I am new to the topic, so I'm trying to get an overview. I am aware of the relation between modular forms and $L$-series (but don't know what that does) and FLT.
Are there other applications of modular forms other than counting problems (by obtaining the coefficients of a series) in number theory?
A short list would be sufficient but a little more detail with that would be helpful.
EDIT: I am aware of this post but my question is specifically on number theory.
Fibonacci and
Lucas Perfect Powers
Modular forms are used to demonstrate that the only perfect powers in
the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect
powers in the Lucas sequence are 1 and 4.
The
Lebesgue-Nagell equation
Modular forms are used to solve the Lebesgue-Nagell equation.
Densest sphere packing
Modular forms find the lattice with the densest sphere packing problem in dimensions 8 ($E_8$ lattice) and in dimension 24 (Leech lattice).
Ramanujan's constant
Modular forms explain why $e^{\pi\sqrt{163}}$ is so close to an integer.
Modular forms are used to solve Fermat's last theorem, Mock modular forms are used in black holes theory. I read also in Quanta magazine that Eisenstein series are used to compute what we call Monster group. https://d2r55xnwy6nx47.cloudfront.net/uploads/2017/08/symmetry-algebra-and-the-monster-20170817.pdf Here is an interesting article to see the beautiful application of modular forms in astronomy https://m.facebook.com/story.php?story_fbid=795545447309716&id=247304225467177
