1 Introduction↩︎

Let \(k\) be an algebraically closed field, \(d\ge2\) an integer such that char\((k)\) does not divide \(d\), and let \(f\in k[x]\) be a polynomial of degree \(d\). The (finite) critical values of \(f\), denoted by \(C_f\), are those \(y\in k\) such that the equation \(f(x)=y\) has less than \(d\) distinct solutions—that is, the branching points of the ramified cover \(f\colon\mathbb{A}^1_k\longrightarrow\mathbb{A}^1_k\) induced by \(f\). They are obtained by evaluating \(f\) at its critical points \(Z_f\), which are the roots of the derivative \(f'\). These notions account for multiplicity, so \(Z_f\) and \(C_f\) are in principle multisets, and they are clearly finite. By compactifying, we can extend the definition of critical values to rational functions and, more generally, to ramified covers of \(\mathbb{P}^1=\mathbb{P}^1_{k}\) (to which we will refer just as “covers"):

Notice how this makes \(\infty\in\mathbb{P}^1\) a critical value of all polynomials. There is a natural action of \(\text{Aut}_{k}(X)\) on the set of covers \(\phi\colon X\longrightarrow\mathbb{P}^1\), given by \(\gamma\cdot \phi = \phi\circ\gamma^{-1}\); critical values behave well under this action as we clearly have \(\text{Branch}(\gamma\cdot \phi)=\text{Branch}(\phi)\). Of special interest for us is the case \(X=\mathbb{P}^1\), with \(\text{PGL}_2(k)\) acting via fractional linear transformations. Observe that the subset of covers consisting of polynomials, which will be our focus, is preserved under this action precisely by the subgroup Aff\((k)\) of affine transformations. Following Arnold [1], we define:

We refer to the equivalence classes of this relation as topological types, adopting the notation \([f]\) for the topological type of \(f\); if, moreover, \(f\in K[x]\) for a subfield \(K\subset k\), we also set \([f]_{K}=\{f\circ\lambda, \;\lambda\in\text{Aff}(K)\}\), referring to any such set as a \(K\)-topological type. So, polynomials of the same topological type have the same critical values. In the following, when we refer to two polynomials as (in)equivalent, we mean with respect to topological equivalence. We also call a polynomial \(f\) normalized if it is monic and \(f(0)=0\) (the normalization with \(f'\) monic is usually chosen in the literature, see e.g\(.\) [2]: our choice is motivated by purely expository reasons and does not affect the nature of the results).

The study of critical values turns out to be crucial in understanding various phenomena, ranging from holomorphic dynamics [3] to the inverse Galois problem [4] to, somewhat surprisingly, exponential sums: while investigating polynomial value sets over finite fields and rings, Kowalski and Soundararajan formulated [5] a Fourier-analogue to the Davenport pair problem (see e.g\(.\) [6] for an overview of this classical problem), observing that two integral polynomials whose reductions modulo a prime \(p\) are equicritical (and have all critical points in \(\mathbb{F}_p\)) give rise to a Fourier-Davenport pair over \(\mathbb{Z}/p^2\mathbb{Z}\). They then point out that there are inequivalent such pairs in any degree at least \(4\). It is then natural to ask (see 5.3 for the application) if there exist inequivalent polynomials with the same global critical values, and not just modulo some prime.

We saw that critical values are naturally attached to a topological type, rather than just a single polynomial; indeed, we can speak of equicritical topological types in the same way. Observe that \(\text{Aff}(k)\) acts on the set of topological types of polynomials over \(k\) by \[\begin{align} [f]^{\lambda}=[\lambda(f)],\label{outa} \end{align}\tag{1}\] with the resulting topological type having finite critical values \(\lambda(C_f)\). This yields an action of \(\text{Aff}(K)\) on the set of \(K\)-topological types which, when char\((k)=0\), is generically free (for complex quartics there is only one exception, see Corollary 5). So, in this case, varying \(\lambda\in\text{Aff}(K)\) allows us to produce infinitely many pairs of distinct equicritical \(K\)-topological types starting from a single one. In light of these remarks, we give the following definition:

One may then ask the following:

This work deals with the case \(d=4\) over number fields: we give, for any number field \(K\), an explicit parametrization of a basis for \(\text{EC}_4(K)\), showing in particular that this set is infinite.

Given an ordered pair \(\mathcal{c}=(f,g)\) of polynomials, we let \(\overline{c}\) be the pair \((g,f)\) and, for \(\lambda\in \text{Aff}(k)\), we let \(\lambda(\mathcal{c})\) be the pair \((\lambda(f),\lambda(g))\). Set \(\rho=1+\sqrt3, \;C=-720\sqrt3-1248, \;R=362+209\sqrt3\) and, for an algebraic number \(\kappa\) of degree \(2\), let \(\overline{\kappa}\) be the other root of its minimal polynomial. Moreover, let \(\omega\in\overline{\mathbb{Q}}\) be a primitive third root of unity and, for a number field \(K\), define \(K'=K\setminus\{\omega,\omega^2\}\) and \(K''=K'\setminus\{0,1,-2,-2\omega,-2\omega^2,\rho,\overline{\rho},\omega\rho,\omega^2\rho,\omega\overline{\rho},\omega^2\overline{\rho}\}\). We can now state our main result:

Kuhn [7] (then generalized by Kristiansen [8]) has shown that, for each ordered tuple \((y_1,...,y_{d-1})\) of real numbers such that \((-1)^{j+1}y_j+(-1)^jy_{j+1}>0\) for \(j\in\{1,...,d-2\}\), there exists exactly one \(\mathbb{R}\)-topological type \([f]\) with real critical points \(Z_f=\{x_1<...<x_{d-1}\}\) such that \(f(x_j)=y_j\). Observe how, if we drop the alternating sign condition on the differences \(y_{j+1}-y_j\), choosing \(K=\mathbb{Q}\) in Theorem 1 gives an infinite family of counterexamples to the uniqueness part: indeed, one easily verifies that for \(t>1\), \(f_t\) and \(g_t\) have real critical points. In other words, two inequivalent rational quartics with real critical points can share the same critical value set, but not the same critical ordered triple \((f(x_1),f(x_2),f(x_3))\), once the critical points \(x_1<x_2<x_3\) are labeled in increasing order. Concerning the existence part, it is key that for \(\mathbb{R}\) the absolute Galois group \(\mathrm{Gal}(\mathbb{C}/\mathbb{R})\simeq\mathbb{Z}/2\mathbb{Z}\) is small, so there is no hope for such a result to hold over number fields. Still, a natural question one can ask is:

Call a three-element subset \(S\subset\overline{\mathbb{Q}}\) critical over \(K\) if there exists a quartic \(f\in K[x]\) with \(C_f=S\). The following result settles this question for \(d=4\):

The paper is structured as follows: in 2 we review the literature on critical values of polynomials, setting up the algebro-geometric picture of the problem; in doing so, we prove a generalization of a result of Arnold, which may bear some independent interest. A key ingredient in the proof of Theorem 1 is a thorough study of the (reduced) Hurwitz space \(\mathcal{H}_4\) associated to quartic covers of \(\mathbb{P}^1\) with monodromy of polynomial type: in 3, after introducing Hurwitz spaces, we construct the aforementioned reduced space for any degree \(d\ge3\), and we study its geometry and arithmetic. 4 is centered around \(\mathcal{H}_4\): we prove that it is a certain well-known modular curve, and use this to establish Theorem 2. 5 is devoted to the equicriticality problem for quartics over number fields: we first give a nonconstructive proof—which may adapt to similar questions—for the existence of a parametrization like that of Theorem 1, and then go on to compute the explicit equations for the polynomials appearing in the statement, by exploiting the rich structure arising from the link to the arithmetic of elliptic curves.

