Dependencies

For \(E=\mathbb {Q}(\sqrt{-3})\) and \(\xi =\zeta _E\),

This is [ 7 , Theorem 4, (80) ] , with \([E:\mathbb {Q}]=2\) and \(\sigma =1/2\).

Let \(a_K(n)\) count integral ideals of norm \(n^3\) in a fixed quadratic field \(K\). If \(F\) is analytic at \(1\) and agrees with \((s-1)^2\sum _{n\geq 1}a_K(n)n^{-s}\) to its right, then

This is the residue form of [ 21 , Corollary 2 ] . The removal of the factor at \(3\) and the comparison with the original count are proved in Lean.

The integral-point counting theorem [ 28 , Theorem 8.7 ] , with its measure normalizations and auxiliary results, is used in the rank-two permutation-Picard setting. The Lean input is a general computational corollary for smooth complete Galois fans, integral subalgebras, and compatible metrics. Its hypotheses include the geometric positivity conditions and the conditions on integral points; its constant is the dual-cone volume times the normalized local integrals. Appendix B verifies those hypotheses and evaluates the constant for this surface. The linked proposition gives the full quantified input.

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

where

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

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

For each \(\eta {\gt}0\) there is a constant \(A_\eta \), depending only on \(\eta \), such that the following holds for all \(\sigma \geq 1/2+\eta \).

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

Let \(f\in L^1(T(\mathbb {A}))\) and suppose \(\widehat f|_{T(\mathbb {Q})^\perp }\in L^1(T(\mathbb {Q})^\perp )\). With counting measure on \(T(\mathbb {Q})\) and the normalized dual measure, the following equality holds for almost every \(y\in T(\mathbb {A})\).

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.

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.

Here \(r=\operatorname {Res}_{z=1}\xi (z)=\pi /(3\sqrt3)\) and \(\mu \) is the Euler–Kronecker constant defined above.

Let \(K_{\text{spl}} = \mathbb {Q}^3\), the split cubic algebra. Then \(C_3\subset S_3 = \mathrm{Aut}_{\mathbb {Q}\text{-alg}}(K_{\text{spl}})\) and \(K_{\text{spl}}\), equipped with its canonical \(C_3\)-action, is a \(C_3\)-algebra. Any transposition gives an isomorphism of \(C_3\)-algebras from \(K_{\text{spl}}\) to its twist \(K_{\text{spl}}'\).

An element \(x\) of trace one in the split \(C_3\)-algebra \(K_{\text{spl}}\) is normal if and only if \(x\) either has distinct coordinates or exactly two identical coordinates. For such a normal element \(x\), the pairs \((K_{\text{spl}},x)\) and \((K_{\text{spl}}',x)\) are equivalent if and only if \(x\) has exactly two identical coordinates (swapping the identical coordinates gives the required isomorphism); in particular, if \(x\) has distinct coordinates then \((K_{\text{spl}},x)\) and \((K_{\text{spl}}',x)\) determine different rational points of \(\mathcal G_1/C_3\), even though \(K_{\text{spl}}\) and \(K_{\text{spl}}'\) are isomorphic as \(C_3\)-algebras.

Let \(K\) be a cubic abelian number field. Then \(K\), equipped with its canonical Galois action, is a \(C_3\)-algebra. The twist \(K'\) of the \(C_3\)-algebra \(K\) by the outer automorphism \(g \mapsto g^{-1}\) of \(C_3\) (with twisted action \(g \ast x = g^{-1}x\)) is not isomorphic to \(K\) as a \(C_3\)-algebra. ¹

If an adele has component in the maximal compact subgroup at every prime outside a finite set \(S\), its finite logarithm is the sum of the local logarithms over \(S\).

For \(\operatorname {Re}(s_0){\gt}0\) and \(\operatorname {Re}(\sigma ){\gt}0\), the Fourier integral in the logarithm coordinate, with Lebesgue measure, is

The five components of the reduced boundary, in chain order, have self-intersections \(-2,-2,-1,-2,-2\).

On the minimal resolution, \(D_{\mathrm{red}}^2=-1\).

