By Roger Weng ’22
Want to learn how some of the most abstract knowledge of mathematics is related to the new frontiers in cryptography? Time to read this article.
Elliptic curves first captured my attention because of their expanding importance in number theory and cryptography. Elliptic curves were used in the proof for Fermat’s Last Theorem, and elliptic curve cryptography provides us with a better alternative to RSA (One of the oldest public-key cryptosystem that is widely used for secure data transmission). I am particularly interested in the elliptic curve L-function since it can convert many difficult problems into conceptually straightforward ones.
During the 2020-2021 school year, I worked with Dr. Steven J. Miller, professor of mathematics at Williams College, to conduct research on elliptic curves. I immediately became interested in elliptic curves because it is part of a mathematical tradition going back thousands of years: counting how many rational points satisfy a polynomial. Although it is relatively easy to find the rational points on a circle, it turns out that it is far more intriguing to find such points on elliptic curves. For background, I read the book “Elliptic curves” by Anthony Knapp, which gave a detailed explanation on the elementary theory of elliptic curves and its connections with modular forms. As I gradually picked up more knowledge in this area by reading mathematical papers, I started to understand why elliptic curves are so fascinating for mathematicians. They are simple enough for us to make real progress and come up with some good results, but they are also complicated enough for us to ask interesting questions and wander into the unknown.
My research paper, “Biases in Moments of the Dirichlet Coefficients in One-Parameter Families of Elliptic Curves,” was recently published by the PUMP Journal of Undergraduate Research. Please check it out.
The PUMP (Preparing Undergraduates through Mentoring toward Ph.D.s) Journal of Undergraduate Research publishes articles written by undergraduate students that want to pursue doctoral studies in the Mathematical Sciences.