1.1 Acknowledgments↩︎

I am grateful to my advisor Emmanuel Kowalski, for recommending this topic and for the helpful advice he offered during the development of this work; to Davide Lombardo, for working out the details of the proof of Lemma 11 and for his suggestion of following an explicit approach to the construction of the Hurwitz space; to Jordan Ellenberg, for sketching the ideas behind the fiber product formulation of the equicriticality problem and for pointing me to the paper of Rubin and Silverberg on elliptic curves with constant mod \(p\) representations. Finally, I thank the anonymous referees for their many helpful suggestions.

The author was partially supported by the Swiss National Science Foundation (grant number 219220). This work is part of ongoing research within the scope of the author’s PhD thesis.

2 Critical values of polynomials: arithmetic and geometry↩︎

From now on, we assume that either char\((k)=0\) or char\((k)>d\). Given a normalized polynomial \(f\in k[x]\), we attach to it the multiset of its critical points, as a (closed) point \(\{x_i\in Z_f\}\in\mathbb{A}^{d-1}/S_{d-1}\). Since with any point \(\widehat{x}\in\mathbb{A}^{d-1}/S_{d-1}\) we can associate the normalized formal integral (always assumed to have \(0\) constant term) of \(\prod_{\widehat{x}=\{x_i\}}(x-x_i)\) multiplied by \(d\), we get a bijection: \[\begin{align} \begin{aligned} \{\text{normalized polynomials } f\in k[x]\text{ of degr}&\text{ee} \;d\} \overset{1:1}{\longleftrightarrow} \mathbb{A}^{d-1}/S_{d-1}\\ &f \;\;\;\;\;\longmapsto \;\;\;\;\;\;Z_f. \end{aligned}\label{bij} \end{align}\tag{2}\] If one does not care about the subfield of definition of the coefficients, the study of polynomials with given critical values can be carried out just for normalized polynomials: the leading coefficient of \(f(ax+b)\) is \(a^d\) times that of \(f\), and its constant term is \(f(b)\). Therefore, in light of 2 , the starting point for our analysis will be the relation between critical points and critical values. We will then recover information about fields of definition via the study of rational points on Hurwitz spaces.

If the normalized polynomial \(f\) corresponds to \(\widehat x=\{x_1,...,x_{d-1}\}\in\mathbb{A}^{d-1}/S_{d-1}\), its finite critical values \(y_1,...,y_{d-1}\) are given by: \[\begin{align} y_j=\sum_{i=0}^{d-1}(-1)^i\frac{d}{d-i}e_i(\widehat x)x_j^{d-i}=:\theta_d(x_1,...,x_{d-1})_j,\label{var} \end{align}\tag{3}\] where \(e_i(\widehat x)\) is the elementary symmetric function of degree \(i\) in \(d-1\) variables (and \(e_0\equiv1\)). This translates the study of critical values for polynomials of degree \(d\) to that of the covering \(\theta_d\colon\mathbb{A}^{d-1}\longrightarrow\mathbb{A}^{d-1}\) defined by 3 , whose fiber above a point \((y_1,...,y_{d-1})\) of \(\mathbb{A}^{d-1}\) corresponds under 2 (and the quotient) to the set of normalized polynomials of degree \(d\) having as finite critical values its coordinates. An elementary but important observation to make is that, for any subfield \(K\subset k,\) the finite critical values of \(f\in K[x]\), taken with their multiplicity, are the (only) roots of a polynomial defined over \(K\): this can be seen directly by letting \(G_{K}=\text{Gal}(\overline{K}/K)\) act on both sides of 3 , since \(\theta_d\) is defined over \(K\).

Let \(U_d\) be the affine open set \(\{\prod_{1\le i < j\le d-1}(y_i-y_j)\neq 0\}\subset\mathbb{A}^{d-1}\) and let \(V_d\subset U_d\) be the closed subscheme given by \(y_1=0\). Polynomials with finite critical values in \(U_d\)—that is, pairwise distinct—are called Morse polynomials. Arnold [1], following Davis [9], Thom [10] and Zdravkovska [11], shows that for \(d\ge3\) there are \(d^{d-3}\) topological types of complex Morse polynomials of degree \(d\) with given critical values. We generalize his result to our fields \(k\) which, we recall, are algebraically closed and have characteristic either \(0\) or larger than \(d\):

Arnold’s proof reduces the question to that of counting trees on \(d\) vertices, which had already been solved by Cayley. Our strategy builds on Beardon, Carne and Ng’s study [2], over \(\mathbb{C}\), of the Jacobian \(J_d\) of \(\theta_d\) above nonzero distinct critical values, extending it to our fields \(k\) and to the case where one of the critical values is \(0\) (for a generalization of their study in the case of equal but nonzero critical values, see [12]). Let the morphism \(\widetilde{\theta}_d\) be given by 3 except for the equation for \(y_1\), where the right-hand side is divided by \(x_1\).

 If $x_1=0$, @eq:int still holds, so
 $F_{\lambda}(x)=\int_0^xf_{\lambda}(w) dw$ vanishes at
 $0,x_2,...,x_{d-1}$; therefore, if it does not vanish
 identically---in which case we conclude as above---we must have
 $$\begin{align} F_{\lambda}(x)=Cx\prod_{2\le j\le d-1}(x-x_j)\label{nv}
 \end{align}$$ {#eq:nv}  for some $C\in k^*$. On the other hand, a
 simple direct computation with @eq:var yields
 $\widetilde{J}(P)_{(1,1)}=(-1)^{d-1}d\frac{\prod_{2\le j\le d-1}x_j}{sec:2}\neq0,$
 while $\widetilde{J}(P)_{(1,j)}=0$ for $2\le j\le d-1$ as $x_1$
 appears in each monomial of these partial derivatives. This
 immediately gives $\lambda_1=0$. But then, we
 find:$$\begin{align} 0\overset{\eqref{int}}{=}f_{\lambda}(x_1)\overset{x_1=0}{=}F'_{\lambda}(0)\overset{\eqref{nv}}{=}(-1)^{d-2}C\prod_{2\le j\le d-1}x_j\neq0,
 \end{align}$$ which yields the desired contradiction.

We are now ready to prove Proposition 4:

3 The Hurwitz space of Morse polynomials↩︎

Hurwitz spaces are, roughly speaking, moduli spaces of covers \(X\longrightarrow\mathbb{P}^1\) with given ramification properties. We assume for simplicity that the field of definition \(k\) of our covers is \(\mathbb{C}\) (the following constructions generalize to any algebraically closed field of characteristic \(0\), but the nice topological setting offered by Riemann surfaces is missing; see [13] for an account), keeping in mind that compact Riemann surfaces come from algebraic curves by Serre’s GAGA. So, let \(\mathbb{P}^1=\mathbb{P}^1_{\mathbb{C}}\) and consider all objects as base-changed to \(\mathbb{C}\) when necessary. Moreover, let \(\mathcal{U}_r\) be the moduli space for \(r\) distinct unordered points in \(\mathbb{P}^1\) equipped with its natural algebraic structure of affine variety over \(\mathbb{Q}\), whereby the field of definition of the point \((y_1,...,y_r)\) is that of the polynomial \(\prod_{y_i\neq\infty}(y-y_i)\).

3.1 Moduli spaces of \(G\)-covers↩︎

We begin by giving an outline of the constructions in Hurwitz theory that are relevant for our study; a more comprehensive introduction can be found, e.g., in the work of Fried and Völklein [14]. Let \(r\) and \(d\) be positive integers, let \(G\) be a subgroup of \(S_d\) and let \(C=(C_1,...,C_r)\) be an unordered tuple of conjugacy classes in \(G\). We will refer to such a triple \((r,G,C)\) as a datum, where the embedding of \(G\) in \(S_d\) is implicit.

