[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.


(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 


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