To every real quadratic number field KK of discriminant Δ\Delta, Zeytin (2016; publ. 2018) attached an affine çark surface CK\mathcal{C}_K, cut out by a multivariate Lucas polynomial via the equation L2k(x1,y1,…,xk,yk)2−4=z2Δ\mathcal{L}_{2k}(x_1,y_1,\dots,x_k,y_k)^2-4=z^2\Delta, whose integral points are in bijection with the narrow ideal classes of KK. The projective closures of these surfaces are conjectured to be Kobayashi hyperbolic, so Lang's conjecture on rational points of varieties of general type would give finiteness of their Q\mathbb{Q}-rational points, suggesting a geometric route toward Gauss's two-century-old class number problems, most prominently the class number one problem for real quadratic fields. The open core: prove the conjectured hyperbolicity (or refute it), develop the Lang-conjecture viewpoint that connects the resulting finiteness of rational points to Gauss's class number problems, and establish the announced extension of the bijection to non-maximal orders. The underlying class number one problem for real quadratic fields remains open.

Background

A real quadratic field is K=Q(d)K=\mathbb{Q}(\sqrt{d}) for a squarefree integer d>1d>1; its ring of integers ZK\mathbb{Z}_K is the subring of elements that are roots of monic integer polynomials. Unique factorization of ideals always holds in ZK\mathbb{Z}_K, but unique factorization of elements is measured by the ideal class group H(K)H(K) (fractional ideals modulo principal ideals), whose order hKh_K is the class number; hK=1h_K=1 exactly when ZK\mathbb{Z}_K is a unique factorization domain. The narrow class group H+(K)H^{+}(K) refines this by modding out only principal ideals generated by elements of positive norm, and its order h+(K)h^{+}(K) is the narrow class number; H+(K)≅H(K)H^{+}(K)\cong H(K) precisely when KK has a unit of norm −1-1. Since Gauss's Disquisitiones, class groups of quadratic fields have been computed through equivalence classes of binary quadratic forms aX2+bXY+cY2aX^2+bXY+cY^2 of the same discriminant; in the real (indefinite) case the automorphism group of each form is infinite cyclic, generated by a hyperbolic element of PSL2(Z)\mathrm{PSL}_2(\mathbb{Z}) (the group of 2×22\times 2 integer matrices of determinant 11 modulo ±I\pm I), and is governed by the solutions of the Pell equation X2−ΔZ2=4X^2-\Delta Z^2=4. The imaginary quadratic class number one problem was settled completely: hK=1h_K=1 occurs exactly for the nine discriminants −3,−4,−7,−8,−11,−19,−43,−67,−163-3,-4,-7,-8,-11,-19,-43,-67,-163. For real quadratic fields the analogous Gauss class number one problem — determine all squarefree d>1d>1 with hK=1h_K=1, in particular decide whether infinitely many exist — is open, with heuristics predicting infinitely many. Çarks, introduced by Uludağ, Zeytin and Durmuş in their work on binary quadratic forms as dessins d'enfants (Grothendieck's bipartite ribbon graphs encoding covers of the projective line minus three points), are the infinite bipartite planar quotient graphs obtained by dividing the Farey graph (the PSL2(Z)\mathrm{PSL}_2(\mathbb{Z})-invariant bipartite graph on the upper half-plane boundary data) by the cyclic automorphism group of an indefinite form. Zeytin's 2016 paper encodes these quotients algebraically: writing a product of matrices M(x1,y1,…,xk,yk)=M(x1,y1)⋯M(xk,yk)M(x_1,y_1,\dots,x_k,y_k)=M(x_1,y_1)\cdots M(x_k,y_k), with M(xi,yi)=(1+xiyixiyi1)M(x_i,y_i)=\begin{pmatrix}1+x_iy_i & x_i\\ y_i & 1\end{pmatrix}, in the Pauli-matrix basis, its trace defines the multivariate Lucas polynomial L2k(x1,y1,…,xk,yk)\mathcal{L}_{2k}(x_1,y_1,\dots,x_k,y_k) (in the paper's notation L2k=2Ak\mathcal{L}_{2k}=2A_k, where AkA_k is the coefficient of the identity matrix), which specializes to the classical Lucas polynomials (the sequence LnL_n with L0=2L_0=2, L1=xL_1=x, Ln=xLn−1+Ln−2L_{n}=xL_{n-1}+L_{n-2}) in one variable. The main theorem (Theorem 3.4 and Corollary 3.7 there) shows that every narrow ideal class of KK yields an integral solution of L2k(x1,y1,…,xk,yk)2−4=z2Δ\mathcal{L}_{2k}(x_1,y_1,\dots,x_k,y_k)^2-4=z^2\Delta, where k=kKk=k_K is the maximum of the çark lengths kak_{\mathfrak{a}} as a\mathfrak{a} runs through the narrow ideal classes in H+(K)H^{+}(K), and where zz depends only on KK: it is the fundamental-solution value zoz_o of the Pell equation X2−ΔZ2=4X^2-\Delta Z^2=4 (the proof of Corollary 3.7 concludes that the relevant discriminant is square-free precisely because zoz_o is minimal). The solution set of this equation in the 2kK2k_K variables (x1,y1,…,xkK,ykK)(x_1,y_1,\dots,x_{k_K},y_{k_K}), with zz fixed at zoz_o, is the affine çark hypersurface CK⊂C2kKC_K\subset\mathbb{C}^{2k_K} (Definition 3.5), and after quotienting it by the natural cyclic action of Z/kKZ\mathbb{Z}/k_K\mathbb{Z} that sends (x1,y1,…,xk,yk)(x_1,y_1,\dots,x_k,y_k) to (xk,yk,x1,y1,…,xk−1,yk−1)(x_k,y_k,x_1,y_1,\dots,x_{k-1},y_{k-1}) and leaves L2k\mathcal{L}_{2k} invariant, the resulting affine çark surface CK=CK/(Z/kKZ)\mathcal{C}_K=C_K/(\mathbb{Z}/k_K\mathbb{Z}) has its integral points in bijection with the narrow ideal classes of KK. Treating zz instead as a free variable, the same equation cuts out an affine hypersurface in C2k+1\mathbb{C}^{2k+1} whose integral points correspond to non-fundamental Pell solutions and, as the paper's concluding remarks explain, to non-maximal orders of ZK\mathbb{Z}_K. The author observes that the high-degree projective closures of these surfaces are conjecturally Kobayashi hyperbolic — a complex variety is Kobayashi hyperbolic when it admits no nonconstant holomorphic maps from C\mathbb{C}, forcing its entire curves and, conjecturally, much of its arithmetic to be degenerate — and that Lang's conjecture (varieties of general type should have rational points not Zariski dense, in particular finitely many on hyperbolic surfaces) would then force finiteness of their Q\mathbb{Q}-rational points. Since the narrow class number h+(K)h^{+}(K) is, by the bijection, the number of integral points of CK\mathcal{C}_K, a proven hyperbolicity together with the Lang-conjecture viewpoint would open a genuinely geometric route to Gauss's class number problems. The companion survey 'Class Number Problems and Lang Conjectures' (2017) elaborates this program without resolving any part of it.

