We now present two modular versions of Schanuel’s conjecture. Both follow from the Grothendieck–André generalised period conjecture applied to powers of the modular curve (see [AEK23, §6.3] and references therein).

Clearly MSC follows from MSCD. Also, in view of the differential equation 2.2, MSCD only needs to involve up to the second derivative of $j$. We also point out that MSCD is somewhat incomplete as it does not say anything about the transcendence degree of $\mathbb{Q}(z,j(z),j^{\prime}(z),j^{\prime\prime}(z))$ over $\mathbb{Q}$ when $z\in\Sigma$. In fact, a stronger conjecture is implied by the generalised period conjecture (see [AEK23, Conjecture 6.13]) which does account for points in $\Sigma$. However, in the proofs of our main results we will show that we can move away from points that have coordinates in $\Sigma$, and for this reason, MSCD will suffice for us.

Given a matrix $g=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)$ in $\mathrm{GL}_{2}(\mathbb{C})$, we define the polynomial $M_{g}(X,Y):=Y(cX+d)-(aX+b)$ and the rational function $gX:=\frac{aX+b}{cX+d}$. Note that $M_{g}(X,gX)=0$.

We remark that if $S$ and $T$ are (weakly) special subvarieties of $\mathrm{Y}(1)^{n}$, then the irreducible components of $S\cap T$ (if there are any) are again (weakly) special suvbarieties of $\mathrm{Y}(1)^{n}$. In fact, by Hilbert’s basis theorem the irreducible components of any non-empty intersection of (weakly) special subvarieties is again a (weakly) special subvariety.

Following [Asl21], we also make the following definition.

Now we introduce some notation for coordinate projections. Let $n$ and $\ell$ denote positive integers with $\ell\leq n$, and let $\mathbf{i}=(i_{1},\ldots,i_{\ell})$ denote a point in $\mathbb{N}^{\ell}$ with $1\leq i_{1}<\ldots<i_{\ell}\leq n$. Define the projection map $\mathrm{pr}_{\mathbf{i}}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{\ell}$ by

In particular we distinguish between the natural number $n$ and the tuple $\mathbf{n}=(1,\ldots,n)$. We also use the notation $\mathbf{n}\setminus\mathbf{i}$ to denote the tuple of entries of $\mathbf{n}$ that do not appear in $\mathbf{i}$.

Define $\mathrm{Pr}_{\mathbf{i}}:\mathbb{C}^{n}\times\mathrm{Y}(1)^{n}\rightarrow\mathbb{C}^{\ell}\times\mathrm{Y}(1)^{\ell}$ by

For example, if $\pi_{\mathbb{C}}(V)$ is Zariski dense in $\mathbb{C}^{n}$, then $V$ is broad.

Here we will define the (EC) condition, give a conjecture for the EC problem for $j$ (see Conjecture 3.5), and review some known results.

Given an integer $n\geq 1$ we denote the graph of $j:\mathbb{H}^{n}\rightarrow\mathbb{C}^{n}$ as

We recall the following result which gives examples of varieties satisfying (EC).

We also recall the following statement (although it falls short from giving examples of varieties satisfying (EC) as it does not prove Zariski density).

These theorems and the main results of [AEK21] give evidence for the following conjecture (cf [AK22, Conjecture 1.2]).

The reader may be wondering why, in light of Theorem 3.3, Conjecture 3.5 does not ask for the stronger condition of $V\cap\mathrm{E}_{j}^{n}$ being Zariski dense in $V$. In fact, as we will see in Lemma 3.8, Conjecture 3.5 already implies the this stronger condition.

Conjecture 3.5 can be thought of as a version of Hilbert’s Nullstellensatz for systems of algebraic equations involving the $j$-function (cf [DMT10] for the discussion in the case of the exponential function). The examples of [EH21, §3] show that if $V$ has constant coordinates or is contained in a proper subvariety of the form $M\times\mathrm{Y}(1)^{n}$, with $M$ a Möbius subvariety, then it can happen that $V\cap\mathrm{E}_{j}^{n}$ is empty.

We also point out that the condition of $V\cap\left(\mathbb{H}^{n}\times\mathrm{Y}(1)^{n}\right)$ being Zariski dense in $V$ is needed, as otherwise the subvariety of $\mathbb{C}^{2}\times\mathrm{Y}(1)^{2}$ defined by the single equation $X_{1}=iX_{2}$ is free and broad, but cannot have points in the graph of the $j$ function since there is no point $z$ in $\mathbb{H}$ for which $iz$ is also in $\mathbb{H}$. One could alternatively resolve this issue by extending the domain of $j$ to the lower half-plane using Schwarz reflection.

The following result is an approximate version of Conjecture 3.5 using what is known as a blurring of the $j$ function.

We finish this subsection by recalling a useful trick.

From now on, we will abbreviate this conjecture as MZP. This conjecture is sometimes presented in terms of optimal varieties, which we discuss next.

As shown in [HP16, Lemma 2.7], MZP is equivalent to the statement that any subvariety of $\mathrm{Y}(1)^{n}$ contains only finitely many optimal subvarieties, i.e. $\mathrm{Opt}(V)$ is a finite set.

Pila showed that MZP implies the following uniform version of itself.