Picard number one and minimality for the elliptic surfaces in Nikolaev's continued-fraction correspondence

In the preprint 'Noncommutative geometry of elliptic surfaces' (arXiv:2106.11392), Igor Nikolaev attaches to an elliptic surface $E(\mathbb{Q}(t))$ over the projective line a one-parameter family of periodic continued fractions $[b_1(t),\dots,b_N(t);a_1(t),\dots,a_k(t)]$ and a noncommutative torus, and proves a formula $\rho(E(\mathbb{Q}(t)))=N+k$ for the Picard number. Version 1 of the preprint asserts in Remark 3.3 that the minimal model satisfies $\rho(E_{\min}(\mathbb{Q}(t)))=1$ for all constructed examples and conjectures this for 'a majority' of the surfaces, while 'a formal proof is yet to be written'; version 2 (2024) silently deletes that remark and instead asks whether the complex-multiplication examples $E_D(t)$ are minimal. The combined question is still open: prove, refute, or delimit the assertion of Remark 3.3 that the minimal model of a surface in the correspondence has Picard number 1, and decide minimality of the CM examples.

Background

An elliptic surface over the projective line $\mathbb{P}^1$ is a smooth projective surface $E$ together with a morphism to $\mathbb{P}^1$ whose generic fiber is an elliptic curve, here the curve $E(\mathbb{Q}(t))$ defined over the rational function field $\mathbb{Q}(t)$ ; one thinks of $E$ as the family of plane cubic curves parametrized by $t$ . The Néron–Severi group $NS(E)$ is the group of divisors modulo algebraic equivalence, a finitely generated abelian group, and its rank $\rho(E)$ is the Picard number, which measures how many independent curve classes the surface carries. A minimal model $E_{\min}$ of a surface is a birational representative containing no rational curves of self-intersection $-1$ ; by Castelnuovo's contraction theorem every surface that is not ruled reaches its minimal model after finitely many contractions of such $(-1)$ -curves. For an elliptic surface with a section, the class of a fiber and of the zero section are independent in $NS$ , so a relatively minimal such surface has Picard number at least $2$ , and Picard number $1$ on the minimal model can only occur when the minimal model is the projective plane $\mathbb{P}^2$ , i.e. when the surface is rational. The noncommutative torus $\mathscr{A}_\theta$ is the universal $C^*$ -algebra (a topological algebra of 'quantized' functions) generated by two unitaries $U,V$ with the commutation relation $VU=e^{2\pi i\theta}UV$ ; when the parameter $\theta$ is an irrational quadratic number it has a periodic continued-fraction expansion $\theta=[b_1,\dots,b_N;\overline{a_1,\dots,a_k}]$ with a finite tail of length $N$ and a repeating block of minimal period $k$ . Nikolaev's program (developed in his 2018 paper on rational elliptic curves and his 2022 De Gruyter monograph) builds a correspondence $\mathscr{F}$ under which an elliptic surface $E(\mathbb{Q}(t))$ determines, and up to complex isomorphism is encoded by, a one-parameter family of such periodic continued fractions, equivalently by sub-bundles $U_{b_1,\dots,b_N;a_1,\dots,a_k}$ of the Brock–Elkies–Jordan varieties $V_{N,k}(\mathbb{C})$ — the affine varieties of integer tuples solving the fixed-point equations of periodic continued fractions, studied by Brock, Elkies and Jordan via Skolem's $p$ -adic method (a classical technique that locates integer solutions of exponential Diophantine systems by computing in the $p$ -adic numbers). In arXiv:2106.11392 Nikolaev proves (Theorem 1.4) that every surface $E(\mathbb{Q}(t))$ arises from such a section of a $V_{N,k}(\mathbb{C})$ -bundle, and (Theorem 1.5) that under his hypotheses the Picard number is $\rho(E(\mathbb{Q}(t)))=N+k$ , the Mordell–Weil rank (the rank of the group of sections of the fibration, i.e. of the rational points of the generic elliptic curve over $\mathbb{Q}(t)$ ) equals the fiber dimension, and the minimal model is complex-isomorphic to one of the complex-multiplication surfaces $E^{CM}_{p(t)}$ (surfaces whose fibers have endomorphism ring an order in an imaginary quadratic field). For CM surfaces of discriminant $-D$ he proves $\rho(E_D(t))=1+k$ (Corollary 4.2). Against this backdrop, version 1 of the preprint records (Remark 3.3) that the construction forces $\rho(E_{\min}(\mathbb{Q}(t)))=1$ and that the author expects this to persist for a majority of the surfaces $E(\mathbb{Q}(t))$ , with no formal proof written; version 2 deletes this remark and poses instead the question of whether the CM surfaces for the class-number-one discriminants $D=2,3,7,11,19,43,67,163$ (the Heegner numbers, for which the imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$ has class number one, i.e. a trivial ideal class group) are minimal. What is missing is any criterion, proof, or refutation: the passage from continued-fraction data $(N,k)$ to the birational geometry of the surface — its minimality and the Picard number of its minimal model — is asserted through examples but not classified, and the two arXiv versions even leave an internal tension, so part of the task is to pin down the precise classical content of the claim.

