## EVENT DETAILS AND ABSTRACT

**Number Theory**
**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 Hilbertâ€™s Tenth Problem has a
negative answer over the ring of integers of every number field.

**Date:** Monday, April 13, 2009

**Time:** 3:00PM

**Where:** Lunt 107

**Contact Person:** Florian Herzig

**Contact email:** herzig@math.northwestern.edu

**Contact Phone:** 847-467-1898

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