Uniform Manin-Mumford for a family of genus 2 curves. (English) Zbl 1452.14027

The authors prove the existence an absolute effective positive constant \(B\) with the following property. Let \(X\) be a smooth bielliptic curve over \({\mathbb{C}}\) of genus \(2\) (bielliptic means that it admits a degree-two branched covering to an elliptic curve) and let \(P\) be a Weierstrass point on \(X\). Let \(j_P:X\hookrightarrow J(X)\) be the Abel-Jacobi embedding of \(X\) into its Jacobian based at \(P\) and \(J(X)^{\mathrm{tor}}\) the set of torsion points of \(J(X)\). Then \(\left| j_P(X)\cap J(X)^{\mathrm{tor}}\right|\le B\). The example due to M. Stoll [“Another new record”, http://www.mathe2.uni-bayreuth.de/stoll/torsion.html] of the hyperelliptic curve \(y^2 = x^6 + 130 x^3 + 13\) shows that \(B\ge 34\). This result answers a question raised by B. Mazur [Bull. Am. Math. Soc. (N.S.) 14, No. 2, 207–259 (1986; Zbl 0593.14021)]. The authors remark that there is no uniform bound for the order of the torsion points on \(X\) in its Jacobian. For the proof, the authors answer a special case of a conjecture by F. Bogomolov and Y. Tschinkel [in: Diophantine geometry. Selected papers of a the workshop, Pisa, Italy, April 12–July 22, 2005. Pisa: Edizioni della Normale. 73–91 (2007; Zbl 1142.14016)] and F. Bogomolov et al. [in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 1. Oxford: Oxford University Press. 19–37 (2018; Zbl 1423.14214)]<. For \(t\in{\mathbb{C}}\setminus\{0,1\}\), let \(E_t\) be the Legendre curve \(y^2=x(x-1)(x-t)\) and \(\pi:(x,y)\mapsto x\) the standard projection on \(E_t\). The authors prove the existence of a uniform constant \(B\) such that, for all \(t_1\not=t_2\) in \({\mathbb{C}}\setminus\{0,1\}\), \(\left|\pi(E_{t_1}^{\mathrm{tor}})\cap \pi(E_{t_2}^{\mathrm{tor}})\right|\le B\). The new tool is a quantification of the approach of L. Szpiro et al. [Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)], E. Ullmo [Ann. Math. (2) 147, No. 1, 167–179 (1998; Zbl 0934.14013)], S.-W. Zhang [Ann. Math. (2) 147, No. 1, 159–165 (1998; Zbl 0991.11034)] utilizing adelic equidistribution theory. The authors reduce to the setting where the curve is defined over the field \(\overline{{\mathbb{Q}}}\) of algebraic numbers, where they build on the proof of the quantitative equidistribution theorem for height functions on \({\mathbb P}^1(\overline{{\mathbb{Q}}})\) of C. Favre and J. Rivera-Letelier [Math. Ann. 335, No. 2, 311–361 (2006; Zbl 1175.11029)]. Consider the family of height functions \(\widehat{h}_t\) on \({\mathbb P}^1(\overline{{\mathbb{Q}}})\) induced from the Néron-Tate canonical height on the elliptic curve \(E_t\) for \(t \in{\mathbb{C}}\setminus\{0,1\}\); its zeroes are precisely the elements of \(\pi(E^{\mathrm{tor}}_t)\). The authors prove the existence of \(\delta>0\) such that \(\widehat{h}_{t_1}\widehat{h}_{t_2}\ge\delta\) for all \(t_1\not=t_2\) in \({\mathbb{C}}\setminus\{0,1\}\). They also prove upper and lower bounds for the product \(\widehat{h}_{t_1}\widehat{h}_{t_2}\) depending on the naive logarithmic height \(h(t_1,t_2)\) on \({\mathbb A}^2(\overline{{\mathbb{Q}}})\). In a forthcoming joint work, [“Common preperiodic points of quadratic polynomial”, Preprint, arXiv 1911.02458], the authors implement their strategy for obtaining a uniform bound on the number of common preperiodic points for distinct polynomials of the form \(f_c(z) = z^2 + c\) with \(c \in {\mathbb{C}}\).


14H40 Jacobians, Prym varieties
11G50 Heights
37P50 Dynamical systems on Berkovich spaces
37F44 Holomorphic families of dynamical systems; holomorphic motions; semigroups of holomorphic maps
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