and so geomxi is the geometric mean of xi. The goal of this note is to sketch the proof of the following theorem, concentrating on the iteration scheme from [4] in the two dimensional case. This justifies (19) of [4] without an appeal to [1]. We also include some routine applications of Hölder, Minkowski, etc. that are omitted from [4] for completeness. We… (More)

During the summer between my junior and senior year, I was extremely lucky to be one of the 80 rising seniors who were admitted to the Research Science Institute (RSI) held at MIT. My mentor was Ryan Reich, a graduate student from Harvard University. Before the program, I was contacted by my mentor and was given a couple of project ideas. I could either do… (More)

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL2(Ẑ). In [Gre10], Greicius develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be… (More)