Problem Statement

Within Nikolaev's correspondence $\mathscr{F}$ , which assigns to an elliptic surface $E(\mathbb{Q}(t))$ over $\mathbb{P}^1$ the one-parameter family of periodic continued fractions $[b_1(t),\dots,b_N(t);a_1(t),\dots,a_k(t)]$ (equivalently, a section of a $V_{N,k}(\mathbb{C})$ -bundle over the Fermat–Pell conic, the affine conic $Cu^2-Buv+Av^2=(-1)^k A$ attached to the continued-fraction data in Definition 2.3 of the preprint, over which $V_{N,k}(\mathbb{C})$ fibers by the Brock–Elkies–Jordan theorem), resolve the following two open questions on the birational geometry of the surfaces in the family: (a) prove, refute, or delimit the assertion of v1 Remark 3.3 of arXiv:2106.11392 that $\rho(E_{\min}(\mathbb{Q}(t)))=1$ — decide whether the assertion holds for the surfaces actually constructed by the correspondence, and determine the class of surfaces $E(\mathbb{Q}(t))$ for which the assertion holds; and (b) decide whether the complex-multiplication surfaces $E_D(t)$ for the class-number-one discriminants $D\in\{2,3,7,11,19,43,67,163\}$ (the Heegner numbers, for which $\mathbb{Q}(\sqrt{-D})$ has class number one) are minimal, as posed in Example 4.3 of v2.

Questions (a) and (b) are independent: question (a) concerns the assertion $\rho(E_{\min}(\mathbb{Q}(t)))=1$ of v1 Remark 3.3, both over the surfaces actually constructed by the correspondence and as a delimitation of the class of surfaces $E(\mathbb{Q}(t))$ in the correspondence for which the assertion holds, while question (b) concerns the minimality of the listed complex-multiplication surfaces $E_D(t)$ .

References

I. V. Nikolaev, Noncommutative geometry of elliptic surfaces, arXiv:2106.11392 (v1 submitted 21 Jun 2021; v2 26 Apr 2024), https://arxiv.org/abs/2106.11392
I. V. Nikolaev, Class field towers and minimal models, arXiv:2108.13470 (2021), https://arxiv.org/abs/2108.13470
I. V. Nikolaev, Arithmetic complexity revisited, The Journal of Analysis 31 (2023) 2115–2126, DOI 10.1007/s41478-023-00554-x, arXiv:2002.10854, https://arxiv.org/abs/2002.10854
B. W. Brock, N. D. Elkies, B. W. Jordan, Periodic continued fractions over $S$ -integers in number fields and Skolem's $p$ -adic method, Acta Arithmetica 197 (2021) 379–420, DOI 10.4064/aa191001-7-8, https://doi.org/10.4064/aa191001-7-8
I. V. Nikolaev, Noncommutative Geometry, Second Edition, De Gruyter Studies in Mathematics 66, De Gruyter, Berlin, 2022, DOI 10.1515/9783110788709, ISBN 9783110788709, https://doi.org/10.1515/9783110788709
I. V. Nikolaev, Noncommutative geometry of rational elliptic curves, Annals of Functional Analysis 9 (2018), no. 2, 202–209, DOI 10.1215/20088752-2017-0045, https://doi.org/10.1215/20088752-2017-0045