Over a field of characteristic zero containing a primitive cube root \(\zeta \), write \(g=u e_0+v e_1+w e_2\) in the idempotent basis of the group algebra. The three character evaluations are \(u,v,w\):

For every \(\beta {\gt}1/2\) there is a constant \(C_\beta \) such that

For \(\operatorname {Re}(z){\gt}1/2\),

In a commutative ring, if \(\zeta ^2+\zeta +1=0\), then

For \(\operatorname {Re}(\sigma ){\gt}1\) and \(t\in \mathbb {R}\), the product of the actual local Haar integrals equals the finite Euler product in Lemma 5, with parameter \(\lambda =\sigma -\pi it\).

If a rational torus point has parameter \(\nu =u+v\zeta \), its characteristic polynomial has integral coefficients exactly when

At each finite prime, the local height factor for \(3D_0\) is one on the integral locus.

Substituting \(s=2z\) in the Laurent expansion at \(s=2\) changes its polar part to

The Euler products and prime sums defining the main-term constants converge and agree with their simplified expressions.

At a finite place, the local height for \(3D_0\) is the square root of the maximum of \(1\) and the absolute values of the three nontrivial weight characters.

Two \(C_3\)-algebras with trace-one normal elements that give the same rational quotient point are isomorphic as pairs.

Let \(f=t^3-t^2+at+b\in \mathbb {Q}[t]\) be separable, and suppose that \(f\) either has Galois group \(C_3\) or splits over \(\mathbb {Q}\). Exactly two rational torus points have characteristic polynomial \(f\).

For \(r\ne 0\) in a field of characteristic zero, the substitution \(x_1^\prime =x_1+rx_2\), \(x_2^\prime =x_1-rx_2\) has the inverse \(x_1=(x_1^\prime +x_2^\prime )/2\), \(x_2=(x_1^\prime -x_2^\prime )/(2r)\). The linked lemma checks the composition in this direction.

For the Galois-invariant parameter \((s_0,\sigma ,\sigma )\), interchanging the two conjugate coordinates leaves the support function unchanged.

Each of the four sections displayed in Lemma 1 is a weight vector for its listed character.

Write \(e_2,e_3,\delta \) for the dehomogenized invariants

Then

A rational point \(P\) of \(S\setminus D_0\) is integral if and only if \(e_2(P),e_3(P)\in \mathbb {Z}\).

For \(\operatorname {Re}(\sigma ){\gt}1\) and \(t\in \mathbb {R}\), put \(\lambda =\sigma -\pi it\). Then

and

Here \(\chi _0\) is the trivial character, corresponding to \(m=0\). The Euler product converges absolutely and locally uniformly for \(\operatorname {Re}(\lambda ){\gt}1\) and is bounded on every half-plane \(\operatorname {Re}(\lambda )\geq \tau {\gt}1\).

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

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

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 \(c\in \mathbb {Z}\).

Let \(\mathbf1\) denote the global section in \(H^0(S_E,\mathcal O(3D_0))\) associated to the effective Cartier divisor \(3D_0\), namely the image of \(1\) under the natural inclusion \(\mathcal O_{S_E}\hookrightarrow \mathcal O_{S_E}(3D_0)\). The space \(H^0(S_E,\mathcal O(3D_0))\) is spanned over \(E\) by the following four weight vectors:

where \(\sqrt{\mathrm{disc}} \mathrel {:=}(X-Z)(Y-X)(Z-Y)\). The associated characters of \(T_E\) are, respectively,

where \(a' = 2v_1^\vee - v_2^\vee \) and \(b' = 2v_2^\vee - v_1^\vee \) in the character lattice \(M_E = X^\ast T_E\) and \((v_1^\vee ,v_2^\vee )\) is the dual basis to \((v_1,v_2)\).

Let \(s=(s_0,\sigma ,\sigma )\in (\mathbb {C}^{\Sigma (1)})^\Gamma \) with \(\operatorname {Re}(s_0){\gt}0\) and \(\operatorname {Re}(\sigma ){\gt}1\), and put \(f=H(\cdot ,-s,D_0)\). Then \(f\in L^1(T(\mathbb {A}))\) and \(\widehat f\, |_{T(\mathbb {Q})^\perp }\in L^1(T(\mathbb {Q})^\perp )\).

