@article{d21d5c37869e41209e712f6c4b9ddf39,

title = "Bounds on the number of rational points of curves in families",

abstract = "In this note, we give an alternative proof of uniform boundedness of the number of integral points of smooth projective curves over a fixed number field with good reduction outside of a fixed set of primes. We use that due to Bertin–Romagny, the Kodaira–Parshin families constructed by Lawrence–Venkatesh can themselves be assembled into a family. We then repeat Lawrence–Venkatesh's study of the (Formula presented.) -adic period map, together with the comparison of nearby fibres.",

keywords = "Diophantine geometry, p-adic geometry",

author = "Alex Torzewski and Pedro Lemos",

note = "Funding Information: We are grateful to Alex Betts and Netan Dogra for helpful discussions and especially to the latter for drawing our attention to the work of Caporaso–Harris–Mazur. We also wish to particularly thank Marco Maculan for explaining how a relative Kodaira–Parshin family is a consequence of work of Bertin–Romagny and Matthieu Romagny for being available to explain the subtleties of their construction. Additionally, we are very grateful for the comments of an anonymous referee, which improved both its mathematical content and its readability. Publisher Copyright: {\textcopyright} 2023 The Authors. Bulletin of the London Mathematical Society is copyright {\textcopyright} London Mathematical Society.",

year = "2023",

month = jan,

day = "3",

doi = "http://dx.doi.org/10.1112/blms.12774",

language = "English",

volume = "55",

pages = "1019--1032",

journal = "BULLETIN OF THE LONDON MATHEMATICAL SOCIETY",

issn = "0024-6093",

publisher = "Oxford University Press",

number = "2",

}