[FOM] Shafarevich-Tate group and Hilbert's Tenth Problem
Timothy Y. Chow
tchow at alum.mit.edu
Fri Apr 24 22:15:04 EDT 2009
Barry Mazur and Karl Rubin recently posted a preprint to the ArXiv that
may be of interest to FOM readers. They show that if the Shafarevich-Tate
group of an elliptic curve over a number field is always finite (actually
they assume something weaker than this), then Hilbert's Tenth Problem has
a negative answer over the ring of integers of any number field.
http://arxiv.org/abs/0904.3709
(Note that the acknowledgments suggest that this work started when Poonen
saw a potential connection between a certain question about elliptic
curves and Hilbert's Tenth Problem; Mazur and Rubin then worked out the
number theory. They acknowledge Poonen and Shlapentokh for the argument
that connects their result about elliptic curves to Hilbert's Tenth
Problem.)
Tim
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