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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?

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dataphile
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    This question is very broad. Modular forms are just special cases of automorphic forms, which are used all over in number theory. I think, the wikipedia article gives a good survey and answeres your question. Searching this site you will also find many similar questions with answers, e.g., here. Also MO has several interesting duplicates, e.g., this one. – Dietrich Burde Aug 19 '19 at 14:18
  • It would be nice if you share a link to mentioned youtube lectures in your question :) – Virtuoz Aug 19 '19 at 14:23
  • link added and question clarified – dataphile Aug 19 '19 at 14:26
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    Automorphic forms are essentially generalizations of $\zeta(s)$. For example the Dedekind/Hecke/Artin L-functions of number field appear everywhere in algebraic number theory and many of the useful ones are modular forms. The theory of elliptic curves is important in itself because it leads to tons of new topics and theorems with applications outside of elliptic curves. Modular forms are generating functions of highly complicated sequences whose corresponding analytic function have many simple (but non-trivial) properties. – reuns Aug 19 '19 at 20:04

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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/

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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

  • I’m no expert on this topic. According to that link, the proof uses the geometric Langlands correspondence. Does the proof make use of modular forms (or the arithmetic Langlands correspondence)? – Mathmo123 Aug 20 '19 at 18:36
  • @Mathmo123 it is arithmetic Langlands correspondence for $GL_n$ over function fields (which is related to geometric Langlands correspondence but the exact implication is not entirely clear to me and possibly is not written down anywhere). It does use automorphic forms. –  Aug 20 '19 at 19:29
  • I'm not sure what you mean by the "arithmetic Langlands correspondence over function fields". If the proof uses automorphic forms over function fields, then my understanding is that, at present, these automorphic forms are only related to modular forms (or automorphic forms over number fields) by analogy. There is no way at present to go between the arithmetic and geometric Langlands correspondences. As such, this does not really relate to the OP's question. – Mathmo123 Aug 21 '19 at 08:09
  • @Mathmo123 we seem to be using the words "geometric" in different senses. There is a Langlands correspondence which relates D-modules on the moduli stack of G-bundles over an algebraic curve to quasi-coherent sheaves on the moduli stack of $G$-local systems. I call it geometric. There is a Langlands correspondence relating automorphic representations to representations of the Weil-Deligne group. I call it arithmetic. In what sense are you using these words? –  Aug 21 '19 at 08:29