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Sequences of powers with second differences equal to two and hyperbolicity

Author: Natalia Garcia-Fritz
Journal: Trans. Amer. Math. Soc. 370 (2018), 3441-3466
MSC (2010): Primary 11D41, 32Q45, 14G05
Published electronically: December 27, 2017
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Abstract: By explicitly finding the complete set of curves of genus 0 or $ 1$ in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound $ M>0$ such that there are only finitely many sequences of length $ M$ formed by $ k$-th rational powers with second differences equal to $ 2$. Moreover, we prove the unconditional analogue of this result for function fields, with $ M$ depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.

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Additional Information

Natalia Garcia-Fritz
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street BA6103, Toronto, Ontario, Canada, M5S 2E4
Address at time of publication: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile

Keywords: Curves of low genus, Brody-hyperbolic surfaces, rational points, undecidability
Received by editor(s): March 7, 2016
Received by editor(s) in revised form: August 8, 2016
Published electronically: December 27, 2017
Additional Notes: This work was part of the author’s thesis at Queen’s University and was partially supported by a Becas Chile Scholarship
Article copyright: © Copyright 2017 American Mathematical Society

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