1 Introduction

We count $f(t)=t^3+a_1t^2+a_2t+a_3\in \mathbb {Z}[t]$ with Galois group $C_3$, ordered by $H(f)=\max (|a_1|,|a_2|^{1/2},|a_3|^{1/3})$. Throughout, $E=\mathbb {Q}(\sqrt{-3})$ and $\xi =\zeta _E$. The constants in the main terms are defined first. We write $S=\mathbb {P}^2/C_3$, $\mathscr {S}$ for its integral model, $D_0$ for the image of $X+Y+Z=0$, and $T$ for its open torus. The integral structure and toric height are described in the chapters below.

Equation 1

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\begin{equation*} \mu =\lim _{s\to 1} \left(\frac{\xi '(s)}{\xi (s)}+\frac{1}{s-1}\right), \quad \alpha =\prod _{p\equiv 1\pmod3}\left(1-\frac{2}{p(p+1)}\right), \quad \beta =\sum _{p\equiv 1\pmod3}\frac{(2p+1)\log p}{(p^2-1)(p+2)}. \end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.

Theorem 1 All monic cubics

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For every $\varepsilon {\gt}0$, as $H\to \infty $, the number of monic integral cubic polynomials with Galois group $C_3$ and root height at most $H$ is

\[ H^3P(\log H)+O_\varepsilon (H^{2+\varepsilon }), \]

where

\begin{equation*} P(X)=\frac{220\alpha }{243} \biggl(X+\mu -\frac{\zeta '(2)}{\zeta (2)}+ \beta +\log 2-\frac{5}{88}\log 3 +\frac{\pi \sqrt3}{10}-\frac{17}{15}\biggr) -\frac{20\pi \sqrt3}{81}. \end{equation*}

Proof in Lean ▼

Reduce each trace class to trace zero or one. Substitute the asymptotic formulas for $A_0$ and $A_1$, approximate their main-term sums by integrals, and then sum the errors. The two real integrals give the displayed polynomial $P$.

The proof assumes Poisson summation (Equation 15) and the Chandrasekharan–Narasimhan estimate (Cited theorem 2).

Theorem 2 Trace-one cubics

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For every $\varepsilon {\gt}0$, as $H\to \infty $, the number of monic integral trace-one cubic polynomials with Galois group $C_3$ and root height at most $H$ is

\[ H^2P_1(\log H)+O_\varepsilon (H^{1+\varepsilon }) \]

where

\begin{equation*} P_1(X)=\frac\alpha 3 \biggl(X+\mu -\frac{\zeta '(2)}{\zeta (2)} + \beta +\frac78\log 3-\frac12\biggr) -\frac{\pi }{3\sqrt3}. \end{equation*}

Proof in Lean ▼

Estimate $\sum _{n\leq X}d_n$ by truncated Perron’s formula and contour shifting. Subtract the split contribution and set $X=1+3H^2$. The paper uses Alberts’s theorem for the coefficient estimate; Lean retains the direct proof.

The proof assumes Poisson summation (Equation 15) and the Chandrasekharan–Narasimhan estimate (Cited theorem 2).

Theorem 3 An exact count

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For any $H \geq 1$ let $E_H \subset \mathbb {R}^2$ be the ellipse defined by $E_{H} : x^2+y^2+xy-x-y = \tfrac {1}{3}(H^2-1)$. If $t^3-t^2+at+b\in \mathbb {Z}[t]$ has Galois group $C_3$, then $a\leq 0$. Fix $a \in \mathbb {Z}_{\leq 0}$. The number of polynomials of the form $t^3 -t^2 + at + b \in \mathbb {Z}[t]$ with Galois group $C_3$ is equal to

\begin{equation*} \frac12\sum _{d|(1-3a)} 3^{\omega (Q_1(d))} (-1)^{\Omega (Q_2(d))} -\frac16 \# E_{\sqrt{1-3a}}(\mathbb {Z}) \end{equation*}

where $Q_j(d)$ denotes the largest divisor of $d$ only divisible by primes $\equiv j \pmod3$, and $\omega (n)$ (resp. $\Omega (n)$) denotes the number of prime factors of a positive integer $n$ counted without (resp. with) multiplicity.

