On Viazovska’s modular form inequalities by Dan Romik

Combinatorics and more 2023-12-17

The main purpose of this post is to tell you about a recent paper by Dan Romik which gives a direct proof of two crucial inequalities in Maryna Viazovska’s proof that $E_8$ lattice sphere packing is the densest sphere packing in eight dimensions.

On Viazovska’s modular form inequalities, by Dan Romik

Abstract

Viazovska proved that the sphere packing associated with the $E_8$ lattice, which has a packing density of $\frac {\pi^4}{384}$ , is the densest sphere packing in $8$ dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

This is a wonderful news. Viazovska’s breakthrough was among the startling mathematical news of the past decade and I am always very excited about different proofs for known theorems, new proofs, simplified proofs, and at times even new complicated proofs and complifications.

Here is my original post on Viazovska’s proof (with many links) and a post by John Baez. And here are slides of a recent talk by Dan on his work.

I will say more about this paper below. First let me mention some related and unrelated things that you can learn from Dan Romik homepage.

DRb2v2

The cover of Dan’s new book with Viazkova’s magic function circled.

Topics in Complex Analysis, also by Dan Romik

This graduate-level mathematics textbook provides an in-depth and readable exposition of selected topics in complex analysis. The material spans both the standard theory at a level suitable for a first-graduate class on the subject and several advanced topics delving deeper into the subject and applying the theory in different directions. The focus is on beautiful applications of complex analysis to geometry and number theory. The text is accompanied by beautiful figures illustrating many of the concepts and proofs.

Among the topics covered are conformal mapping and the Riemann mapping theorem; the Euler gamma function, the Riemann zeta function, and a proof of the prime number theorem; elliptic functions, and modular forms. The final chapter gives the first detailed account in textbook format of the recent solution to the sphere packing problem in dimension 8, published by Maryna Viazovska in 2016 — a groundbreaking proof for which Viazovska was awarded the Fields Medal in 2022.

The entire book can freely be downloaded from the publisher, and section 6 gives a full account of the eight-dimensional sphere packing theorem.

Romik’s Ambidextrous Sofa and more from Romik’s homepage.

Romik’s page on the moving sofa problem contains a lot of information on this problem and links to many discussions including one by John Baez linked above. Here is the pages of Dan’s very recommended 2015 book The Surprising Mathematics of Longest Increasing Subsequences and his recent book An Invitation to MadHat and Mathematical Typesetting (both are freely available).

Viazovska’s Magic functions

I had a zoom meeting with Dan hoping to get some more information (that I can share with you) about his proof. Instead Dan did not talk much about his work but rather enthusiastically explained to me Viazovska’s breakthrough so let me tell you about that. Very good places to read about it are Viazovska’s original paper The sphere packing problem in dimension 8; Henry Cohn’s 2016 paper in the Notices of the AMS, A conceptual breakthrough in sphere packing, and a 2016 paper by David de Laat, Frank Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, and Andrei Okounkov’s paper The magic of 8 and 24.

By the work of Cohn and Elkies in order to prove that $E_8$ gives the densest sphere packing in $\mathbb R^8$ one needs to find the following “magic function”:

$f$ is a real function on $\mathbb R ^8$ so that

1) $f$ is radial: $f(x)$ depends only on $\|x\|$ , the distance from $x$ to the origin.

2) $f$ is a Schwartz function: it and all its derivatives decrease faster than the reciprocal of any polynomial.

3) $f(0)=\widehat f(0)=1$ , where $\widehat f$ is the Fourier transform of $f$ .

4) $\widehat f(x)\ge 0$ for all $x \in \mathbb R^8$

5) $f(x) \le 0$ for all $x$ with $\|x\| \ge 2$ .

(Strictly speaking, a non-radial function that satisfies properties (2)-(5) could do the magic, and will also lead to a radial magic function.)

Quite a lot of knowledge of these hypothetical magic functions was gathered and in particular both they and their Fourier transform need to have zeros when the norm of $x$ is an integer of the form $\sqrt{2n}$ . Here is what Dan told me about steps and ideas for Viazovska’s work. Before that Dan reminded me that a modular function of weight $k$ satisfies

$g(-1/\alpha)= \alpha^k g(\alpha).$

(This property is just part of the definition of modular functions, and for a full definition see Dan’s book or Wikipeda page on modular forms under the heading “Modular forms for SL(2,Z)”.)

So here is some of the main steps and ideas in Viazovska’s construction as I understood from my conversation with Dan.

A) First to consider eigenfunctions for the Fourier transform. One way of generating such eigenfunctions is via an integral formula of the form

$\displaystyle \int _0^\infty g(i \alpha) e^{-\pi \alpha |x|^2} d\alpha$ ,

where $g(x)$ is a modular function of weight 4 (for Fourier eigenfunctions in dimension 8).

(A word of explanation: the dual requirement on the zeros of both the function and its Fourier transform is the reason why it’s so useful to consider Fourier eigenfunctions – with such functions, once you manage to force them to have zeros somewhere, you get the same property for the Fourier transform for free.)

B) Then comes the following bold idea: To insist on the locations of zeros by multiplying with $\sin^2(\pi ||x||^2 / 2).$ . At this point Dan shared his screen and started showing how to continue.

A few words of explanation: From Cohn and Elkies’s work it was known that a (radial) magic function must have zeros on spherical shells of radius $sqrt(2n)$ for $n=1,2,\dots$ , and that all of these zeros should be of even order (as a function of ||x||), except for the zero at $\sqrt 2$ , which should be of odd order. The simplest way to satisfy these constraints is to look for a magic function with a zero of first order at $\sqrt 2$ and a double zero at $\sqrt 4, \sqrt 6, \dots$ and the simplest way to achieve that is to multiply by the factor $sin^2(\pi ||x||^2 / 2)$ . (This adds some complication, see below.)

