Curves that change genus can have arbitrarily many rational points

Details Curves that change genus can have arbitrarily many rational points Curves that change genus can have arbitrarily many rational points

1995-06-06

arxiv.org/abs/alg-geom/9506009v1

Mathematics

Algebraic Geometry

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Scientific paper

A singular curve over a non-perfect field K may not have a smooth model over K. Those are said to "change genus". If K is a global field of positive characteristic and C/K a curve that change genus, then C(K) is known to be finite. The purpose of this note is to give examples of curves with fixed relative genus, defined over K for which #C(K) is arbitrarily large. The motivation for considering this problem comes from the work of Caporaso et al. [CHM], where they show that a conjecture of Lang implies that, for a number field K, #C(K) can be bounded in terms of g and K only for all curves C/K of genus g > 1.

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