Abstract: Among the most studied objects in current number theory are Galois groups and their associated representations. Much progress in modern number theory, including the proof of the former Shimura-Taniyama conjecture which resulted in the proof of Fermat's Last Theorem, has depended on a detailed understanding of these types of representations.
A classic result of Serre is that the image of the $l$-adic Galois representation naturally arising from an elliptic curve has image as large as possible for almost all primes $l$. Extensions of this result are of great interest and under development. I extended this result to a rank $3$ case. In this talk, I will explain the background and the methods.