Let \(K^{D_0}=K_{3,2}\times \prod _{v\neq 3}K_v^{\mathrm{max}}\) be the subgroup of \(K^{\mathrm{max}}\) fixing \(1^{D_0}\), as in Proposition 4. The map \(\ell \) induces an isomorphism

Let \(a(u)\in T(\mathbb {A})\) have infinite component \(\iota ^{-1}(e^u)\) and all finite components one, so \(\ell (a(u))=uv_0\). In this coordinate, the pushforward of the quotient measure is \(du/6\).

Let \(K/\mathbb {Q}\) be a \(C_3\)-algebra and let \(x\in K\) have characteristic polynomial \(t^3-e_1t^2+e_2t-e_3\). Then \(x\) is normal if and only if \(e_1(e_1^2-3e_2)\neq 0\).

For \(s\) as in Lemma 6, we have

where both the series defining \(Z(s)\) and the integral converge absolutely.

Let \(d_n\) be the coefficients of \(Z_0(z)=\sum _{n\geq 1}d_n n^{-z}\). For \(X\geq 1\),

Let \(P=(K/\mathbb {Q},x)\) be an integral rational point of \(T\) with characteristic polynomial \(f=t^3-t^2+at+b\in \mathbb {Z}[t]\). Write \(H_{\mathrm{tor}}(f)=H(P,\mathcal O(D_0))\) for its toric height and set \(h(f)=\max (|a|^{1/2},|b|^{1/3})\), so that \(H(f)=\max \{ 1,h(f)\} \). Then \(a\leq 0\) and

so for every \(B\geq 0\),

In particular, \(H_{\mathrm{tor}}(f)\asymp H(f)\). The only polynomial here with \(h(f)=0\) is \(t^3-t^2\), for which \(H_{\mathrm{tor}}(f)=1\).

The height zeta function is also given by

where \(\xi \) is the Dedekind zeta function of the cyclotomic field \(\mathbb {Q}(\sqrt{-3})\). The height zeta function has meromorphic continuation to the region \(\{ s \in (\mathbb {C}^{\Sigma (1)})^{\Gamma } : \mathrm{Re}(s_0+s_1+s_2) {\gt} 1\} \).

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

Let \(\eta {\gt}0\). For \(1/2\leq \sigma \leq 2\) and \(U\geq 2\),

Let \(w\) be a place of \(E\) lying over a place \(v\) of \(\mathbb {Q}\). There is an exact sequence

If \(w\) is infinite then \(\ell _w\) is surjective, and if \(w\) is finite then the image of \(\ell _w\) is the lattice \(X_\ast (T_{E})^{\Gamma (w/v)}\).

The characteristic polynomial \(f = t^3-t^2+at+b \in \mathbb {Q}[t]\) of a rational point \((K/\mathbb Q,x) \in T(\mathbb Q)\) has integer coefficients if and only if \((K/\mathbb Q,x)\) is integral. For any integral rational point \((K/\mathbb {Q},x)\) on \(T\),

The tori \(T\) and \(\mathcal G_1 = R^E_\mathbb {Q}\mathbb {G}_m\) are isomorphic as algebraic groups over \(\mathbb {Q}\). Every rational point \((K/\mathbb {Q},x)\) of \(T\) thereby determines an element of \(\mathbb {Q}(\sqrt{-3})^\times \) which is canonically determined up to the action of \(\mathrm{Aut}(\mathcal G_1)\). For integral rational points of \(T\), the toric height \(H_{\mathrm{tor}}(f)\mathrel {:=}\sqrt{1-3a}\) is identified with the square-root of the norm on \(\mathbb {Q}(\sqrt{-3})^\times \). Let \(\zeta \) be a primitive cube root of unity. If \(u+v\zeta \in \mathbb {Q}(\sqrt{-3})^\times \) has norm \(N\) and trace \(T\), then the characteristic polynomial of the corresponding rational point \((K/\mathbb {Q},x)\) is

