# Notes and Essays

Below is a collection of expository notes in number theory.

- Intro to Iwasawa theory. This post summarizes Iwasawa's Theorem about growth of class numbers in infinite towers of number fields. This is a very beautiful theorem that uses tools from algebraic number theory and communitative algebra. Personally, it is the theorem that got me interested in this subject.
- Intro to $p$-adic $L$-functions. This post introduces $p$-adic $L$-functions, which are a key ingredient in the analytic side of Iwasawa theory. Specifically, it talks about the Kubota-Leopoldt $p$-adic $L$-function and its interpolation properties.
- The Iwasawa main conjecture. This post draws the link between the algebraic side of Iwasawa theory and the analytic side of Iwasawa theory. These two worlds are connected via the so-called "Iwasawa main conjecture".
- Kato's Euler System. This post describes the construction of Kato's Euler system, which was used to prove one direction of the Iwasawa main conjecture for modular forms.

## Other Writing

Here are some writeups I have from courses I've taken in the past.

- The Quantum Unique Erogidicity Conjecture (QUE). This was a final project for a reading course I took in Summer 2022 with Prof. Henry Kim called "Subconvexity Bounds for $L$-functions on the criticial line".
- An overview of the Weil Conjectures. This was a final project for a reading course I took in Summer 2021 with Faisal Al-Faisal on algebraic number theory and elliptic curves.