English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We show that there is a bound depending only on g and [K:Q] for the number ofK-rational points on a hyperelliptic curve C of genus g over a number field Ksuch that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, anexplicit bound is 8 r g + 33 (g - 1) + 1.  The proof is based on Chabauty's method; the new ingredient is an estimatefor 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 fora p-adic field k, the set of k-points on C can be covered by a collection ofdisks and annuli whose number is bounded in terms of g (and k).  We also show, strengthening a recent result by Poonen and the author, thatthe lower density of hyperelliptic curves of odd degree over Q whose onlyrational point is the point at infinity tends to 1 uniformly over familiesdefined by congruence conditions, as the genus g tends to infinity.

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