The point counting problem in
representation varieties of torus knots

Given a manifold $M$, the variety of representations $\rho:\pi_{1}(M)\to G$ of its fundamental group into an algebraic group $G$ contains information on the topology of $M$. It is especially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related [3]. This can be used to study knots $K\subset S^{3}$, by analyzing the representation variety for $G=\mathrm{SL}_{2}(\mathbb{C})$ of the fundamental group of the knot complement $S^{3}-K$. In this paper we focus in the case in which $K$ is a torus knot, which is the first family of knots where the computations are rather feasible. The geometry of $\mathrm{SL}_{2}(\mathbb{C})$-representation varieties of torus knots as been described in [19, 24]. For $\mathrm{SL}_{3}(\mathbb{C})$, it has been carried out in [25], and the case of $\mathrm{SL}_{4}(\mathbb{C})$ has been addressed in [11] through a computer-aided proof.

The case in which $M$ is a closed orientable surface has also been extensively analyzed due to their prominent role in non-abelian Hodge theory [14, 15]. In these cases, the approach has a geometric flavor focused on obtaining explicit descriptions of algebro-geometric invariants of the representation variety. This is at the heart of much recent research that justifies the study of the geometry of representation varieties of surface groups, in particular their Hodge numbers and $E$-polynomials (defined in Section 2.3).

An existing approach to address this problem is the so-called geometric method, initiated by Logares, Muñoz and Newstead in [17], which aims to compute $E$-polynomials of representation varieties of surface groups. In this method, the representation variety is chopped into simpler strata for which the $E$-polynomial can be computed. Following this idea, in the case $G=\mathrm{SL}_{2}(\mathbb{C})$, the $E$-polynomials were computed in a series of papers [17, 21, 22] and for $\mathrm{PGL}_{2}(\mathbb{C})$ in [20].

Additionally, in the papers [17, 21], the authors showed that a recursive pattern underlies the computations. This led to another approach, the quantum method, initiated by González-Prieto, Logares and Muñoz in [8]. This method is based on the existence of a Topological Quantum Field Theory (TQFT) that provides a powerful machinery to compute $E$-polynomials of representation varieties. Furthermore, the quantum method also enables the computation of the motive of the representation variety in the Grothendieck ring of algebraic varieties, a subtler invariant than the $E$-polynomial that will be of interest for the purposes of this work [7, 8].

A different landscape of the problem is drawn by the third method present in the literature, the arithmetic method. This approach, introduced by Hausel and Rodríguez-Villegas in [13], is inspired in the Weil conjectures and aims to compute the number of points of the representation variety over finite fields. In [13] the authors obtained the $E$-polynomials for $\mathrm{GL}_{r}(\mathbb{C})$ in terms of generating functions, and Mereb [23] studied this case for $\mathrm{SL}_{r}(\mathbb{C})$, giving an explicit formula for the $E$-polynomial in the case $\mathrm{SL}_{2}(\mathbb{C})$. Moreover, using this technique, explicit expressions of the $E$-polynomials have been computed [1] for orientable surfaces with $G=\mathrm{GL}_{3}(\mathbb{C})$, $\mathrm{SL}_{3}(\mathbb{C})$ and for non-orientable surfaces with $G=\mathrm{GL}_{2}(\mathbb{C})$, $\mathrm{SL}_{2}(\mathbb{C})$.

The arithmetic method deeply relies in a theorem of Katz contained in [13], that provides a link between arithmetic geometry and complex geometry. Let us describe briefly the main ideas of this result (for a more detailed account, refer to [13]). Let $X$ be a complex variety and suppose that it makes sense to consider $X$ also over finite fields $X(\mathbb{F}_{q})$ (for instance, because the equations defining $X$ have integral coefficients). Hence, $X(\mathbb{F}_{q})$ is a finite collection of points and we can count them. Suppose moreover that there exists a polynomial $P(t)\in\mathbb{Z}[t]$ such that $P(q)=|X(\mathbb{F}_{q})|$ for all large enough $q$. In this case, we will say that $X$ is asymptotically polynomial count and the polynomial $P(t)$ will be called the counting polynomial. In that case, Katz theorem proves that the $E$-polynomial of the complex variety $X$ is actually a polynomial in the product variable $q=uv$ and coincides with the counting polynomial. In this way, the arithmetic approach is indeed a method of computing solutions of equations on finite fields.

