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Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular forms. I understand some sphere packing and issues going with it but my understanding is most upperbounds come from linear programming and the bounds that are currently proven optimal (including at $n=8$ and $n=24$) already come from linear programming bounds.

  1. How do modular forms become part of the story that provide the lower bound (do they arise naturally from some packing related structure)?

  2. Is there a bigger story that this is just a chapter of that may apply to other upper bounds generated from linear programming? What makes modular forms click for this class of linear programming bounds (perhaps Sphere packing and quantum gravity is of utility?)?

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  • $\begingroup$ This by Henry Cohn might answer your question: mathoverflow.net/questions/235347/… $\endgroup$
    – Yoav Kallus
    Commented Feb 13, 2020 at 13:23
  • $\begingroup$ @YoavKallus arxiv.org/abs/1603.05202 says the properties of the functions involved. However does not say why it has to be a modular form. Perhaps modular forms are not the right objects at $n\not\in\{8,24\}$ and something else? $\endgroup$
    – VS.
    Commented Feb 13, 2020 at 14:15
  • $\begingroup$ @VS. What do you mean by right objects for these n? For most of these n, the LP bound is not expected to be tight. Are you asking whether modular forms are involved in the construction of the smallest LP upper bound? $\endgroup$
    – Yoav Kallus
    Commented Feb 13, 2020 at 14:33
  • $\begingroup$ @YoavKallus No. What I am saying is this. There is an achievability result by construction and an upper bound as Harry Richman says below based on some functions. It is intriguing that at $n\in\{8,24\}$ there is a modular form candidate for $f$ showing achievability known is the best. At $n=5$ perhaps a different construction needs to be shown optimal. For that I doubt $f$ will be a modular form if similar strategy works. $\endgroup$
    – VS.
    Commented Feb 13, 2020 at 15:07
  • $\begingroup$ @VS. Numerical results for n=5 suggest there is no f that gives upper bound matching current best lower bound. For n=8, even before the exact construction was given, the numerical results suggested there was such f. Either LP bounds are not powerful enough in n=5 to show optimality or currently densest known packing is not optimal. More likely the former. $\endgroup$
    – Yoav Kallus
    Commented Feb 13, 2020 at 16:35

2 Answers 2

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This is a tough question, and I don’t think there’s a definitive answer yet. For some mathematical details, see the following survey articles:

https://arxiv.org/abs/1611.01685 https://arxiv.org/abs/1603.05202

Instead, I’ll focus on the big picture here. Why modular forms? I can see a couple of potential answers:

(1) Why not modular forms? Before Viazovska’s proof, numerical experiments indicated that there were remarkable special functions in 8 and 24 dimensions that would prove the optimality of $E_8$ and the Leech lattice. However, nobody had any idea how to construct them explicitly, or prove existence at all. Modular forms are by far the most important class of special functions related to lattices (in higher dimensions, since arguably trig functions and the exponential function are the most important special functions related to lattices), so lots of people expected that the magic functions for sphere packing should be connected somehow to modular forms. The proof had to wait for Viazovska to come up with a beautiful integral transform, but the fact that it used modular forms wasn’t such a great surprise. I.e., her contribution wasn’t the idea that modular forms should play a role, but rather figuring out how to use them, which was quite subtle and ingenious.

You’re right that nobody has any idea how to use modular forms to optimize the linear programming bound in other dimensions. However, it’s possible that they will continue to play a role. For example, see the example Felipe Gonçalves and I found at the end of Section 2.1 of our paper https://arxiv.org/abs/1712.04438 (which is not a sphere packing bound, but closely related). It really looks like a small perturbation of a function based on modular forms (see https://arxiv.org/abs/1903.05737), and I wouldn’t be surprised if the optimal function has a nice series expansion based in some way on modular forms. From this perspective, the remarkable thing about 8 and 24 dimensions wouldn’t be the appearance of modular forms, but rather the fact that the series collapses to a single term, with a matching sphere packing. However, this is all speculative.

(2) The other perspective is that we have very little understanding of why 8 and 24 dimensions are special in the first place. For example, why shouldn’t sphere packing in 137 dimensions also admit an exact solution via linear programming bounds? It sure doesn’t look like it does, but perhaps we just don’t know the right sphere packing to use, and some currently unknown packing might match the upper bound. That would be very surprising, since our experience is that exceptional phenomena occur in clusters. We would expect to see some sort of remarkable symmetry group, probably a finite simple group, and there aren’t any candidates acting on 137 dimensions. However, this expectation is just based on our limited experience, and mathematics can confound our expectations. So far, nobody has found even a convincing heuristic argument for why there shouldn’t be an exact solution in 137 dimensions, and that’s a major gap in our understanding. The most we can say is that it would have to differ in some important ways from 8 and 24 dimensions, which is far from an explanation of why it can’t happen.

