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

Masser, David and Zannier, Umberto. (2020) Torsion points, Pell's equation, and integration in elementary terms. Preprints Fachbereich Mathematik, 2020 (01).


Official URL: https://edoc.unibas.ch/77959/

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The main results of this paper involve general algebraic differentials $\omega$ on a general pencil of algebraic curves. We show how to determine if $\omega$ 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 André 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 characterize 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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Zahlentheorie (Masser)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Masser, David
Item Type:Preprint
Publisher:Universität Basel
Last Modified:07 Aug 2020 10:38
Deposited On:07 Aug 2020 10:38

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