EVENT DETAILS AND ABSTRACT

Title: Elliptic Curves and Hilbert's Tenth problem
Speaker: Professor Barry Mazur
Speaker Info: Harvard
Brief Description:
Special Note:
Abstract:

This lecture is about joint work with Karl Rubin regarding the Mordell-Weil group of elliptic curves over arbitrary number fields. As a consequence of this work, under appropriate hypotheses we can find elliptic curves that have many quadratic twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group over an arbitrary number field K. Moreover we can find such elliptic curves E that have the following stability property: for a given cyclic extension-field L of K of prime degree the Mordell-Weil rank of E over L remains equal to 1. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.