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Stoll, Michael:
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank.
In: Journal of the European Mathematical Society.
Bd. 21
(2019)
Heft 3
.
- S. 923-956.
ISSN 1435-9855
DOI: https://doi.org/10.4171/JEMS/857
Abstract
We show that there is a bound depending only on g,r 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 8rg+33(g–1)+1.
The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of an abelian logarithm on 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.