Recall that a \(G\)-cover is a pair \((\phi\colon X\longrightarrow\mathbb{P}^1,\delta)\)—usually denoted just by \(\phi\)—where \(\phi\) is a Galois cover and \(\delta\colon\text{Aut}(\phi)\longrightarrow G\) is a group isomorphism. Let \(\phi\) have \(r_{\phi}=r\) critical values \(y_1,...,y_r\), let \(y_0\in\mathbb{P}^1\) be any other point and, for \(1\le i\le r\), let \(U_i\) be a simply-connected open neighborhood of \(y_i\) that does not contain any other critical value but contains \(y_0\). Then, taking a generator \(\rho_i\) of \(\pi_1(U_i\setminus\{y_i\},y_0)\) with positive orientation and lifting it to \(X\) gives a deck transformation \(g_i\in G\) of \(X\), via the isomorphism \(\delta\). Let \(\text{Cl}(g_i)\) be the respective conjugacy class in \(G\) and denote by \(C(\phi)\) the unordered tuple \((\text{Cl}(g_1),...,\text{Cl}(g_r))\): we say that \(\phi\) branches with local monodromy \(C(\phi)\); this is a good definition since picking different, positively-oriented generators for the fundamental groups only changes the induced deck transformations by inner automorphisms of \(\text{Aut}(\phi)\). We have just described a way to attach to a \(G\)-cover \(\phi\) a datum, namely \((r_{\phi},\text{Aut}(\phi),C(\phi)),\) and by choosing the \(\rho_i\)’s so that \(\prod_{i=1}^r\rho_i=1\) under the inclusions \(\pi_1(U_i\setminus\{y_i\},y_0)\hookrightarrow\pi_1(\mathbb{P}^1\setminus\{y_1,...,y_r\},y_0), \;1\le i\le r\), we have also attached to \(\phi\) the Nielsen class \((g_1,...,g_r).\)

As in the case of polynomials, it is more natural to attach a datum to a suitable equivalence class of \(G\)-covers: we generalize the definition of topological equivalence by saying that two \(G\)-covers \((\phi:X\longrightarrow\mathbb{P}^1, \;\delta)\) and \((\psi:Y\longrightarrow\mathbb{P}^1, \;\xi)\) are equivalent, or isomorphic, if there is an isomorphism of covering spaces \(\gamma:X\longrightarrow Y\) that commutes with the respective actions of \(G\), i.e. \(\delta(g)=\xi(\gamma g\gamma^{-1}) \;\forall g\in\text{Aut}(\phi)\). We adopt the terminology \(G\)-cover class and the notation \((\phi)\) for the equivalence class of \(\phi\) under this relation. Isomorphic covers clearly have the same critical values, while the additional condition on the \(G\)-actions implies that the datum associated to a \(G\)-cover is an invariant of its class. Hence, we have a (noncanonical, due to its dependence on the choice of generators \(\rho_i\)) map \[\begin{align} \Phi: \left\lbrace \begin{aligned} \displaystyle \text{classes of G-covers with datum } \\ \text{(r,G,C) branching at y_1,...,y_r} \end{aligned} \right\rbrace \longrightarrow \;\; \text{Ni}_r(G,C)^{\text{in}}, \label{nc} \end{align}\tag{6}\] sending \((\phi)\) to the inner Nielsen class representative of \((g_1,...,g_r)\) as defined above. The following result (see [15] and [6]) shows that \(\Phi\) is a bijection:

Notice how in Theorem 3 the branching set is fixed, while in the situation that interests us it varies. The Riemann Existence Theorem for families (see again [15]) enables us to construct Hurwitz spaces as parameter families of Nielsen classes over \(\mathcal{U}_r\): in particular (see [13]), the inner Hurwitz space \[\begin{align} \label{inhs} \Psi_r\colon H_r^{\text{in}}(G,C)\longrightarrow \mathcal{U}_r \end{align}\tag{7}\] has the property that for any \(Q=(y_1,...,y_r)\in\mathcal{U}_r\), the fiber above \(Q\) is in bijection with the set of \(G\)-cover classes that branch at \(Q\) with local monodromy \(C\). Moreover, it admits a unique algebraic model as an affine variety over a number field \(k_0=k_0(C)\) such that \(\Psi_r\) is defined over \(k_0\) and, for any algebraic extension \(K/k_0\), any \(\sigma\in G_{K}\) and any \((\phi)\in H_r^{\text{in}}(G,C)(\overline{\mathbb{Q}})\), we have: \[\begin{align} ^{\sigma}(\phi)=(^{\sigma}\phi).\label{galc} \end{align}\tag{8}\]

As mentioned at the end of 1, our focus will mainly lie on a reduced Hurwitz space. Observe that there is an outer \(\text{PGL}_2(\mathbb{C})\)-action on \(G\)-cover classes, given by \((\phi)^{\alpha}=(\alpha(\phi))\). We say that two \(G\)-cover classes in the same orbit are \(\text{PGL}_2(\mathbb{C})\)-equivalent, and refer to an equivalence class as a \(G/\text{PGL}_2\)-cover class. Then, given the same datum as above, the reduced Hurwitz space \(H_r^{\text{red}}(G,C)\) is the affine variety of dimension \(r-3\) obtained as the quotient of \(H_r^{\text{in}}(G,C)\) by this action. Indeed, as we already observed in 1 for polynomials and affine transformations, we have \(\text{Branch}(\alpha(f))=\alpha(\text{Branch}(f)),\) and local monodromy around critical values is clearly preserved by the action. This gives a commutative diagram:

where \(\mathcal{I}_r=\mathcal{U}_r/\text{PGL}_2(\mathbb{C})\) and the maps \(\Psi_r,\Pi_r\) and \(\beta_r\) depend on the datum. A classical result of Mumford and Fogarty [17] shows that both quotient spaces are well-defined in the category of affine \(k_0\)-varieties. In virtue of its construction, \(K\)-rational points on \(H_r^{\text{red}}(G,C)\) correspond with \(\text{PGL}_2(\mathbb{C})\)-orbits of \(G\)-cover classes \((\phi)\) (with \(\phi\) taken to be defined over \(\overline{K}\)) such that for each \(\sigma\in G_{K}\) there exist \(\alpha_{\sigma}\in\text{PGL}_2(\mathbb{C})\) and an isomorphism \(\gamma_{\sigma}\) of \(G\)-covers from \(\alpha_{\sigma}(^{\sigma}\phi)\) to \(\phi\).

If \(r=4\) then \(\mathcal{I}_4\) is the usual moduli space of elliptic curves \(Y(1)\), since the \(j\)-invariant of an elliptic curve \(E\) equals \[\begin{align} j(\lambda)=256\frac{(\lambda^2-\lambda+1)^3}{\lambda^2(\lambda-1)^2},\label{jquad} \end{align}\tag{9}\] with \(\{0,1,\lambda,\infty\}\) a representative in the \(\text{PGL}_2(\mathbb{C})\)-orbit of the \(x\)-coordinates of \(E[2]\). We will hence from now on refer to the \(j\)-invariant of a quadruple of distinct complex numbers without this causing any confusion. A fiber of the morphism \[\begin{align} \beta_4\colon H_4^{\text{red}}(G,C)\longrightarrow Y(1) \end{align}\] is then naturally identified with the set of \(G/\text{PGL}_2\)-cover classes with monodromy \(C\) and representatives that branch over the \(2\)-torsions of the elliptic curves with \(j\)-invariant at the base. We will refer to the latter as the critical \(j\)-invariant of a \(G\)- or \(G/\text{PGL}_2\)-cover class.

By construction, \(\beta_4\) only ramifies above critical \(j\)-invariants of quadruples with “large" stabilizer in \(\text{PGL}_2(\mathbb{C})\), i.e\(.\) the elliptic \(j\)-invariants \(0\) and \(1728.\) It is then well known that \(H_r^{\text{red}}(G,C)\) must be a quotient of the upper half-plane \(\mathbb{H}\) by a finite index subgroup \(\Gamma=\Gamma(G,C)\) of the modular group \(\text{PSL}_2(\mathbb{Z})\), with \(\beta_4\) induced by the inclusion \(\Gamma<\text{PSL}_2(\mathbb{Z})\). In other words, we are dealing with a (not necessarily congruence) modular curve. See again [15] for a general overview of reduced Hurwitz spaces of dimension \(1\) as modular curves.

