## 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 $$c \in {\mathbb{C}}$$.

### MSC:

 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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