I guess I’d summarize it like this. If you accept that lattices in 8 and 24 dimensions play a special role, then modular forms feel naturally connected. However, we’re missing a deeper explanation of the role of these special dimensions.

Let me add one more specific mathematical comment. The magic functions in 8 and 24 dimensions fit into a general picture of building radial functions that vanish on all but finitely many vector lengths in a lattice, and whose Fourier transforms vanish on all but finitely many vector lengths in the dual lattice. If you can do this in full generality, then Poisson summation lets you solve for the number of lattice vectors of each length. These are the coefficients of a modular form, namely the theta series of $E_8$ or the Leech lattice, so the conclusion is that this family of functions somehow “knows about” the theta series. In other words, you can’t expect to construct the whole family without running into modular forms in some way. This leaves a couple of possibilities: maybe the magic functions for sphere packing are simpler than most functions in this family, and could be constructed without modular forms, or maybe these functions are deeper than modular forms (and require some mysterious special functions not yet known to mathematicians). What we know now is that modular forms suffice, and in a sense are necessary because the magic functions are unique.

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  • $\begingroup$ Would there be some connection between $3$, $8$ and $24$ or would it be wild to expect one? Since we know the best bounds in all three has it been predicted that some multiplicative behavior might be possible? Just wondering if there could be some underlying arithmetic to these optimality? $\endgroup$
    – VS.
    Commented Feb 13, 2020 at 16:29
  • $\begingroup$ The linked paper in post on quantum gravity speculates some other deeper construction than modular forms of which modular forms emerge at $n\in\{8,24\}$ (I think). $\endgroup$
    – VS.
    Commented Feb 13, 2020 at 16:37
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    $\begingroup$ Unfortunately I don’t know of an overall reference. “Monstrous moonshine” is probably the most dramatic and important case, but there’s a lot of other numerology scattered in the literature. Figuring out what’s going on can be tricky. For example, suppose we’re counting minimal vectors in lattice. For D_4, E_8, and the Leech lattice, we get 24, 240, and 196560, with each number divisible by the previous one. Is there an explanation in terms of group actions? Yes for the first case: the 24 vectors form the units in the Hurwitz integral quaternions, and they act on E_8. $\endgroup$
    – Henry Cohn
    Commented Feb 13, 2020 at 20:07
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    $\begingroup$ No in the second case: the 240 vectors form the unit integral octonions, but they aren’t associative so the question of whether they act on the Leech lattice doesn’t even make sense (they don’t even act on themselves, let alone other things). Nevertheless, there are octonionic constructions of the Leech lattice that are a little more subtle but explain the divisibility by 240. So sometimes things work the way you’d hope, but sometimes we need more subtle or sophisticated explanations. $\endgroup$
    – Henry Cohn
    Commented Feb 13, 2020 at 20:09
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    $\begingroup$ For the second question about conformal field theory and packing in other settings, I don’t know of a relationship, but I have no grounds for ruling one out. $\endgroup$
    – Henry Cohn
    Commented Feb 13, 2020 at 20:12
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In my understanding, the connection to modular forms came via a result of Cohn and Elkies in their paper "New upper bounds on sphere packings I," Ann. of Math. 157 (2003) 689-714, also on the arxiv.

Theorem 3.1 (p. 694) states that the density $\rho$ of a sphere packing in $\mathbb R^n$ satisfies $$ \rho \leq \frac{\pi^{n/2}}{(n/2)!} \frac{f(0)}{2^n \hat f(0)} \tag{$*$}$$ where $f: \mathbb R^n \to \mathbb R$ is any admissible (i.e. sufficiently decaying) function such that

  1. the Fourier transform $\hat f$ is admissible,
  2. $f(x) \leq 0$ for $\|x\| \geq 1$,
  3. $\hat f(t) \geq 0$ for all $t$.

However, this result gives an upper bound on packing density rather than a lower bound. (So I'm not sure whether this answers your question 1.) In dimensions 8 and 24, the sphere packing problem was later solved by finding a modular-form-inspired function $f$ such that $(*)$ matches the density of the $E_8$ lattice and the Leech lattice, respectively.

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  • $\begingroup$ Ahh ok. somewhat counterintuitive though. So for other cases still have to go through explicit yet unknown constructions? $\endgroup$
    – VS.
    Commented Feb 13, 2020 at 8:15
  • $\begingroup$ Why $f$ should be modular form (it seems it will not be modular forms in $n\not\in\{8,24\}$ or at least beyond some finite $n$)? $\endgroup$
    – VS.
    Commented Feb 13, 2020 at 9:57

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