7 Counting arbitrary trace

Let \(F_s\) be the monic integral cubics of trace \(s\) and Galois group \(C_3\), and put \(n(f)=a_1^2-3a_2\). We use

\[ A_1(X)=\# \{ t^3-t^2+at+b\in F_1:1-3a\leq X\} , \]

\[ A_0(X)=\# \{ t^3-Nt+c\in F_0:1\leq N\leq X\} . \]

For \(N\geq 1\), let \(r(N)=\# \{ c\in \mathbb {Z}:4N^3-27c^2\text{ is a square}\} \). The polynomial \(Q\) is as in Corollary 1. Summing the trace-zero and trace-one counts over \(s\) proves Theorem 1.

Lemma 12 Reducing the trace

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For \(s\in \mathbb {Z}\), define \(\phi _s\) on \(F_s\) by

\[ \phi _s(f)(t)= \begin{cases} f(t+\tfrac {s}{3}),& s\equiv 0\pmod3,\\ f(t+\tfrac {s-1}{3}),& s\equiv 1\pmod3,\\ -f(-t+\tfrac {s+1}{3}),& s\equiv -1\pmod3. \end{cases} \]

Then \(\phi _s\) is a bijection from \(F_s\) to \(F_0\) if \(3\mid s\), and from \(F_s\) to \(F_1\) if \(3\nmid s\), and it preserves \(n\). For \(f\in F_s\), put \(n=n(f)\). Then \(n{\gt}0\), and

\[ H(f)\leq H \quad \Longleftrightarrow \quad |s|\leq H\ \text{ and }\ n\leq s^2+3H^2. \]

If \(3\nmid s\), then \(H_{\mathrm{tor}}(\phi _s(f))^2=n\). If \(3\mid s\), then \(\phi _s(f)=t^3-\frac n3t+c\) for some \(c\in \mathbb {Z}\).

Proof in Lean ▼

Integer translation, followed by reflection in the third residue class, gives the stated bijections and preserves \(n\). The real-root coefficient inequalities reduce the height condition to the two displayed bounds.

Equation 36

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For every \(\varepsilon {\gt}0\), the following asymptotic holds as \(X\to \infty \).

\begin{equation*} A_1(X) =\frac12XQ\left(\tfrac 12\log X\right) -\frac{\pi }{9\sqrt3}X +O_\varepsilon (X^{1/2+\varepsilon }). \end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.

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

Lemma 13 Trace-zero coefficients

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Let \(e\) be the exponent of \(3\) in \(N\), and write \(N=3^eM\) with \(3\nmid M\). Then

\[ r(N)= \begin{cases} d_M,& e=0,\\ 3d_M+1,& N=3u^2\text{ for some }u\in \mathbb {Z}_{{\gt}0},\\ 3d_M,& \text{otherwise}. \end{cases} \]

Proof in Lean ▼

Factor in the Eisenstein integers and separate the valuation at \(1-\zeta \). Count the allowed unit multiples. The vanishing-discriminant cases contribute the extra term when \(N=3u^2\).

The proof assumes Poisson summation (Equation 15).

Proposition 8 The trace-zero count

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For every \(\varepsilon {\gt}0\),

\[ A_0(X) =\frac52XQ\left(\tfrac 12\log X\right) -\left(\frac{\alpha }{4}\log 3+\frac{\pi }{3\sqrt3}\right)X +O_\varepsilon (X^{1/2+\varepsilon }). \]

Proof in Lean ▼

Sum the coefficient formula over the powers of \(3\), apply the asymptotic for the \(d_M\), and remove the reducible cubics. The geometric series in the powers of \(3\) gives the stated constants.

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

Equation 37

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\begin{align*} \# \{ f\in F_{\mathrm{mon}}:H(f)\leq H\} =\sum _{\substack {s\in \mathbb {Z},\ |s|\leq H\\ { 3\nmid s }}} A_1(3H^2+s^2) +\sum _{\substack {s\in \mathbb {Z},\ |s|\leq H\\ { 3\mid s }}} A_0\left(H^2+\frac{s^2}{3}\right). \end{align*}

Proof in Lean ▼

The linked Lean proof establishes this statement.

Equation 38

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\begin{equation*} \begin{aligned} \int _{-1}^1(3+u^2)\, du& =\frac{20}{3},\\ \int _{-1}^1(3+u^2)\log (3+u^2)\, du & =\frac{40}{3}\log 2+\frac{4\pi \sqrt3}{3}-\frac{76}{9}. \end{aligned}\end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.

Equation 39

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\begin{equation*} P(X) =\frac{110}{27}Q(X) +\frac{220\alpha }{243}\left(\log 2-\frac{19}{44}\log 3 -\frac{19}{30}+\frac{\pi \sqrt3}{10}\right) -\frac{20\pi \sqrt3}{81}. \end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.