Proof in Lean ▼

For $n=1-3a$, the torus points are counted by $d_n$. The split points are the free cyclic orbits of $E_{\sqrt n}(\mathbb {Z})$, and each irreducible polynomial contributes two points. Thus the answer is $d_n/2-\# E_{\sqrt n}(\mathbb {Z})/6$.

The proof assumes Poisson summation (Equation 15).

Equation 2

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\begin{equation*} x_0x_1x_2=x_3^3. \end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.

Equation 3

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\begin{equation*} S^{(d)}\colon x_0(x_1^2-dx_2^2)=x_3^3. \end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.

Theorem 4 The one-variable height zeta function

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\begin{equation*} \sum _{P \in (\mathscr {S}\setminus \mathscr {D}_0)(\mathbb {Z})} H(P)^{-s} = \left(1-\frac{1}{3^{s/2}}\right)^2 \xi \left(\frac s2\right)^2 \prod _{q\equiv 2\pmod{3}}\left(1-\frac{1}{q^{s}}\right) \prod _{p\equiv 1\pmod{3}}\left(1-\frac{3}{p^{s}}+\frac{2}{p^{3s/2}}\right) \end{equation*}

where $\xi $ is the Dedekind zeta function of $E$. This height zeta function can be meromorphically continued to the half-plane $\mathrm{Re}(s){\gt}1$ and its only pole in this region is at $s = 2$ with order $2$.

Proof in Lean ▼

Restrict the multivariate formula to the height under consideration and identify its Euler factors with those of $\xi $. The correction product is holomorphic in the required half-plane; the Laurent expansion gives the unique double pole.

The proof assumes Poisson summation (Equation 15).

Corollary 1 Integral points

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For every $\varepsilon {\gt}0$, as $H\to \infty $,

\[ \# \{ P\in (\mathscr {S}\setminus \mathscr {D}_0)(\mathbb {Z}):H(P)\leq H\} =H^2Q(\log H)+O_\varepsilon (H^{1+\varepsilon }), \]

where

\begin{equation*} Q(X)=\frac{2\alpha }{9} \biggl(X+\mu -\frac{\zeta '(2)}{\zeta (2)} +\beta +\frac38\log 3-\frac12\biggr). \end{equation*}

Proof in Lean ▼

The coefficient sum counts the integral points with the prescribed height. Insert its asymptotic formula and the Laurent coefficients of $Z_0$.

The proof assumes Poisson summation (Equation 15) and the Chandrasekharan–Narasimhan estimate (Cited theorem 2).

Remark 1 The alternative ideal count

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We prompted GPT-$6.1$ with the contents of an earlier version of this paper and asked it to identify the Dirichlet series in Theorem 4. Using our Eisenstein parametrization (§4) it proved that

\[ \sum _{P \in (\mathscr {S}\setminus \mathscr {D}_0)(\mathbb {Z})} H(P)^{-s} = \sum _{\substack {0\ne J\subseteq \mathcal{O}_E\\ J+3\mathcal{O}_E=\mathcal{O}_E\\ \mathrm N J\text{ a cube}}} N^E_\mathbb {Q}(J)^{-s/6} \]

where $J \subset \mathcal{O}_E$ runs over integral nonzero ideals. With the help of this description using ideals, we found another result in the literature [ 21 ] that we previously missed. After translating the $C_3$-polynomial counting problem into an integral Diophantine problem on $S^{(-3)}$ (explained in the next section), Kühleitner–Nowak’s [ 21 , Corollary 2 ] can be used to obtain an alternative proof of Corollary 1 and Theorem 2 without using toric harmonic analysis, and with an error term $O\bigl(H(\log H)^3(\log \log H)^2\bigr)$. We did not incorporate this argument as the toric approach of this paper has the merit of generalizing readily to other abelian groups where there is no analogue of [ 21 ] .

Proof in Lean ▼

The Eisenstein parametrization identifies integral torus points with ideals of cube norm prime to $3$. Apply the cube-ideal summatory estimate, remove the Euler factor at $3$, and compare the constants and heights.

The proof assumes the Kühleitner–Nowak estimate (Cited theorem 3).

Equation 4

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\begin{equation*} w_f=\frac{|\mathrm{Aut}(C_3)|}{|\mathrm{Aut}(K_f,x_f)|}= \begin{cases} 1 & \text{if $f$ has a double root,}\\ 2 & \text{otherwise,} \end{cases}\end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.