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I've just read some materials of modular forms on $SL_2(\mathbb{Z})$, and find some interesting application.

  1. Deal with Ramanujan $\tau$ function. I saw it in

Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?.

And I think is really a good example.

  1. Prove Ramanujan conjecture. I have read the proof of Ramanujan conjecture in

Ahlgren S, Boylan M. Arithmetic properties of the partition function[J]. Inventiones mathematicae, 2003, 153(3): 487-502.

And I think it is not only a good example of application on modular forms of $SL_2(\mathbb{Z})$ but an amazing proof. (I just read the first half part.)

I really want more such examples to help me understand modular forms on $SL_2(\mathbb{Z})$! Can anybody show me more like this? Or recommend me some references?

Thanks a lot for any help.

1 Answers 1

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I think a really down to earth application was in the 2005 Hanke-Bhargava Theorem.

The proof showed that if a positive definite quadratic form with integral entries represents the first 290 positive integers, then it represents all of them.

A quadratic form might be able to "represent a number" (has integral solutions) in multiple ways: $Q(\vec{x})=n$ may have multiple solutions, which we will count, calling them "representation number," denoted $r_{Q}(n)$.

The proof essentially considers a "theta series," which is a generating function with $r_Q(n)$ as the fourier coefficients:

$$1+\sum_{n=1}^{\infty} r_Q(k) e^{2 \pi i k z}.$$

However, unlike the usual sense of a generating function, it is not so important that this is a formal algebraic object, but viewed analytically, it is a modular form!

Using the theory, we know that the space of modular forms (the full modular group) decomposes into the space of Eisenstein series and Cusp forms (with respect to a technical inner product.)

The point here, is that Eisenstein series are increasing and nonnegative, while the cusp series can be negative (but vanishes as $n \to \infty$.)

Using the theory of Siegel, Deligne, Hanke, and many others, we have ways to compute the Eisenstein series efficiently, and bound the cusp form.

In other words, we can decompose our representation number in the following sense:

$$r_Q(n)=r_{E}(n)+r_{C}(n),$$

so if we can show that this value is nonnegative for sufficiently large $n$, we will be done. The theory allows us to show this via modular form decomposition.

Other relevant papers using similar methods:

http://www.fen.bilkent.edu.tr/~franz/mat/15.pdf

https://arxiv.org/abs/1111.0979

https://arxiv.org/abs/1608.01656

Here is a more detailed exposition (in slide form): http://college.wfu.edu/mathreu/wp-content/uploads/AMS_slides.pdf

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    But this application requires modular forms with level and half integral weight, so not SL(2,Z). – 2017-02-20
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    Well, we consider the natural map $SL_2(\mathbb Z) \to SL_2(\mathbb Z/n \mathbb Z)$ which has kernel $\Gamma (N)$, where $N$ is the level. The weight is also $2$ throughout, since the theorem did not require any $5$ dim quadratic forms. – 2017-02-20