3.2 Lifting points on Hurwitz spaces, and the case of polynomials↩︎

Given \(x\in H_r^{\text{red}}(G,C)\), we call any of its preimages in \(H_r^{\text{in}}(G,C)(K)\) a “lift" of \(x\). The question of which points \(x\in H_r^{\text{red}}(G,C)(K)\) lift to \(H_r^{\text{in}}(G,C)(K)\) is a rich and open one, whose main application is in the domain of the inverse Galois problem: indeed, the IGP reduces to finding rational points on inner Hurwitz spaces, thanks to Hilbert’s Irreducibility Theorem. Looking for \(K\)-rational points on \(H_r^{\text{red}}(G,C)\) is easier due to it being a rational quotient of lower dimension, especially in the case \(r=4\) when it is a curve. However, these points rarely lift to points of \(H_r^{\text{in}}(G,C)(K)\); this behavior is largely controlled by two factors:

(a) the lifting of points from \(\mathcal{I}_r(K)\) to \(\mathcal{U}_r(\overline{K})\);

(b) the base invariant of the \(G/\text{PGL}_2(\mathbb{C})\)-cover class \(x\), i.e. the conjugacy class in \(\text{PGL}_2(\mathbb{C})\) of the stabilizer of any of its lifts to \(H_r^{\text{in}}(G,C)\).

In [13], Cadoret gives a thorough account of the theory and proves some strong lifting results, especially for \(G/\text{PGL}_2\)-cover classes with nontrivial base invariant: see for instance [13]. In point (1) of the same result, the author observes that, for \(r=4\), a point in \(\mathcal{I}_r(K)\) always lifts to a point in \(\mathcal{U}_r(K)\), as this is equivalent to fields of moduli being fields of definition for elliptic curves. Notice that a stronger condition than \(x\) having trivial base invariant is the \(\text{PGL}_2(\mathbb{C})\)-stabilizer of the branching set of any (i.e\(.\) all) of its lifts being trivial. For our purposes, it is enough to prove the following lifting result:

Let us now go back to polynomials. A Morse polynomial \(f\in k[t]\) defines a cover \(f\colon\mathbb{P}^1\longrightarrow\mathbb{P}^1\) of the same degree \(d\), branching at \(C_f\cup\{\infty\}\) with local monodromy \(C_{\text{pol}}\) given by a transposition at each of the finite critical values and a \(d-\)cycle at \(\infty\) (see [6]). It is a classical result going back to Hilbert that any choice of representatives for the classes in \(C_{\text{pol}}\) whose product is the identity must generate the full symmetric group (see [18]), so the Galois group of the cover \(f(t)-y\) is \(S_d\). In order to apply Hurwitz theory to our question, we set \(r=d, \;G=S_d\) and \(C=C_{\text{pol}}\). Then necessarily \(X=\mathbb{P}^1\) (see Remark 3), and we get the inner Hurwitz space\[\begin{align} \Psi^{\text{pol}}_d\colon H_d^{\text{in}}(S_d,C_{\text{pol}})\longrightarrow\mathcal{U}_d, \end{align}\]whose fiber above \(\{y_1,...,y_d\}\in\mathcal{U}_d\) is in bijection with the set of classes of \(S_d\)-covers \(\phi\colon\mathbb{P}^1\longrightarrow\mathbb{P}^1\) branching over the \(y_i\)’s with local monodromy \(C_{\text{pol}}\). We call these covers polynomial \(S_d\)-covers. Since \(C_{\text{pol}}^m=C_{\text{pol}}\) for odd \(m\), the results in [14] imply that our inner Hurwitz space is defined over \(\mathbb{Q}\).

Notice the similarity between \(\Psi_d^{\text{pol}}\) and the map \(\theta_d\) of Section 2: if we denote by \(\iota_d\) the inclusion \(U_d/S_{d-1}\hookrightarrow\mathcal{U}_d\) given by \(\{y_1,...,y_{d-1}\}\mapsto\{y_1,...,y_{d-1},\infty\}\), and by \(\mathcal{W}_d\) its image, then \(\Psi_d^{\text{pol}}|_{\Psi_d^{-1}(\mathcal{W}_d)}\) and \(\iota_d\circ\theta_d\) are both rational maps from an affine variety to \(\mathcal{W}_d\) whose fibers convey information about the polynomials branching at the base. Nevertheless, it is not the same information: \(\Psi_d^{\text{pol}}\) accounts for all classes of covers with polynomial monodromy, not just (normalized) polynomials, and more importantly, it carries data about the field of definition of the covers by 8 (see Theorem 4). While polynomials are not Galois covers themselves, we have:

In other words, Lemma 8 tells us that once we pass to the reduced space we are, up to equivalences, dealing just with polynomials. The choice of not focusing directly on absolute Hurwitz spaces is motivated by our, albeit short, treatment of the issue of lifting rational points, which the literature usually covers in the inner case, in virtue of the connection to the inverse Galois problem.

We refer to \(\beta_d\colon H_d^{\text{red}}(S_d,C_{\text{pol}})\longrightarrow\mathcal{I}_d\) as the Hurwitz space of (Morse) polynomials of degree \(d\). Let us compute its degree:

As we just observed, a point \(x\in H_d^{\text{red}}(S_d,C_{\text{pol}})(\mathbb{C})\) always lifts to a complex polynomial \(f\): from now on, we will use the expression “topological type" also for \((f)\), whenever this does not create ambiguity. In order to shed light on the equicriticality problem, we need to be able to relate the minimal field of definition of \(f\) to that of \(x\); our spaces have the following lifting property:

If $\alpha$ is affine we are done, so suppose
$\alpha=\begin{pmatrix}a & b \\ c & d\end{pmatrix}, \;c\neq0$. Then
$\alpha$ can be written as $\beta\alpha'$ with $\beta$ affine and
$\alpha'$ having top-left entry $0$. Therefore, up to replacing $f$
with $f\circ\beta$, $\gamma_{\sigma}$ with
$\beta^{-1}\gamma_{\sigma}{^\sigma\beta}$ and $\alpha$ with
$\alpha'$, which does not affect @eq:fmod and @eq:fdef , we can take
$a=0$, so say $\alpha=\begin{pmatrix}0 & 1 \\ c & d\end{pmatrix}$.
Let us now combine @eq:fmod and @eq:fdef ; for any
$\sigma\in G_{K}$, we have $$\begin{align}
\label{sides} f=(f\circ\alpha)\circ\alpha^{-1}=\;^{\sigma}(f\circ\alpha)\circ\alpha^{-1}=f\circ(\gamma_{\sigma}({^{\sigma}\alpha})\alpha^{-1}),
\end{align}$$ {#eq:sides}  which implies that the fractional linear
transformation on the right-hand side is the identity, since it
induces an automorphism of the cover $f(t)-y$. Therefore, since
$\gamma_{\sigma}$ is affine by @eq:fmod , so must be
$^{\sigma}\alpha\alpha^{-1}=\begin{pmatrix}1 & 0 \\ -d\frac{^{\sigma}c}{c}+{^{\sigma}}d & \frac{^{\sigma}c}{c}\end{pmatrix}$.

