The main results of this paper involve general algebraic differentials $\omega$ on a general pencil of algebraic curves. We show how to determine whether $\omega$ is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck–Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin–Mumford type allied to the Zilber–Pink conjectures: we characterise torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least $2$. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations $A^2 - DB^2 = 1$ over a polynomial ring. We determine whether the Pell equation (with squarefree $D$) is solvable for infinitely many members of the pencil.

中文翻译：

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扭转点，佩尔方程和基本积分

本文的主要结果涉及到一般代数曲线上的一般代数微分$ \ omega $。我们展示了如何确定$ \ omega $在铅笔上对于无限多的成员是否可积分。特别是，这更正了1981年对James Davenport的主张，并提供了第一个证据，即使形式相当坚固。我们还指出了与Andre和Hrushovski的工作以及Grothendieck-Katz猜想的类比。为了达到这个目标，我们首先提供独立结果的证明，这些结论扩展了与Zilber-Pink猜想相关的相对Manin-Mumford类型的结论：我们描述了任意相对维度至少为$ 2的一般阿贝尔方案中位于一般曲线上的扭点。 $。反过来，我们将后者结果的另一个应用应用于多项式环上相当普通的Pell方程$ A ^ 2-DB ^ 2 = 1 $。我们确定Pell方程（带有无平方的$ D $）对于无限多的铅笔成员是否可求解。