Number Theory Seminar
Alina Carmen Cojocaru
University of Illinois at Chicago
One parameter families of elliptic curves with maximal Galois representations
Abstract: Let E be an elliptic curve over Q and let Q(E[n]) be its n-th division field. In 1972, Serre showed that if E is without complex
multiplication, then the Galois group of Q(E[n])/Q is as large as possible, that is, GL_2(Z/n Z), for all integers n coprime to a
constant integer c(E, Q) depending (at most) on E/Q. Serre also showed that the best one can hope for is to have
|GL_2(Z/n Z) : Gal(Q(E[n])/Q)| at most 2 for all nonzero integers n.
I will discuss the frequency of this optimal situation in a one-parameter family of elliptic curves over Q.
This is joint work with David Grant and Nathan Jones.
This talk will be followed by additional discussions at 4pm.
Wednesday November 10, 2010 at 2:00 PM in SEO 636