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

Title data

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. Vol. 21 (2019) Issue 3 . - pp. 923-956.
ISSN 1435-9855
DOI: https://doi.org/10.4171/JEMS/857

Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Keywords: Rational points on curves; uniform bounds; Chabauty’s method; p-adic integration; Mordell–Lang conjecture; Zilber–Pink conjectures
Subject classification: Mathematics Subject Classification (2010): Primary 11G30, 14G05; Secondary 14G25, 14H25, 14H40
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - Univ.-Prof. Dr. Michael Stoll
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 19 Aug 2019 08:03
Last Modified: 19 Aug 2019 08:03
URI: https://eref.uni-bayreuth.de/id/eprint/51924