Looking at the bottom-left coefficient we get
$^{\sigma}(\frac{d}{c})=\frac{d}{c}\;\forall \sigma\in G_{K}\Longrightarrow l=\frac{d}{c}\in K$,
and hence $f\left(\frac{c^{-1}}{t+l}\right)\in K(t)$ by @eq:fdef .
But now, as the denominator in the variable is $K$-rational itself,
simply taking common denominator $(t+l)^4$ shows that the
coefficients $a_i$ of $t^i$ in $f$ satisfy $a_ic^{-i}\in K$, so we
finally get: $$\begin{align} f\bigg(\frac{t}{c}\bigg)\in K[t].
\end{align}$$

4 The case of quartics: modularity↩︎

4.1 The Hurwitz space of quartics as a modular curve↩︎

From now on, \(K\) will denote a number field. We consider the case \(d=4\): compactifying our reduced Hurwitz space, we get a rational modular curve\[\begin{align} \beta_4\colon\mathcal{H}_4\longrightarrow X(1), \end{align}\] to which we refer simply as the Hurwitz space of quartics, since the information about those that are non-Morse is carried by the fiber above the cusp. We adopt the notation \(j_{\mathrm{CV}}\) for our coordinate on the level \(1\) modular curve, corresponding to the critical \(j\)-invariant of the (classes of) covers above it.

Let us focus on the geometry of our modular curve \(\mathcal{H}_4=:\Gamma\backslash\mathbb{H}\). Recall the formula for the genus of quotients of the upper half plane by modular subgroups [22]: \[\begin{align} g(X(\Gamma))=1+\frac{[\text{PSL}_2(\mathbb{Z}):\Gamma]}{12}-\frac{\epsilon_2}{sec:4}-\frac{\epsilon_3}{sec:3}-\frac{\epsilon_{\infty}}{sec:2}, \end{align}\] where the \(\epsilon_i\)’s count points of period \(2\) and \(3\), and cusps, respectively. Since \([\text{PSL}_2(\mathbb{Z}):\Gamma]=\deg\beta_4=4\), the index term gives a contribution of \(\frac{sec:1}{sec:3}\). As the genus is nonnegative, \(\epsilon_{\infty}\ge1\), and \(3\) is coprime with \(4\), surely \(\epsilon_3=1\). Then necessarily \(g(\mathcal{H}_4)=0\). Notice how a priori we have the two possibilities \((2,1)\) and \((0,2)\) for \((\epsilon_2,\epsilon_{\infty})\).

Having degree less than \(7\), our \(\Gamma\) is a congruence subgroup by a result of Wohlfahrt [23], so \(\mathcal{H}_4\) is a congruence modular curve. By the list of genus \(0\) congruence subgroups in the Cummins-Pauli database [24], it is either \(X_0(3)\) or \(4A^0\backslash\mathbb{H}=X(4A^0)\) as a curve over \(\mathbb{C}\) (and hence over \(\overline{\mathbb{Q}}\)). Observe how these groups realize the two aforementioned possibilities for the number of elliptic points above \(j_{\mathrm{CV}}=1728\) and of cusps.

To determine whether \(\mathcal{H}_4\) is \(X_0(3)\) or \(X(4A^0)\) we just need to compute \(\epsilon_2\). Recall that quadruples \(\widehat y=\{y_1,y_2,y_3,y_4\}\in\mathcal{U}_4\) with elliptic \(j\)-invariant have stabilizer larger than the usual Klein group (in [13], Cadoret gives explicit descriptions for the Legendre representatives), in particular \(D_8\) for \(j_{\mathrm{CV}}=1728\): \(\mathcal{H}_4\) has no elliptic point above \(j_{\mathrm{CV}}=1728\), i.e. \(\epsilon_2=0\), precisely if this stabilizer acts freely on topological types in the fiber. Indeed, the elliptic point above \(j_{\mathrm{CV}}=0\) corresponds to (the orbit of) the topological type of \(p_0(x)=x^4-x\), which is fixed by multiplication by \(\omega\). The following result shows that our reduced Hurwitz space is the modular curve \(X_0(3):\)

Notice how Proposition 9 together with Lemma 10 imply the following nice result:

At this point, we have a rational curve \(\mathcal{H}_4\) that fits into a diagram:

where we take \(\pi_3\) (not to be confused with one of the maps in the bottom row of the diagram in 3.1) as a rational map making \(X_0(3)\) into the moduli space of isomorphism classes \([(E,C)]\) of elliptic curves \(E\) with a cyclic \(3\)-torsion subgroup \(C\) (so that \(j(E)=\pi_3([(E,C)])\)). A priori, the isomorphism \(\psi_4\) is only defined over \(\overline{\mathbb{Q}}\): one may ask whether it is forced to be defined over \(\mathbb{Q}\). The answer is affirmative:

4.2 A direct method↩︎

Our Hurwitz space \(\mathcal{H}_4\) has genus \(0\) and a certain algebraic structure over \(\mathbb{Q}\) coming from Hurwitz theory. Since a point on \(\mathcal{H}_4\) represents a \(\text{PGL}_2(\mathbb{C})\)-orbit of topological types, it is natural to ask if we can describe its coordinate explicitly as a function of the orbit. Recall that the morphism \(\theta_d\) of Section 2 maps critical points to critical values: one may then guess that \(\beta_4\) maps the \(j\)-invariant of the critical points of the orbit to that of the critical values. Indeed, this map is well-defined: the critical points of a topological type form a \(\text{PGL}_2(\mathbb{C})\)-orbit, while \(Z_{\alpha(f)}=Z_f\) for \(\alpha\in\text{PGL}_2(\mathbb{C})\) (basically, their behavior under the two \(\text{PGL}_2(\mathbb{C})\)-actions we introduced is the mirror image of that of critical values). Moreover, to any \(j\in\mathbb{C}\) we can associate the \(\text{PGL}_2(\mathbb{C})\)-orbit of the (unique, by the above discussion) topological type of \[\int_0^x(u-u_1)(u-u_2)(u-u_3) du,\] with \(\{u_1,u_2,u_3,\infty\}\) any quadruple with \(j\)-invariant \(j\). In summary, by sending an orbit to the \(j\)-invariant of its critical points, we get a bijection: \[\begin{align} \left\lbrace \begin{aligned} \displaystyle \text{\text{PGL}_2(\mathbb{C})-orbits of topological types of complex Morse quartics} \; \end{aligned} \right\rbrace \overset{1:1}{\longleftrightarrow} \; \mathbb{A}^1(\mathbb{C}). \label{cp} \end{align}\tag{14}\]

Identify \(H_4^{\text{red}}(S_4,C_{\text{pol}})\) with \(\mathbb{A}^1\) via the \(j\)-invariant \(j\) of the critical points: by compactifying, the above correspondence yields a map \(\widetilde{\beta_4}\colon\mathcal{H}_4\longrightarrow X(1)\), \(j\mapsto j_{\mathrm{CV}}\). Note that even if we know that such a map exists, it need not be \(\beta_4\) a priori: nonetheless, we will now prove that \(\widetilde{\beta_4}\) is indeed an algebraic morphism defined over \(\mathbb{Q}\) by computing it explicitly, so necessarily \(\widetilde{\beta_4}\) and \(\beta_4\) are the same up to a change of variable in \(\text{PGL}_2(\mathbb{Q})\) by the uniqueness property for algebraic models of Hurwitz spaces (see the paragraph before 8 ). In the following, the variable \(j\) will represent a point in \(\mathcal{H}_4\) with the above identification.

Before working out an explicit model for \(\beta_4\), we remark that the above discussion shows how any point \(j\in H_4^{\text{red}}(S_4,C_{\text{pol}})(K)\) lifts to a quartic defined over \(K\): we can just integrate the Weierstrass polynomial of any \(K\)-rational elliptic curve with \(j\)-invariant \(j\) (the existence of such a curve is well known, see [26]). This completes the proof of Theorem 4.

