Abstract
A hyperelliptic curve over Q is called "locally soluble" if it has a point over every completion of Q. In this paper, we prove that a positive proportion of hyperelliptic curves over Q of genus g≥1 are locally soluble but have no points over any odd degree extension of Q. We also obtain a number of related results. For example, we prove that for any fixed odd integer k>0, the proportion of locally soluble hyperelliptic curves over Q of genus g having no points over any odd degree extension of Q of degree at most k tends to 1 as g tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the BrauerManin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers.
Original language  English 

Pages (fromto)  451493 
Number of pages  43 
Journal  Journal of the American Mathematical Society 
Volume  30 
Issue number  2 
Early online date  27 Jul 2016 
DOIs  
Publication status  Published  Apr 2017 
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Profiles

Professor Tim Dokchitser
 School of Mathematics  Heilbronn Chair in Algebraic/Arithmetic Geometry
 Pure Mathematics
 Number theory and combinatorics
Person: Academic , Member