Such a polynomial either has Galois group \(C_3\) or splits into three linear factors over \(\mathbb {Q}\), with at most two linear factors being the same. Conversely, a monic trace-one polynomial \(f=t^3-t^2+at+b \in \mathbb {Q}[t]\) which either has Galois group \(C_3\) or splits into three linear factors over \(\mathbb {Q}\), with at most two linear factors being the same, can be expressed in this way for precisely two rational points of \(T\) if \(f\) has no repeated roots, or for precisely one rational point of \(T\) if \(f\) has a double root which is not a triple root. The elements \(u+v\zeta \in \mathbb {Q}(\sqrt{-3})^\times \) corresponding to \(f=t^3-t^2+at+b\) will be the roots of the quadratic polynomial

The polynomial \(f\) will have integral coefficients if and only if

Let \(s \in (\mathbb {C}^{\Sigma (1)})^{\Gamma }\) and assume \(\mathrm{Re}(s_e){\gt}0\) for each \(e \in \{ 0,1,2\} \). Let \(w\) be the infinite place of \(E\). Let \(\chi \in T(\mathbb {R})^\vee \) be a unitary character. If \(\chi \) is ramified then \(\widehat{H_\infty }(\chi ,-s,D_0)\) is identically zero. If \(\chi \) is unramified, then \(\chi (x)= e(\langle \ell _w(x) ,m\rangle )\) for all \(x \in T(\mathbb {R})\) for a unique \(m \in \mathfrak t^\vee \), and

Next let \(v\) be a finite place of \(\mathbb {Q}\). For any unitary character \(\chi \in T(\mathbb {Q}_v)^\vee \), the integral defining \(\widehat{H_v}(\chi ,-s,D_0)\) converges absolutely to a holomorphic function of \(s\) in the region

Assume \(v \neq 3\). Let \(w\) be any place of \(E\) lying over \(v\). The local characteristic function \(1_v^{D_0}\) is \(K_v^{\mathrm{max}}\)-invariant. If \(\chi \) is ramified, then \(\widehat{H_v}(\chi ,-s,D_0)\) is identically zero. If \(\chi \) is unramified then we may regard \(\chi \) as a character on \(X_\ast (T_{E})^{\Gamma (w/v)}\) (Proposition 1) and

If \(v = 3\) then the support of \(x \mapsto H_3(x,-s,D_0)\) is the congruence subgroup \(K_{3,2}\) of \(K_3^{\mathrm{max}}\) of index six. Under the isomorphism \(\iota :T(\mathbb {Q}_3)\to E_w^\times \) the support corresponds to the subgroup \(1+3O_{E,w}\) of \(O_{E,w}^\times \) where \(w\) is the unique place of \(E\) lying over \(3\).

The only pole of \(Z_0(z)\) in the half-plane \(\mathrm{Re}(z){\gt}1/2\) is a double pole at \(z=1\). Its Laurent expansion there is

where \(\alpha \) and \(\beta \) are as in 1.

There is an exact sequence

where \(K^{\mathrm{max}}\) is the maximal compact subgroup of \(T(\mathbb {A})\) and \(\mu ^{\mathrm{max}} = T(\mathbb {Q}) \cap K^{\mathrm{max}}\).

Let \(v\) be a finite place of \(\mathbb {Q}\) and let \(w\) be a place of \(E\) lying over \(v\).

1. If \(v \neq 3\), then \(1_v^{D_0}(x) = 1\) if and only if \(\ell _w(x) \in \mathbb R_{\geq 0}v_1+\mathbb R_{\geq 0}v_2\). In particular \(K_v^{D_0}=K_v^{\mathrm{max}}\).

2. Let \(v = 3\), let \(w\) be the unique place of \(E\) over \(3\), and let \(\mathfrak w = (1-\zeta )O_{E,w}\) be the maximal ideal of \(O_{E,w}\), so that \(3O_{E,w} = \mathfrak w^2\). The support of \(1_3^{D_0}\) is the subgroup

   \begin{equation*} K_{3,2}\mathrel {:=}\iota ^{-1}(1+\mathfrak w^2) =\iota ^{-1}(1+3O_{E,w})\subset K_3^{\mathrm{max}}. \end{equation*}

