Independence of rational points on twists of a given curve

By Michael Stoll. Compositio Mathematica, 2006

85 citations recorded by the source corpus.

In this paper, we study bounds for the number of rational points on twists $C'$ of a fixed curve $C$ over a number field ${\mathcal K}$, under the condition that the group of ${\mathcal K}$-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 $J'({\mathcal K})$ 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 ${\mathcal K}_v$, where $v$ is a place of bad reduction for $C'$.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c5cab9e13a7795dfc8c844ad12bad5148abf45dc

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Citations in the corpus

The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, methodology): “To obtain explicit bounds on the number of rational points on these curves, we follow the arguments in [49].”
Linear series on metrized complexes of algebraic curves (2012, background): “Stoll [43] improved this bound by replacing 2g − 2 with 2r .”
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank (2013, background): “t ℓbe a power series whose derivative is w. Then it is well-known that for ξ0,ξ1 ∈ D0(k) we have I ϕ(ξ 1) ϕ(ξ0) ω= Z ξ 1 ξ0 w(z)dz= ℓ(ξ1)−ℓ(ξ0). Using Newton polygons, one then shows (see for example [Sto06]) that the number of zeros of λωon ϕ(D0(k)) is bounded by 1 plus the number nof zeros of ω(co”
Twists of X(7) and primitive solutions to x^2+y^3=z^7 (2005, background): “If p > ν + 1, there can be at most ν + 1 rational points on C reducing to Q mod p: see, e.g., [Stoll2004preprint].”
Degeneration of Linear Series from the Tropical Point of View and Applications (2015, background): “Stoll observed in [Sto06] that one could do better than this by adapting the differential ω to the point Q rather than using the same differential ω for all residue classes.”
Uniform bounds for the number of rational points on curves of small Mordell–Weil rank (2015, methodology): “Coleman’s method has been refined by many authors: these authors [36], [38], [39], [48] allow X to have bad reduction at p and improve the 2g 2 to 2r ; [41], [46] generalize to symmetric powers of curves; and a large body of work by many authors allow one to explicitly execute this method in Magma f”
Most odd degree hyperelliptic curves have only one rational point (2013, background): “Finally, we prove N L ≤ n w + 1 + δ ( p, n w ): By Remark 3.6, we may reduce to the case g = 1, with coeﬃcients now in an unramiﬁed extension; this case can be deduced easily by considering the slopes of the (now standard) Newton polygon; cf. [Sto06, Prop.”
Explicit Chabauty over number fields (2010, methodology): “Coleman’s approach has been adapted to give bounds (assuming some reasonable conditions) for the number of solutions of Thue equations [Lorenzini and Tucker 2002], the number of rational points on Fermat curves [McCallum 1992; 1994], the number of points on curves of the form y2 = x5+ A [Stoll 2006b”
Integral points on hyperelliptic curves (2008, methodology): “To determine the integral points on the affine model C given by an equation (1) there are four available methods: (I) The first is Chabauty’s elegant method which in fact determines all rational points on C in many cases, provided the rank of the Mordell–Weil group of its Jacobian is strictly less t”
Chabauty for symmetric powers of curves (2009, background): “…to give bounds (assuming some reasonable conditions) for the number of solutions of Thue equations [33], the number of rational points on Fermat’s curves [34], [35], the number of points on curves of the form y2 = x5 + A [52], and the number of rational points on twists of a given curve [51].”