Bounds for the number of rational points on twists of a given curve

Michael Stoll

     stoll@math.uni-duesseldorf.de, stoll@mpim-bonn.mpg.de
           http://www.math.uni-duesseldorf.de/~stoll

We present a result giving a bound on the number of rational points mapping into a given small-rank subgroup of the Mordell-Weil group of a twist of a given curve over a number field. Among others, this implies the following.

Theorem. Let C : y^{2} = f(x) be a hyperelliptic curve of genus g ≥ 2 over ℚ. Then for all but finitely many d ∈ ℚ^{×}/(ℚ^{×})^{2}, the following is true. Any set of n ≤ g rational points on the quadratic twist C_{d} that is disjoint from its image under the hyperelliptic involution generates a subgroup of rank n in the Jacobian of C_{d}.

Theorem. Let f(x, y) ∈ ℤ[x, y] be homogeneous of degree n ≥ 3 and squarefree. Then for all but finitely many nth power free h ∈ ℤ, the Thue equation f(x, y) = h has at most r rational solutions, provided the Mordell-Weil rank r of the Jacobian of the curve defined by this equation is at most n − 3.