3. Let \(K^{\mathrm{max}}=\prod _v K_v^{\mathrm{max}}\) and \(K^{D_0}=\prod _v K_v^{D_0}\subset K^{\mathrm{max}}\). The latter is the subgroup which fixes the characteristic function \(1^{D_0}=\otimes _v1_v^{D_0}\) for integral points in \(T(\mathbb {A})\), with \(1_\infty ^{D_0}=1\). Then

   \[ K_3^{D_0}=K_{3,2},\qquad K^{D_0}=K_{3,2}\times \prod _{v\neq 3}K_v^{\mathrm{max}}. \]

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 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

The invariant cubics

identify \(\mathscr {S}\) with the cubic hypersurface in \(\mathbb {P}^3_\mathbb {Z}\) defined by

The divisor \(\mathscr {D}_0=\{ e_1=0\} \) corresponds to the line \(\{ A=B=0\} \) in these coordinates. On the generic fiber,

In particular, this cubic embedding is anticanonical.

Over \(\mathbb {Z}[1/6]\), \(\mathscr {S}\) is isomorphic to the cubic hypersurface

in \(\mathbb {P}_{\mathbb {Z}[1/6]}^3\). Its generic fiber \(S\) is a del Pezzo surface of degree \(3\) and geometric Picard rank \(1\). Its three geometric singularities are of type \(\mathbf A_2\): one is rational, and the other two are conjugate over \(E=\mathbb {Q}(\sqrt{-3})\). Over \(E\), the coordinates \(x_1'=x_1+\sqrt{-3}x_2\) and \(x_2'=x_1-\sqrt{-3}x_2\) give the split toric cubic \(x_0x_1'x_2'=x_3^3\). In these coordinates, \(D_0=\{ x_0=x_3=0\} \).

The surface \(S\) admits a minimal resolution \(\pi \colon X\to S\) over \(\mathbb {Q}\). Put \(D_{\mathrm{red}}=(\pi ^{-1}D_0)_{\mathrm{red}}\). Then \(X\) is a weak del Pezzo surface of degree \(3\), of geometric Picard rank \(7\). The geometric components of \(D_{\mathrm{red}}\) form a chain with self-intersections

The divisor \(L=-K_X-D_{\mathrm{red}}\) is nef and not numerically zero, but is not big. More precisely,

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

Comparing Euler factors in 27 gives the coefficients explicitly: \(d_n=0\) if \(3\mid n\), and if \(3\nmid n\) then

where \(d=d_1d_2\) with \(d_1\) divisible only by primes \(p\equiv 1\pmod3\) and \(d_2\) only by primes \(q\equiv 2\pmod3\). Equivalently, \(d_n\) is the product of \(3k+1\) over prime powers \(p^k\parallel n\) with \(p\equiv 1\pmod3\) if every prime \(q\equiv 2\pmod3\) divides \(n\) to an even power, and \(d_n=0\) otherwise.

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

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 ] .

The local Fourier transforms — and therefore the entire Poisson summation argument — must be computed before restricting to the line in \(\mathrm{Pic}^T(S) \otimes \mathbb {C}\) spanned by the \(T_E\)-line bundle \(\mathcal O(D_0)\) of interest since \(x \mapsto H_v(x,-s,D_0)\) will not be integrable for any place \(v \neq 3,\infty \) once either of \(s_1\) or \(s_2\) vanishes, no matter how large and positive \(\mathrm{Re}(s_0)\) is.

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

where

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

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.

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

where

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 \(2\).

Fix any \(s \in (\mathbb {C}^{\Sigma (1)})^{\Gamma }\) with \(\mathrm{Re}(s_0+s_1+s_2) {\gt} 2\). Then the multivariate Dirichlet series

is absolutely convergent and equals

where \(z = \frac{1}{2}(s_0+s_1+s_2)\). This multivariate Dirichlet series admits a meromorphic continuation to \(\{ s \in (\mathbb {C}^{\Sigma (1)})^{\Gamma } : \mathrm{Re}(s_0+s_1+s_2) {\gt} 1\} \).

There is a bijection between rational points of \(T\) and equivalence classes of \(C_3\)-algebras \(K/\mathbb {Q}\) equipped with a trace-one normal element \(x\in K\).