How are Modular Forms used in number theory I've been reading up on Fermat's last theorem and the Beal conjecture and in that context watched some of Edward Frenkel's lectures on Youtube. I understand how periodic trigonometric functions like $\sin$ and $\cos$ can be used in number theory e.g. selecting integers with $y=\sin(\pi x)$. But how are modular forms used in number theory?
I'll do the further reading myself but I'm just trying to visualise the approach. Does the topology visually represent integer or prime numbers e.g. make them stand out on a multi-dimensional plane or is it "simply" the intersection of elliptic curves like the analytic continuation of the Riemann Zeta function?
How would you use Modular Forms to pick out integers from a set of real numbers in the way $y=\sin(\pi x)$ does, do they simply use the rotation of quaternions instead of the trigonometric unit circle?
 A: Hilbert modular forms can be used to construct a curious number field, namely a non-solvable finite Galois extension of $\mathbb{Q}$ that is ramified at $p=2$ only (well, this extension is also ramified at infinity if you count it). I guess it would be difficult to find this number field from scratch. See https://arxiv.org/abs/0811.4379, also https://galoisrepresentations.wordpress.com/2019/03/27/dembele-on-abelian-surfaces-with-good-reduction-everywhere/
A: Automorphic forms, a generalization of modular forms, can be used to prove a conjecture of Deligne stating that for a normal variety $X$ over a finite field $F_q$, a prime number $l$ relatively prime to $q$, an irreducible lisse $l$-adic sheaf $\sigma$ on $X$ whose determinant has finite order and a closed point $x\in |X|$, the roots of the polynomial $\mathrm{det}_{\sigma}(\mathrm{Id}-y\:\mathrm{Frob}_{x}^{-1})$ are algebraic numbers of absolute value $1$. This is a somewhat geometric statement but I would consider it to be a part of number theory. As far as I am aware, there is no known proof not using automorphic forms, see https://mathoverflow.net/a/323564/144105