C) With the extra sine-squared factor, the condition on the weight function g(x) that would ensure the resulting function is a Fourier eigenfunction is no longer that of a simple modular form of weight 4, and needs to be rethought. By expanding the sine in terms of complex exponentials, one can work out that g(x) needs to satisfy a much more unusual functional equation. Viazovska realized that functions of this type can be constructed using combinations of modular forms of different weights as well as a more exotic type of function known as a quasimodular form.

D) To make everything work it is also necessary to have an analytic continuation argument. This requires, so Dan explained to me, a certain deformation of complex contours that replaces three parallel lines with a figure that looks like a rake.

(Another word of explanation: multiplying by the factor $\sin^2(\pi ||x||^2 / 2)$ also forces a double zero at $\sqrt 2$ , which is undesirable, and needs to be compensated for by multiplying by a function that has a pole at $\sqrt 2$ – this is why the extra step of analytic continuation is needed.)

E) After constructing two separate Fourier eigenfunctions, associated with the respective eigenvalues +1 and -1 and each constructed using the appropriate modular and quasimodular forms, Viazovska took their sum to obtain the required magic function.

Viazovska’s magic functions as presented in Dan’s book (and its cover) is:

$\displaystyle \varphi_8(x) = \displaystyle -4\sin^2\left(\frac{\pi \lVert x \rVert^2}{2}\right) \times$ $\displaystyle \int_0^\infty\left(108 \frac{(i t E_4'(it) + 4 E_4(it))^2}{E_4(it)^3-E_6(it)^2}+ 128 \left( \frac{\theta_3(it)^4 + \theta_4(it)^4}{\theta_2(it)^8}+ \frac{\theta_4(it)^4 - \theta_2(it)^4}{\theta_3(it)^8}\right)\right)$ $\displaystyle e^{-\pi \lVert x \rVert^2 t}\,dt$

The definitions of the Eisenstein series $E_2, E_4, E_6$ , and Jacobi thetanull functions $\theta_2, \theta_3$ and $\theta_4$ is given below. Enjoy these beautiful formulas!

How to prove positivity?

Viavzkova’s brilliant ideas led to a function with zeroes at the right locations. The nonnegativity requirements 4) and 5) were not obvious at all. To complete the proof it was necessary to prove them. It was clear to Viazovska that this part can be done numerically and this consists of the computational part of her proof.

The divisor function, Eisenstein series and Jacobi thetanull functions

Here are the definitions of the functions that appear in Viazovska’s magic function. Let us start with the divisor function

$\sigma_\alpha(n) = \sum_{d\,|\,n} d^\alpha$ .

We will also need the definitions of the the Eisenstein series $E_2, E_4, E_6$ , and Jacobi thetanull functions $\theta_2, \theta_3$ and $\theta_4$ . Put $q=e^{\pi i z}$ .

$E_2(z) = \displaystyle 1 - 24 \sum_{n=1}^\infty \sigma_1(n) q^{2n},$

$E_4(z) = \displaystyle 1 + 240 \sum_{n=1}^\infty \sigma_3(n) q^{2n},$

$E_6(z) = \displaystyle 1 - 504 \sum_{n=1}^\infty \sigma_5(n) q^{2n},$

$\theta_2(z) = \displaystyle \sum_{n=-\infty}^\infty q^{(n+1/2)^2},$

$\theta_3(z) = \displaystyle \sum_{n=-\infty}^\infty q^{n^2},$

$\theta_4(z) = \displaystyle \sum_{n=-\infty}^\infty (-1)^n q^{n^2}.$

Viazovska’s inequalies and Dan’s proof

Dan took it as a challenge to prove Viazovska’s two inequalities without using numerical calculations. This was not an easy project and after several unsuccessful attempts he succeeded.

Viazovska’s inequalities

Now we need two functions: $\displaystyle \phi_0(z) = \displaystyle 1728 \frac{(E_2(z) E_4(z) - E_6(z))^2}{E_4(z)^3 - E_6(z)^2},$

and $\displaystyle \psi_I(z) = \displaystyle 128 \left( \frac{\theta_3(z)^4 + \theta_4(z)^4}{\theta_2(z)^8} + \frac{\theta_4(z)^4 - \theta_2(z)^4}{\theta_3(z)^8} \right),$

and need further to define functions $A(t)$ , $B(t)$ of a real variable $t>0$ by $\displaystyle A(t) = -t^2 \phi_0(i/t) - \frac{36}{\pi^2} \psi_I(it),$

and

$\displaystyle B(t) = -t^2 \phi_0(i/t) + \frac{36}{\pi^2} \psi_I(it).$

Theorem [Viazovska’s modular form inequalities]:

The functions $A(t)$ , $B(t)$ satisfy

$\displaystyle A(t) < 0 \qquad (t>0),$
$B(t) > 0 \qquad (t>0).$

I will not say much here about Dan’s proof. The proof of the first inequality is fairly short and it relies on various well-known (but not to me) modular function identities like this standard (!) one

$E_4^3 - E_6^2 = \displaystyle 1728 q^2 \prod_{n=1}^\infty (1-q^{2n})^{24},$

and an identity by Jacobi

$\displaystyle \theta_2^4+\theta_4^4 = \theta_3^4$ .

The proof of the second inequality is considerably harder.