Fix an isomorphism \(u\colon\mathbb{P}^1\longrightarrow X_0(3)\) and let us take for \(\pi_3\) the model given by \(u\mapsto \frac{(u+3)^3(u+27)}{u}\) (see [27]). For a complex cubic curve given by a Weierstrass equation \(E: y^2=x^3+Ax+B\), let¹: \[\begin{align} f_{E}(x)=12\int_0^x(s^3+As+B) \;ds - A^2.\label{normp} \end{align}\tag{15}\] Moreover, let \(\Delta_E=-16(4A^3+27B^2)\) be the discriminant of \(E\) (note that we are allowing \(\Delta_E\) to vanish), let \(j=-1728\frac{(4A)^3}{\Delta_E}\in\mathbb{P}^1(\mathbb{C})\) be its \(j\)-invariant and \(j_{\mathrm{CV}}\in\mathbb{P}^1(\mathbb{C})\) be that of the cubic \(E^{\mathrm{CV}}:z^2=p_E(x)\), where \(p_E(x)\) is the monic polynomial vanishing at the critical values of \(f_{E}\). Since \(j\) and \(j_{\mathrm{CV}}\) represent respectively the \(j\)-invariant of the critical points and of the critical values of a variable quartic, there is no ambiguity with our coordinates for \(\mathcal{H}_4\) and \(X(1)\), as the latter stand for the same things in our moduli problem.

Observe how the second-to-last expression in 17 is our explicit model for \(\beta_4\), so we find \(\psi_4(j)=\frac{j}{64}-27\) in the diagram in 4.1. Moreover, Lemma 12 implies that the integrals of the Weierstrass polynomials of \(\mathbb{C}-\)isomorphic elliptic curves have the same critical \(j\)-invariant, which is not a surprise since it is equivalent to the right direction of 14 . Finally, the lemma allows us to characterize the critical \(j\)-invariants of quartics over \(K\): let \[\begin{align} CJ(K)=\{v\in K\cup\{\infty\}:\exists f\in K[x]\text{ s.t. \deg f=4 and }j(C_f)=v\}. \end{align}\]

Let us examine the fiber of \(\beta_4\) above \(j_{\mathrm{CV}}=0\): since \(\pi_3\) vanishes only for \(u=-3\) and \(u=-27\) (with order respectively \(3\) and \(1\), in accordance with the ramification structure we previously found), Lemma 12 tells us that the values of \(j\) corresponding to these two preimages are \(0\) and \(1536\): in other words, the elliptic curves with \(j\)-invariant \(0\) and \(1536\) are precisely those with Weierstrass integrals having critical \(j\)-invariant \(0\). What about elliptic curves with \(j=1728\)? Since \(\beta_4(1728)=\infty\), they have non-Morse Weierstrass integrals: this is indeed immediate to check with \(E:y^2=x^3-x\).

We are now almost ready to prove Theorem 2: we only need the following strengthening of Theorem 4
r(i):

5 Equicritical quartics and elliptic curves↩︎

We have come to the question of inequivalent equicritical quartics: the goal of this section is to prove Theorem 1. Let \(\mathcal{C}_4\) be the non-diagonal irreducible component of \(\mathcal{H}_4\underset{X(1)}{\times}\mathcal{H}_4\) (there may be a priori up to three such components, but the results of 5.1 will show that there is only one) and let \(\nu:C_4\longrightarrow\mathcal{C}_4\) be its normalization; we refer to a point of \(P\in C_4(\overline{\mathbb{Q}})\) such that \(\nu(P)\) is singular (or to a pair of quartics to which \(P\) lifts) as “exceptional". A pair of inequivalent equicritical quartics \((f,g)\) over a number field \(K\) clearly defines a \(K\)-point on \(C_4\). Moreover, Lemma 14 implies that, generically, such a point lifts to a pair of \(K\)-rational equicritical quartics. Therefore, our task is reduced, at least generically, to that of finding rational points on \(C_4\).

5.1 Elliptic curves with constant mod \(3\) representation↩︎

As \(H_4^{\text{red}}(S_4,C_{\text{pol}})\simeq_{\mathbb{Q}}Y_0(3)\) as covers of \(Y(1)\) by Lemma 11, \(C_4\) turns out to be \(\mathbb{Q}\)-isomorphic to the curve \(X_3\) studied by Rubin and Silverberg [28] in the context of parametrizing elliptic families with given mod \(p\) Galois representation. Indeed, in virtue of Corollary 7 we have \[\begin{align} H_4^{\text{red}}(S_4,C_{\text{pol}})\underset{Y(1)}{\times}H_4^{\text{red}}(S_4,C_{\text{pol}})\simeq Y_0(3)\underset{Y(1)}{\times}Y_1(3), \end{align}\]so removing the diagonal we precisely obtain the second equivalent definition for \(Y_3\) given in [28]. This is enough for us to give a proof of a nonconstructive version of Theorem 1:

The curve \(X_3\) is itself a moduli space of elliptic curves, with the additional structure induced by the self-fiber product of \(X_0(3)\). In the aforementioned work of Rubin and Silverberg, the authors give a certain rational model \(t\colon\mathbb{P}^1\longrightarrow X_3\) of \(X_3\), and write down the respective affine equation for the universal elliptic curve over \(X_3\) as \[\begin{align} E_t\colon Y^2=X^3-27t(t^3+8)X+54(t^6-20t^3-8), \end{align}\]having \(j\)-invariant \[\begin{align} j_t=j(E_t)=27\left(\frac{t(t^3+8)}{t^3-1}\right)^3.\label{juni} \end{align}\tag{19}\]

We take as our model for \(X_3\) the one obtained by compactifying the self-fiber product of \((X_0(3),\pi_3)\). Thanks to Lemma 12 and 19 , we can explicitly parametrize \(C_4(K)\): this amounts to finding two rational functions \(x_1(t), x_2(t)\) such that the pairs \((x_1(t),x_2(t))\) as \(t\) ranges in \(K\cup\{\infty\}\) parametrize \(X_3(K)\), and then pulling them back via \(\psi_4\). Searching for rational functions such that \[\begin{align} \pi_3(x(t))=j_t\label{sect} \end{align}\tag{20}\] by comparing the right-hand side of 19 with \(\pi_3\), it is not particularly hard to figure out that \[\begin{align} x_1(t)=\frac{27}{(t^3-1)}\text{ and }x_2(t)=\frac{3(t-1)^3}{t^2+t+1}=x_1\left(\frac{t+2}{t-1}\right)\label{xis} \end{align}\tag{21}\] work. We then have:

Proposition 16 and 21 give: \[\begin{align} C_4(K)=\biggl\{\left(1728\frac{t^3}{t^3-1},1728+192\frac{(t-1)^3}{t^2+t+1}\right), \;t\in K\cup\{\infty\}\biggl\},\label{jequicq} \end{align}\tag{22}\] where we treat pairs appearing with multiplicity as distinct points. Recall the notation \(\rho=1+\sqrt3, \;\overline{\rho}=1-\sqrt3\), let \(\widetilde{K}=K\setminus\{\omega,\omega^2,-2\omega,-2\omega^2,\rho,\overline{\rho}\}\) and set \(j_1(t)=1728\frac{t^3}{t^3-1}\), \(j_2(t)=1728+192\frac{(t-1)^3}{t^2+t+1}\). Then 22 means precisely that, for \(t\in K\cup\{\infty\}\), the integrals of the Weierstrass polynomials of any two cubic curves \(E_t, F_t\) with \(j(E_t)=j_1(t)\) and \(j(F_t)=j_2(t)\) are two quartics with critical \(j\)-invariant \(j_t\), and they are inequivalent if \(t\in\widetilde{K}\cup\{\infty\}\). This, along with Lemma 14, gives a nice criterion for a (generic) quartic over \(K\) to have an inequivalent equicritical “twin":

In order to parametrize a basis for \(\text{EC}_4(K)\), we need to find an equicritical pair over \(K\) for each \(t\) (if there is one), that is, we need to explicitly go through the procedure described in the proof of Lemma 14 for nonelliptic \(j_{\mathrm{CV}}=j_t\), while taking care of the excluded values separately. Specifically, given \(t\in \widetilde{K}\setminus\{-2,0,1\}\), we:

1. pick elliptic curves \(E_t,F_t\) over \(K\) with \(j(E_t)=j_1(t), \;j(F_t)=j_2(t)\), yielding us quartic lifts \(f^t=f_{E_t}, \;g^t=f_{F_t}\in K[x]\) of \(j_1(t)\) and \(j_2(t)\), with the notation of 15 ;

2. find the lift \(\widetilde{g^t}\) of \(j_2(t)\) such that \(C_{\widetilde{g^t}}=C_{f^t}\), by computing \(\lambda_t(g^t)\) with \(\lambda_t\in\text{Aff}(K)\) the transformation mapping \(F^{\mathrm{CV}}_t[2]\) to \(E^{\mathrm{CV}}_t[2]\) (we will not need to pick another representative of the topological type since, as we will see, the family \(\lambda_t\) is defined over \(\mathbb{Q}(t)\)).

