Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank

By Michael Stoll. Journal of the European Mathematical Society (Print), 2013

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We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an explicit bound is 8 r g + 33 (g - 1) + 1. The proof is based on Chabauty's method; the new ingredient is an estimate for the number of zeros of a logarithm in a p-adic `annulus' on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli whose number is bounded in terms of g (and k). We also show, strengthening a recent result by Poonen and the author, that the lower density of hyperelliptic curves of odd degree over Q whose only rational point is the point at infinity tends to 1 uniformly over families defined by congruence conditions, as the genus g tends to infinity.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1de81c2e94807800a55952d01724519affa71d4e

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Citations in the corpus

Equidistribution in Families of Abelian Varieties and Uniformity (2021, unknown): “The bounds in [44, 85] are very explicit and rather small.”
Uniformity in Mordell–Lang for curves (2020, unknown): “For example, if Jac( C )( F ) has rank at most g − 3, Stoll [Sto19] showed that # C ( F ) is bounded solely in terms of [ F : Q ] and g if C is hyperelliptic; Katz–Rabinoﬀ–Zureick-Brown [KRZB16] later, under the same rank hypothesis, removed the hyperelliptic hypothesis.”
Degeneration of Linear Series from the Tropical Point of View and Applications (2015, background): “In [Sto13], Stoll proved that a uniform bound M(g,K) exists for hyperelliptic curves provided that one assumes in addition that the Mordell–Weil rank of Jac(C)(K) is at most g−3.”