In this work, we focus on the group $G=\mathrm{AGL}_{r}(\mathbf{k})$ of affine automorphisms of the $r$-dimensional affine space $\mathbf{k}^{r}$ over an arbitrary field $\mathbf{k}$ (possibly not algebraically closed). We will study the space of $\mathrm{AGL}_{r}(\mathbf{k})$-representations of torus knots of type $(m,n)$, denoted $\mathfrak{X}_{m,n}(\mathrm{AGL}_{r}(\mathbf{k}))$, and we shall provide an explicit computation of their motives for ranks $r=1,2$. This result extends the previous work [10] where the motive of the representation variety in the case $\mathbf{k}=\mathbb{C}$ was computed using the geometric method. Explicitly, we prove the following result:

As a by-product of the computation of this motive, just by seeing $q$ as a formal variable, we directly obtain the number of points of $\mathfrak{X}_{m,n}(\mathrm{AGL}_{r}(\mathbf{k}))$ over a finite field $\mathbf{k}=\mathbb{F}_{q}$ as well as its $E$-polynomial over $\mathbf{k}=\mathbb{C}$ (where we have $\xi_{l}^{\mathbb{C}}=l$ and the result agrees with [10]). In some sense, this result jeopardizes the arithmetic method. Indeed, the number of $l$-th roots of unity $\xi_{l}^{\mathbb{F}_{q}}$ in a finite field $\mathbb{F}_{q}$ strongly depends on the divisibility of $l$ and $q$. For this reason, Theorem 1.1 shows that, instead of a single counting polynomial that recovers the number of points on finite fields, several counting polynomial arise depending on the arithmetic between $n,m$ and $q$, reflecting the fact that several trends in the growth of the number of points co-exist.

Among these trends, only one of them is ‘the right one’ in the sense that its counting polynomial agrees with the $E$-polynomial. Nevertheless, unraveling it may be a difficult task. Indeed, as we shall show in Section 3, this ‘right trend’ is a minority: in terms of density, for most of the prime powers $q$ the number of points of the representation variety over $\mathbb{F}_{q}$ lies in a wrong trend. This poses an epistemological problem: unless you know the $E$-polynomial beforehand, you cannot detect the right trend that computes it.

Furthermore, Theorem 1.1 also evidences that the gap between polynomial counting and motivic theory is bigger than expected. Given a complex variety $X$, it may happen that $X$ lies in the subring of $\mathrm{K}\mathbf{Var}_{\mathbb{C}}$ generated by the Lefschetz motive, that is, $[X]=P(q)$ for some polynomial $P(t)\in\mathbb{Z}[t]$. In that case, we shall say that $X$ is polynomial motivic and $P$ will be its motivic polynomial (for further information, refer to Section 3). This is the natural analogue of the polynomial count property in the motivic framework. In some sense, the motivic polynomial is reflecting some kind of transcendental-arithmetic property that instantiates as the number of points over finite fields and as the motive over $\mathbb{C}$. For this reason, it makes sense to pose the following conjecture which, in some form or another, has been implicitly considered in many previous works on representation varieties.

However, despite that Conjecture 1 is valid for all the previously known examples of representation varieties of surface groups, it turns out that it does not hold true for the varieties of Theorem 1.1. In fact, in Section 3 we prove the following corollary of Theorem 1.1.

