Title:
Prime torsion order on elliptic curves over number fields of small
degree

Abstract:
This is joint work with Maarten Derickx, Sheldon Kamienny and William
Stein.

For a positive integer d, we define S(d) to be the set of all prime
numbers that occur as the order of a K-rational torsion point on an
elliptic curve over a number field K of degree d. Prior to our work,
the following sets S(d) were known:
   S(1) = {2, 3, 5, 7}  (Mazur)
   S(2) = {2, 3, 5, 7, 11, 13}  (Kamienny)
   S(3) = {2, 3, 5, 7, 11, 13}  (Parent)
We determine S(4), S(5) and S(6), and come close to determining S(7).