Arithmetic of the Legendre curve

Series: 
Algebra Seminar
Monday, March 7, 2011 - 15:00
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
Georgia Tech
Organizer: 
Let k be a field (not of characteristic 2) and let t be an indeterminate.  Legendre's elliptic curve is the elliptic curve over k(t) defined by y^2=x(x-1)(x-t).  I will discuss the arithmetic of this curve (group of solutions, heights, Tate-Shafarevich group) over the extension fields k(t^{1/d}).  I will also mention several variants and open problems which would make good thesis topics.