Independence of rational points on twists of a given curve

Details Independence of rational points on twists of a given curve Independence of rational points on twists of a given curve

2006-03-23

arxiv.org/abs/math/0603557v1

Compositio Math. 142, 1201-1214 (2006)

Mathematics

Number Theory

16 pages; to appear in Compositio Math (in a slightly different version)

Scientific paper

10.1112/S0010437X06002168

In this paper, we study bounds for the number of rational points on twists C' of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J' of C' has rank smaller than the genus of C'. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J'(K) and c is a constant depending on C. For the proof, we use a refinement of the method of Chabauty-Coleman; the main new ingredient is to use it for an extension field of K_v, where v is a place of bad reduction for C'.

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