An asymptotic formula for the number of non-Serre curves in a two-parameter family of elliptic curves

An elliptic curve E over Q has attached to it a Galois representation phiE : Gal( Q&d1;/ Q ) → GL2( Z&d4; ). The image of the Galois representation is always contained in an index two subgroup HE of GL 2( Z&d4; ). An elliptic curve is called a Serre curve if the image of the associated Galois representation is equal to HE, i.e. is as large as possible. A natural question one is interested in is the following: How many elliptic curves over Q are non-Serre curves, i.e. have Galois representations whose image has index greater than two? In this thesis, we compute an asymptotic formula for the number of such non-Serre elliptic curves in a natural two-parameter family of elliptic curves over Q . This strengthens a bound of Jones which shows almost all elliptic curves are Serre curves.