As we will discuss in Section 3, the previous result does not fully prevent Katz theorem to work in this context. A slightly stronger version of this result is proven in [13] that allows us to discard the wrong countings. The idea is the following. Suppose that we manage to find a ring $A$ such that, for any embedding $A\hookrightarrow\mathbb{F}_{q}$ into a finite field, the variety $X(\mathbb{F}_{q})$ does admit polynomial count (in the sense that there exists a polynomial $P(t)\in\mathbb{Z}[t]$ such that $P(q)=|X(\mathbb{F}_{q})|$). The choice of the initial ring $A$ allows us to get rid of the ‘bad prime powers’ so the ‘right trend’ is precisely those prime powers $q$ that admit an embedding $A\hookrightarrow\mathbb{F}_{q}$. For this reason, this property is called strongly polynomial count.

In our case, it turns out that taking $A=\mathbb{Z}[\xi_{n},\xi_{m}]$, where $\xi_{n},\xi_{m}\in\mathbb{C}$ are primitive $n$-th and $m$-th roots of unity, we are able to dismiss all the finite fields lying in a ‘wrong trend’ and to select precisely the prime powers whose counting polynomial agrees with the $E$-polynomial. In this sense, we shall show in Section 3 the following.

The combination of Corollaries 1.2 and 1.3 provides a counterexample to Conjecture 1. For this reason, instead of the naïve Conjecture 1, it makes sense to pose the following subtler conjecture which, to the best of our knowledge, remains open.

In the seminal paper [13], Hausel and Rodríguez-Villegas introduced an arithmetic method for computing the $E$-polynomial of representation varieties. The key idea was to count the number of points of a complex variety over finite fields, as suggested by the Weil conjectures.

Suppose that $A$ is also a finitely generated $\mathbf{k}$-algebra. For any morphism $A\to A^{\prime}$ to a finitely generated $\mathbf{k}$-algebra, we will denote by $\mathcal{X}(A^{\prime})$ the set of closed points of $\mathcal{X}\times_{\textrm{Spec}(A)}\textrm{Spec}(A^{\prime})$ after the extension of scalars. In particular, if $A^{\prime}=\mathbb{F}_{q}$ is a finite field, then $\mathcal{X}(\mathbb{F}_{q})$ is a finite set of cardinal $|\mathcal{X}(\mathbb{F}_{q})|$.

A related notion is the following, where the key difference is that, now, the ground ring $A$ of the spreading out may be any finitely generated subring of $\mathbb{C}$.

The very remarkable feature of these varieties is that this counting process actually captures their Hodge structure.

A related property to polynomial count can be read in the Grothendieck ring of algebraic varieties, $\mathrm{K}\mathbf{Var}_{\mathbb{C}}$.

The interplay between polynomial count and polynomial motivic varieties is widely presented in the literature about representation varieties. In [13], the authors showed that the twisted $\mathrm{GL}_{r}(\mathbb{C})$-representation varieties of Riemann surfaces are polynomial count and computed their counting polynomials. This work on polynomial count has been extended in [23] for $\mathrm{SL}_{r}(\mathbb{C})$ and in [12] for parabolic representation varieties. At the other side, in the paper [21] it was shown that the $\mathrm{SL}_{2}(\mathbb{C})$-representation varieties of Riemann surfaces are also polynomial motivic and the motivic polynomial agrees with the counting polynomial. Additionally, the recent developments on Topological Quantum Field Theories has opened the door to show that general representation varieties are also polynomial motivic and to compute their motives [8, 7].

Based on these results, and the similarity between these concepts, the following conjecture has been considered in one or another form in the aforementioned works.

