Torsion points, Pell’s equation, and integration in elementary terms
Volume 225, Issue 2 (2020), pp. 227–312
Pub. online: 21 January 2021
Type: Article
Open Access
Open Access
Received
4 March 2018
4 March 2018
Revised
26 June 2020
26 June 2020
Accepted
1 August 2020
1 August 2020
Published
21 January 2021
21 January 2021
Notes
This paper (with some devilish difficulties) is dedicated to Enrico Bombieri in celebration of his 80th birthday.
Abstract
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.