Rational points on families of curves with genus at least 2
Let \({\mathcal C}\) and \({\mathcal T}\) be two projective varieties defined over a number field \(k\). Suppose that there exists a flat morphism \(\pi:{\mathcal C} \to {\mathcal T}\) of \(k\)-schemas such that the generic fibre \({\mathcal C}_\eta ={\mathcal C} \times_{\mathcal T} \text{Spec} ({\mathcal T})\) is a smooth curve of genus \(\geq 2\) over \(k(\eta) \cong k({\mathcal T})\). We denote by \({\mathcal T}^0\) the open of \({\mathcal T}\) over which the morphism \(\pi\) is smooth. For every \(t\in {\mathcal T}^0 (\overline k)\) we consider the curve \({\mathcal C}_t\) embedded in its Jacobian \({\mathcal J}_t\) and we denote by \(| \cdot|\) the Euclidean norm associated to the theta divisor. In this paper the author proves that there exist two positive constants \(\gamma_1 ({\mathcal C})\) and \(\gamma_2 (g)\) such that for every point \(t\in {\mathcal T}^0 (\overline k)\) and every finite extension \(k'\) of \(k(t) \) we have \[ \text{Card} \biggl\{x\in {\mathcal C}_t(k')/ | x| \geq\gamma_1 ({\mathcal C}) \bigl( \sqrt {h_\tau (t)}+1 \bigr)\biggr\} \leq\gamma_2 (g)7^{\text{rank} {\mathcal J}_t (k')}\leq {55\over 2} 7^{\text{rank} {\mathcal J}_t (k')}. \] The constant \(\gamma_1 ({\mathcal C})\) depends only on \({\mathcal C}\) and the choice of the height \(h_\tau\) over \({\mathcal T}\) associated to an ample divisor \(\tau\) and the constant \(\gamma_2 (g)\) is a decreasing function of \(g\). This result is a uniform version of the well-known Falting's theorem which asserts that every algebraic curve of genus \(\geq 2\) defined over a number field has a finite number of rational points. The proof of this theorem by Vojta and Bombieri uses the inequalities of Mumford and Vojta [\textit{E. Bombieri}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 615--640 (1990; Zbl 0722.14010); ibid. 18, 473 (1991; Zbl 0763.14007)]. The author, using Vojta-Bombieri methods, proves uniform versions of the inequalities of Mumford and Vojta which imply its result. As an application, she obtains, under the hypothesis that Szpiro's conjecture is valid, a bound of the above type for every family of one parameter curves of genus \(\geq 2\) such that the family of their Jacobians is isogenous to a product of non isotrivial families of elliptic curves.
scientific article; zbMATH DE number 1097549