We shall disprove this conjecture with our computations of $\mathrm{AGL}_{r}(\mathbf{k})$-representation varieties of torus knots. Here, $\mathrm{AGL}_{r}(\mathbf{k})$ is the group of affine transformations of the $r$-dimensional affine space. In other words, the elements of $\mathrm{AGL}_{r}(\mathbf{k})$ are matrices of the form

with $\alpha\in\mathbf{k}^{r}$ and $A_{0}\in\mathrm{GL}_{r}(\mathbf{k})$. Multiplication in $\mathrm{AGL}_{r}(\mathbf{k})$ is given by matrix multiplication, so we have a natural description as semi-direct product $\mathrm{AGL}_{r}(\mathbf{k})=\mathbf{k}^{r}\rtimes\mathrm{GL}_{r}(\mathbf{k})$.

In this section, we compute the motive of the $\mathrm{AGL}_{1}(\mathbf{k})$-representation variety of the $(m,n)$-torus knot by describing it explicitly. Fix a field $\mathbf{k}$ whose characteristic is $\mathrm{char}(\mathbf{k})=0$ or $\mathrm{char}(\mathbf{k})>0$ not dividing both $n$ and $m$. Suppose that we have an element $(A,B)\in\mathfrak{X}_{m,n}(\mathrm{AGL}_{1}(\mathbf{k}))$ with matrices of the form

A straightforward computation shows that

Under the isomorphism of Lemma 4.1, the representation variety can be explicitly described as

where, for $l\geq 1$, $\Phi_{l}$ is the polynomial $\Phi_{l}(x)=1+x+\ldots+x^{l-1}\in\mathbf{k}[x]$. Written in a more geometric fashion, the morphism $(t,\alpha,\beta)\mapsto t$ defines a regular map

The fiber over $t\in\mathbf{k}^{*}$ is the orthogonal complement of the vector $(\Phi_{n}(t^{m}),-\Phi_{m}(t^{n}))\in\mathbf{k}^{2}$. This complement is $\mathbf{k}$ if $(\Phi_{n}(t^{m}),\Phi_{m}(t^{n}))\neq(0,0)$ and is $\mathbf{k}^{2}$ otherwise.

Recall that if $\mathrm{char}(\mathbf{k})$ does not divide $l$, then the roots of $\Phi_{l}$ are the $l$-th roots of unit different from $1$. Denote by $\mu_{l}^{\mathbf{k}}$ the group of $l$-th roots of units in $\mathbf{k}$ (including $1$). In our case, the assumptions on the characteristic of $\mathbf{k}$ imply that $(\Phi_{n}(t^{m}),\Phi_{m}(t^{n}))=(0,0)$ if and only if $t\in\mu_{nm}^{\mathbf{k}}-\left(\mu_{n}^{\mathbf{k}}\cup\mu_{m}^{\mathbf{k}}\right)$. Set

and notice that it is a finite set. To count its elements, set $\xi_{l}^{\mathbf{k}}=|\mu_{l}^{\mathbf{k}}|$. Since $m,n$ are coprime $\mu_{n}^{\mathbf{k}}\cap\mu_{m}^{\mathbf{k}}=\left\{1\right\}$ and thus

Note that if $\mathbf{k}$ is algebraically closed, then $\xi_{l}^{\mathbf{k}}=l$ and, thus, $|\Omega_{m,n}^{\mathbf{k}}|=nm-n-m+1=(n-1)(m-1)$.

Therefore, (3) decomposes into the two Zariski locally trivial fibrations

with $\mathfrak{X}_{m,n}(\mathrm{AGL}_{1}(\mathbf{k}))=\mathfrak{X}_{m,n}^{(1)}(\mathrm{AGL}_{1}(\mathbf{k}))\sqcup\mathfrak{X}_{m,n}^{(2)}(\mathrm{AGL}_{1}(\mathbf{k}))$. Thus, this implies that the motive of the whole representation variety is

In this section, we give a counterexample to Conjecture 3. In fact, the counterexample will be the $\mathrm{AGL}_{1}(\mathbf{k})$-representation variety of a $(m,n)$-torus knot with $n$ and $m$ coprime numbers. The core of this fact is the following easy lemma, whose proof is provided for completeness.