What math should I learn to get in depth with Elliptic Curve Cryptography research? My background is computer scientist. I have done applied cryptography research for a while. Currently, I'm working on Elliptic curve cryptography.

To understand the idea and how to use Elliptic curve cryptography (ECC) is not that complicated. I understand basic algebra, EC point addition, finite field, Discrete Logarithm problem, etc.

However, When it comes to improving ECC research, it requires a lot of mathematical knowledge. For instance, if I want to introduce a new curve (like when Bernstein, Birkner, Joye, Lange, and Peters introduced Twisted Edwards curve in 2008) and evaluate its security, or trying to solve Elliptic Curve Discrete Log Problem (ECDLP), what math field(s) should I study?
An Introduction to the Theory of Elliptic Curves (Brown University)

Elliptic Curve Cryptography (Wikipedia)

Performance analysis of identity management in the Session Initiation Protocol (SIP) (IEEE)

Overview of Elliptic Curve Cryptosystems. (RSA.com)

Lecture by Dan Boneh (Youtube)