The first step is easy: it is well known that the elliptic curve \(E_j:y^2=x^3+\frac{3j}{1728-j}x+\frac{2j}{1728-j}\) has \(j\)-invariant \(j\) for all \(j\neq0,1728\), so we can just take \[\begin{align} E_t: y^2=x^3-3t^3x-2t^3\text{ and } F_t: y^2=x^3-3{\left(\frac{t+2}{t-1}\right)^3}x-2\left(\frac{t+2}{t-1}\right)^3, \;t\in\widetilde{K}\setminus\{-2,0,1\}, \end{align}\] corresponding to \[\begin{align} f^t(x)=3x^4-18t^3x^2-24t^3x-9t^6\text{ and }g^t(x)=3x^4-18{\left(\frac{t+2}{t-1}\right)^3}x^2-24{\left(\frac{t+2}{t-1}\right)^3}x-9\left(\frac{t+2}{t-1}\right)^6.\label{pols} \end{align}\tag{23}\] The second step is in principle more complex, but a careful examination of the critical values for the first few negative integer values for \(t\) suggests for \(\lambda_t\) the form \[\begin{align} z\mapsto-\frac{t^4(t-1)^6}{3(t+2)^4}z-36t^4(t^2+t+1),\label{2map} \end{align}\tag{24}\] which we easily verify with SageMath [29]. As anticipated, the \(\lambda_t\)’s turn out to be defined over \(K\), so there is no need to look for a rational representative in \([g^t]\). Let us remark that it is always the case that, for two elliptic curves \(E,F\) defined over \(K\), isomorphic over \(\overline{\mathbb{Q}}\) and with nonelliptic \(j\)-invariant, their \(2\)-torsions are related by an affine transformation defined over \(K\). Indeed, any \(\overline{\mathbb{Q}}\)-isomorphism \(\phi:E\longrightarrow F\) descends to an isomorphism \(\lambda:E[2]\longrightarrow F[2]\) of the \(2\)-torsions as Galois modules, so in particular it is affine. If \(\lambda\) is not defined over \(K\), there is \(\sigma\in G_K\) such that \(\lambda^{-1}\circ{^{\sigma}}\lambda\neq I\), so in particular \(\phi^{-1}\circ{^{\sigma}}\phi\in\text{Aut}_{\overline{\mathbb{Q}}}(E)\setminus\langle\iota\rangle\), with \(\iota\) the elliptic involution; in other words, \(E\) has nontrivial automorphisms, and hence elliptic \(j\)-invariant.

Putting together Proposition 16, 23 and 24 , we have proved:

More specifically, substituting in 22 we see that the values \(t=-2,0,1,\infty\) we left out correspond to the pairs \((1536,0),(0,1536),(\infty,1728),(1728,\infty)\in C_4(K)\), while Remark 8 tells us that the other points in \(C_4(K)\) which we might be missing representatives for are the eight exceptional ones, along with the two diagonal points \((1152\pm384\sqrt3,1152\pm384\sqrt3)\).

5.2 Elliptic fibers and cusps↩︎

Let us now deal with equicritical pairs lifting the remaining eight points in \(C_4(K)\). We start by examining the situation above \(j_{\mathrm{CV}}\in\{0,1728\}\).

1. \(j_{\mathrm{CV}}=0\): let us deal with the points \((0,1536)\) and \((1536,0)\) first; we already found a representative \((p_0)\) for the \(\text{PGL}_2(\mathbb{C})\)-orbit of topological types corresponding to \(j=0\), and we can do the same for \(j=1536\) simply by integrating the Weierstrass polynomial of any elliptic curve with \(j\)-invariant \(1536\). Still, the affine transformation relating the critical values of the two representatives is not guaranteed to be rational (recall that this is why Lemma 14 fails for elliptic \(j_{\mathrm{CV}}\)), but it happens to be so in some cases: choosing the right elliptic curve with the help of the L-functions and modular forms database (LMFDB) [30], we obtain the two additional equicritical pairs \(\mathcal{c}_0=(f_0,g_0)\) and \(\mathcal{c}_{-2}=(g_0,f_0)\) with \[\begin{align} f_0=p_0=x^4-x, \;g_0=-\frac{sec:1}{48}x^4-\frac{sec:1}{sec:4}x^2+\frac{sec:1}{6}x-\frac{sec:1}{sec:2}.\label{j610} \end{align}\tag{25}\]

   Let us now look at the two exceptional points in this fiber, both represented by the coordinates \((1536,1536)\) in our parametrization: we know that, for any \(\widehat y\in\mathcal{U}_4(K)\) such that \(j(\widehat y)=j_{\mathrm{CV}}\), there are three topological types with critical values \(\widehat y\) that lift \(j=1536\). We have just constructed a rational one \([g_0]\); the other two will be given by \([\lambda(g_0)]\) and \([\lambda^{(2)}(g_0)]\) with \(\lambda\) the nontrivial affine transformation preserving \(C_{g_0}\) (we know by Corollary 5 that these are distinct topological types), which, since \(C_{g_0}=C_{p_0}\), is simply the multiplication by \(\omega\). As \(\mathbb{Q}(\omega)\) is clearly the minimal field of definition of any conjugate of \(\lambda\), we get two additional pairs \[\begin{align} \mathcal{c}_{-2\omega}=(g_0,\omega g_0)=\overline{\mathcal{c}_{-2\omega^2}}\label{j0double} \end{align}\tag{26}\] if and only if \(\omega\in K\) (and no other, since the pairs \((g_0,\omega^2 g_0)\) and \((\omega g_0,\omega^2 g_0)\) belong to the orbit of one of those appearing in 26 ).

2. \(j_{\mathrm{CV}}=1728\): we consider the diagonal points \((1152\pm384\sqrt3,1152\pm384\sqrt3)\) first; they correspond to the values \(t=\rho,\overline{\rho}\) in our parametrization, so the quartics \(f_{\rho},f_{\overline{\rho}}\) (following the notation of 5.1) are lifts for \(j=1152+384\sqrt3\) and \(j=1152-384\sqrt3\) respectively. As we just saw in the case \(j_{\mathrm{CV}}=0\), to find the other lifts making up the pairs we need to apply to \(f_{\rho}\) and \(f_{\overline{\rho}}\) the affine transformations preserving their respective critical value sets. SageMath [29] shows that \(f_{\rho}=x^4-6\rho^3x^2-8\rho^3x\) satisfies \(C_{f_{\rho}}=(-\upsilon+C,C,\upsilon+C)\) for \(\upsilon=72\sqrt{362\sqrt3 + 627}\) and \(C=-720\sqrt3 - 1248\). As \(f_{\rho}\) and \(f_{\overline{\rho}}\) are Galois conjugates, so are \(C_{f_{\rho}}\) and \(C_{f_{\overline{\rho}}}\), and hence we get the additional pairs \[\begin{align} \mathcal{c}_{\rho}=(f_{\rho}, \;-f_{\rho}+2C), \;\mathcal{c}_{\overline{\rho}}=(f_{\overline{\rho}}, -f_{\overline{\rho}}+2\overline{C})\label{j611728} \end{align}\tag{27}\] if and only if \(\sqrt3\in K\).

