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I have seen the definition of a modular form, but it seems obscure to me. I get the impression that if I were to read a lot about them, eventually I would see how they can be used. I am curious about the ways in which modular forms are applied. How are they used? What are some important theorems of intrinsic interest that can be (relatively easily) obtained by using them? Are there any that I should look at in particular?

JessicaB
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  • I'm not an expert, but Wiles's proof of Fermat's Last Theorem involved showing that certain spaces of modular forms are zero. – Cheerful Parsnip May 17 '13 at 20:31
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    You mind find this and this interesting. – Stahl May 17 '13 at 20:32
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    Stein has a nice list of applications here: http://wstein.org/books/modform/modform/modform.html#applications-of-modular-forms – Qiaochu Yuan May 17 '13 at 20:34
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    Modular forms can be seen as certain space of functions which provides you with concrete models of infinte dimensional representations of locally compact groups (SL(2,R) say). Modular forms can be geometrically thought of as certain sections on moduli spaces parametrizing elliptic curves and some extra data. This connects them to arithmetic geometry and as Deligne has shown one can use them to construct Galois representations with certain desirable properties (and if I am not wrong think this is the only known way to construct interesting Galois representations). – DBS Jul 10 '13 at 08:04
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    This is perhaps little vague because the technical prerequisite behind all this is somewhat heavy. You may want to take a look at the beautiful book by Peter Sarnak called Some Applications of Modular Forms , published by CUP for some very concrete (and beautiful) applications. – DBS Jul 10 '13 at 08:07

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One of the simplest applications (and quickest to get to) is to representation numbers of quadratic forms. E.g. Jacobi's formula, that the number of ways of writing a natural number $n$ as the sum of four squares is equal to $8 \sum_{d|n, 4 \not\mid d} d$, was originally proved using modular forms, and I think this is still the most versatile method of proof.

For more general quadratic forms, one can't necessarily get as precise formulas (so-called cuspforms introduce error terms which don't admit explicit formulas), but one gets approximations (and the Ramanujan--Petersson conjecture on growth of Fourier coeffs. of cuspforms plays a role in bounding the error terms coming from cuspforms).

Some of this (although not Jacobi's formula itself) can be found in Serre's Course in arithmetic, which is the nicest treatment for a beginner.

There are also the applications to the theory of elliptic curves (and then to FLT) mentioned in the comments. For example, the best results in the direction of BSD (such as Gross--Zagier, or Kato's results) rely on the connection between modular forms and elliptic curves.

Matt E
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  • Saying that Serre is "nicest treatment for a beginner" is insane... Not only does he write with an extremely terse style, he also uses many unforgivable notational conventions in the modular forms chapter of that book, that are different from the rest of the world. – Zongshu Wu Jun 05 '25 at 13:41