9 Comparison with Santens

We use the minimal resolution \(\pi :X\to S\) of the preceding chapter. The proposition first specifies the integral points and height. Its proof verifies the hypotheses of Santens’s theorem and then computes the leading constant.

Proposition 12 The Santens constant

✓

Let \(\pi \colon X\to S\) be the minimal resolution, let \(D\subset X\) be the reduced preimage of \(D_0\), and put \(U=X\setminus D\). The coordinate ring of the affine chart \(\mathscr {S}\setminus \mathscr {D}_0\) pulls back to a finitely generated \(\mathbb {Z}\)-subalgebra

\[ A:=\pi ^*\Gamma (\mathscr {S}\setminus \mathscr {D}_0,\mathcal O) \subset \Gamma (U,\mathcal O). \]

A point \(P\in U(\mathbb {Q})\) is integral if \(g(P)\in \mathbb {Z}\) for every \(g\in A\). With this integral condition and the height \(H(\pi (-),1)\), [ 28 , Theorem 8.7 ] gives

\[ \# \{ P\in (\mathscr {S}\setminus \mathscr {D}_0)(\mathbb {Z}):H(P,1)\leq B\} \sim cB^2\log B,\qquad c=\frac{2\alpha }{9}. \]

Proof in Lean ▼

Use the pullback of the affine coordinate ring to express the integral condition. Verify the fan, divisor, and positivity hypotheses, then evaluate the dual-cone volume and the normalized local integrals. Their product is \(2\alpha /9\).

The proof assumes Santens’s theorem (Cited theorem 1).

Equation 47

✓

For this surface the exponents are \(a=b=2\) and the dual-cone factor is \(\theta =2\). Write \(\rho _E=\operatorname {Res}_{s=1}\xi (s)\). For a finite prime \(p\), \(\tau _p^\prime \) is the local volume of the integral points multiplied by \(L_p(1,\operatorname {Pic}U_{\overline{\mathbb {Q}}})^{-1}\); \(\tau _\infty ^\prime \) is the residue volume of the real boundary component. The Brauer factor is one.

\begin{equation*} c=\frac{\theta }{a(b-1)!}\, \tau =\tau ,\qquad \tau =2\rho _E\prod _v\tau ’_v, \end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.

Equation 48

✓

For \(P\in T(\mathbb {Q}_v)\), put \(\nu =\iota (P)\in (E\otimes _\mathbb {Q}\mathbb {Q}_v)^\times \) and \(n=\nu \overline\nu \). Each \(|\cdot |_w\) is applied to the \(w\)-component. The height for \(3D_0\) is given by the following formula.

\begin{equation*} H_v(P,3)=\prod _{w\mid v} \max \bigl(1,|n|_w,|\nu ^2\overline\nu |_w, |\nu \overline\nu ^2|_w\bigr)^{1/2}. \end{equation*}

Proof in Lean ▼

The linked Lean proof establishes this statement.