Non-Serre elliptic curves

Vinod Radhakrishnan, University of Colorado at Boulder
An elliptic curve over the rationals without complex multiplication is called a Serre curve if its Galois representation is as large as possible. A natural question is the following: How many elliptic curves overs the rationals are Serre curves? Than answer was provided by Jones: Almost all curves are Serre curves. In this talk we provide an asymptotic estimate for the number of non-Serre curves of bounded height. This strengthens Jones' bound.