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Journal of the European Mathematical Society
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Published online: 2018-12-12
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank
Michael Stoll[1] (1) Universität Bayreuth, Germany
We show that there is a bound depending only on $g, r$ and [$K : \mathbb Q$] 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 $K = \mathbb Q$, 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 $\mathbb Q$ 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.
Keywords: Rational points on curves, uniform bounds, Chabauty’s method, $p$-adic integration, Mordell–Lang conjecture, Zilber–Pink conjectures
Stoll Michael: Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank. J. Eur. Math. Soc. 21 (2019), 923-956. doi: 10.4171/JEMS/857