   The previous paragraph enables us to also find pairs lifting the four exceptional points in this fiber, corresponding to \(t=\omega\rho,\omega^2\rho,\omega\overline{\rho},\omega^2\overline{\rho}\), as their orbits in \(\text{EC}_4(K)\) are necessarily obtained by taking one orbit from each of the pairs in 27 . Again with the help of SageMath [29], we find that the linear map sending \(C_{f_{\overline{\rho}}}\) to \(C_{f_{\rho}}\) is \(z\mapsto iRz-iR\overline{C}\), with \(R=362+209\sqrt3\), yielding the four additional pairs\[\begin{align} \mathcal{c}_{\omega\rho}=(f_{\rho}, \;iRf_{\overline{\rho}}-iR\overline{C})=\overline{\mathcal{c}_{\omega\overline{\rho}}}, \;\mathcal{c}_{\omega^2\rho}=(f_{\rho}, \;iR(-f_{\overline{\rho}}+2\overline{C})-iR\overline{C})=\overline{\mathcal{c}_{\omega^2\overline{\rho}}}\label{exc1728} \end{align}\tag{28}\] if and only if \(\omega,\sqrt3\in K\). Note that the other four possible combinations do not appear in the basis, as they all belong to the \(\text{PGL}_2(\mathbb{C})\)-orbit of one of those in 28 .

Let us now look above \(j_{\mathrm{CV}}=\infty\). As anticipated, the two cusps \(j=\infty,1728\) of \(\mathcal{H}_4\) correspond to the \(\text{PGL}_2(\mathbb{C})\)-orbits of topological types \((f)\) of non-Morse quartics; indeed, there are two of these: if \(f\) branches over less than three points, then either \(f'\) has a double root—in which case \(f\) is of the form \(A(x-a)^3(x-b)+B, \;(A,B)\in K^*\times K, \;a,b\in\overline{\mathbb{Q}}\) (type 1)—or \(f\) takes the same value at two of the distinct roots of \(f'\), i.e. \(f=A(x-a)^2(x-b)^2+B\) for \(A,B,a,b\) as above (type 2) (we can assume \(a\neq b\), as we otherwise get the \(\text{Aff}(K)\)-orbit of a fourth power, which is, above each choice for the critical value \(B\), the class of a unique topological type). When we mod out by the outer action of \(\text{Aff}(\overline{\mathbb{Q}})\), type 1 and type 2 are clearly distinct topological types. Observe that if a multiset \(\{w,w,x\}\) is fixed by an affine transformation, this must necessarily fix \(w\) as the element of multiplicity \(2\), and hence also \(x\), so it must be the identity: therefore, there are no equicritical non-Morse quartics arising from the same outer \(\text{Aff}(\overline{\mathbb{Q}})\)-orbit. In other words, the two exceptional points with coordinates \((\infty,\infty)\) in our parametrization, which correspond to \(t=\omega,\omega^2\), do not lift to an inequivalent equicritical pair.

All we have to do now is find, if it exists, an equicritical pair \((f_1,g_1)\) with \(f\) of type 1 and \(g\) of type 2. It is not hard to see that \[\begin{align} (f_1,g_1)=(-3x^3(x+2), \;x^2(x+3)^2),\label{j61inf} \end{align}\tag{29}\] with critical values \(\{0,0,\frac{81}{16}\}\), works, and this gives the last two ordered pairs \(\mathcal{c}_1=(f_1,g_1)\) and \(\mathcal{c}_{\infty}=(g_1,f_1)\). Finally, the parametrization part of Theorem 1 follows from Lemma 17 together with 25 , 26 , 27 , 28 and 29 , while the last part follows from Remark 9.

5.3 Application to Weyl sums mod \(p^2\), and one example↩︎

As we anticipated in 1, our main result has an application in the theory of exponential sums attached to polynomials. Specifically, given \(a\in\mathbb{Z}\) and a prime power \(q\), in [5] Kowalski and Soundararajan consider the mod \(q\) Weyl sum\[\begin{align} W_{f}(a,q)=\frac{sec:1}{\sqrt q}\sum_{x\in\mathbb{Z}/q\mathbb{Z}}e\left(\frac{af(x)}{q}\right) \end{align}\]attached to \(f\in\mathbb{Z}[x]\), investigating pairs \((f,g)\) satisfying \(W_f(a,q)=W_g(a,q)\) (or other weaker variants). For \(q=p^2\) they observe [5] that, when \((a,p)=1\), one has\[\begin{align} W_{f}(a,q)=\sum_{\substack{f'(v)=0\\v\in\mathbb{F}_p}}e\left(\frac{af(v)}{p}\right), \end{align}\] so equicritical polynomials \(f,g\in\mathbb{Z}[x]\) whose leading coefficients are invertible mod \(p\) satisfy \(W_f(a,p^2)=W_g(a,p^2)\) for \((a,p)=1\). Therefore, following its notation, Theorem 1 implies:

Finally, let us test Theorem 1 with an example: take, for instance, \(t=42\). Then, the simple implementation

from sage.all import QQbar, polygen, sqrt

def critical_values_exact(f):
    x = f.parent().gen()
  
    fp = f.derivative()
    critical_points = fp.roots(multiplicities=False)
    critical_values = []
    
    for cp in critical_points:
        cv = f(cp)
        cv_radical = cv.radical_expression()
        critical_values.append((cv==cv_radical))

    return critical_values

def main():
    x = polygen(QQbar)
    t = 42
    v = t**4*(t-1)**3/(t+2)
    f = x**4 - 6*t**3*x**2 - 8*t**3*x
    g = -(t-1)**3*v/(3*(t+2)**3)*x^4 + 2*v*x**2 + 8/3*v*x - 8*t**4*(t**2+t+1)
    
    print("Polynomial f(x):") 
    print(f)
    print("Polynomial g(x):") 
    print(g)
    
    critical_values_f = critical_values_exact(f)
    critical_values_g = critical_values_exact(g)
    
    print("\nCritical values of f:")
    for i, cv in enumerate(critical_values_f, 1):
        print(f"Critical value {i}:\n {cv}")
        print()
        
    print("\nCritical values of g:")
    for i, cv in enumerate(critical_values_g, 1):
        print(f"Critical value {i}:\n {cv}")
        print()
        
if __name__ == "__main__":
    main()

in SageMath [29] returns the same critical values both in numerical and radical form:

Polynomial f(x): 
x^4 - 444528*x^2 - 592704*x
Polynomial g(x): 
-307935007631307/234256*x^4 + 107230600008/11*x^2 + 142974133344/11*x - 44982677376

Critical values of f:
Critical value 1: 
-4.912195494217215?e10 == -1/2*(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3)*(I*sqrt(3) + 1) - 
135597252859697147904*(-I*sqrt(3) + 1)/(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3) - 32934190464

Critical value 2: 
197568.1975316542? == (482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3) + 
271194505719394295808/(482155633265041602566848512*I*sqrt(74087) + 
4464098178074162551805159964672)^(1/3) - 32934190464

Critical value 3: 
-4.968081401802539?e10 == -1/2*(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3)*(-I*sqrt(3) + 1) - 
135597252859697147904*(I*sqrt(3) + 1)/(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3) - 32934190464

Critical values of g:
Critical value 1: 
-4.912195494217215?e10 == -1/2*(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3)*(I*sqrt(3) + 1) - 
135597252859697147904*(-I*sqrt(3) + 1)/(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3) - 32934190464

Critical value 2: 
-4.968081401802539?e10 == -1/2*(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3)*(-I*sqrt(3) + 1) - 
135597252859697147904*(I*sqrt(3) + 1)/(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3) - 32934190464

Critical value 3: 
197568.1975316542? == (482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3) + 
271194505719394295808/(482155633265041602566848512*I*sqrt(74087) +
4464098178074162551805159964672)^(1/3) - 32934190464

as expected.

References↩︎