Problem Statement

The overarching question is Gauss's class number one problem for real quadratic fields, in its standard form: determine all squarefree integers d>1d>1 such that Q(d)\mathbb{Q}(\sqrt{d}) has class number one; in particular, decide whether infinitely many such fields exist. Zeytin's program proposes a derived geometric attack on this problem and leaves the following concrete questions open. First, prove — or refute — the conjectural Kobayashi hyperbolicity of the projective closures of the çark surfaces CK\mathcal{C}_K (the affine quotients of the solution sets of L2kK(x1,y1,…,xkK,ykK)2−4=zo2Δ\mathcal{L}_{2k_K}(x_1,y_1,\dots,x_{k_K},y_{k_K})^2-4=z_o^2\Delta in the 2kK2k_K variables (xi,yi)(x_i,y_i), with zoz_o the fundamental-solution value of X2−ΔZ2=4X^2-\Delta Z^2=4) for real quadratic fields KK of discriminant Δ\Delta. Second, make the Lang-conjecture bridge between the geometry of the çark surfaces and Gauss's class number problems effective: by Lang's conjecture, hyperbolic projective closures have only finitely many Q\mathbb{Q}-rational points, so prove an effective or uniform bound on the narrow class numbers h+(K)h^{+}(K) across real quadratic fields that follows from this finiteness — or show that no such bound follows. Third, establish in full detail the announced extension of the integral-point bijection to non-maximal orders of ZK\mathbb{Z}_K: treat zz as a free variable in the affine hypersurface L2k(x1,y1,…,xk,yk)2−4=z2Δ⊂C2k+1\mathcal{L}_{2k}(x_1,y_1,\dots,x_k,y_k)^2-4=z^2\Delta\subset\mathbb{C}^{2k+1}, whose integral points give binary quadratic forms whose discriminant's square-free part equals Δ\Delta and hence correspond to non-maximal orders of ZK\mathbb{Z}_K, and analyze its slices z=λz=\lambda, λ∈Z\lambda\in\mathbb{Z}, whose integral points give narrow ideal classes in an appropriate class group, identifying precisely which class group of the order each integral point represents. These three questions form the program and are all required; each may be pursued independently, and resolving any one of them in full settles that part of the task. Gauss's class number one problem itself is the overarching motivation and background, not a required deliverable.

References

  1. Ayberk Zeytin, Multivariate Lucas polynomials and ideal classes in quadratic number fields, Turkish Journal of Mathematics 42(4) (2018), DOI: 10.3906/mat-1608-65; preprint arXiv:1608.05544, https://arxiv.org/abs/1608.05544
  2. Ayberk Zeytin, Class Number Problems and Lang Conjectures, in: Algebraic Geometry and Number Theory (CIMPA-CMI-TÜBİTAK Summer School, Galatasaray University, Istanbul, 2014), Progress in Mathematics, Springer, Cham (2017), pp. 211–232, DOI: 10.1007/978-3-319-47779-4_7, https://link.springer.com/chapter/10.1007/978-3-319-47779-4_7
  3. A. Muhammed Uludağ, Ayberk Zeytin, Merve Durmuş, Binary quadratic forms as dessins, Journal de théorie des nombres de Bordeaux 29(2) (2017), 445–469, DOI: 10.5802/jtnb.987, https://doi.org/10.5802/jtnb.987
  4. Ayberk Zeytin, Binary quadratic forms: modern developments, arXiv:2501.08830 (2025), https://arxiv.org/abs/2501.08830