Some notes of mine

October 29, 2017

I wrote these notes a while back for my own reference. I don't know how useful they will be to others but they are my attempt to understand the subjects they talk about. They have no references to where I got the material since they are only intended for private use (despite my publishing them here...).

1. Modular forms over finite characteristic

These notes contain material from Serre's article on modular forms mod-p and at the very end, a little bit about modular forms over the p-adics as in Serre's paper culminating in a definition of the Kubota-Leopoldt p-adic zeta function using this theory.

After a brief summary of classical modular forms, the results should be more or less self contained. The most interesting section (in my opinion...) is Section 4 where I briefly discuss Katz' perspective on modular forms and use it to show that the Hasse invariant is a modular form and compute it's q-expansion using the Tate curve. This section is also scarce on details and I understood this stuff by reading Prof. Emerton's wonderful expository article here.

The article is here: Modular_Forms_mod_P.

2. Complex Multiplication

This is my attempt to streamline and summarize the main results of complex multiplication as I see them. This is short (about 5-6 pages) and the final section is my answer to the question here.

The article is here: Complex_Multiplication_notes.

3. Growth of Class number in $\mathbb{Z}_p$ extensions

A summary of the relevant chapter in Washington's books. Nothing new here.

The article is: Growth_of_class_groups_in_Z_p_extensions.