Torsion points, Pell’s equation, and integration in elementary terms

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.

2010 Mathematics Subject Classification

11G10, 11G50, 12H05, 14K15, 14K20

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This paper (with some devilish difficulties) is dedicated to Enrico Bombieri in celebration of his 80th birthday.

Received 4 March 2018

Received revised 26 June 2020

Accepted 1 August 2020

Published 21 January 2021