Extremal even unimodular lattices

A lattice in $\mathbb{R}^n$ is the set of all integer combinations of $n$ linearly independent vectors $b_1,\dots,b_n$:

$$\Lambda=\{\,c_1b_1+\dots+c_nb_n : c_1,\dots,c_n\in\mathbb{Z}\,\}.$$

The simplest example is $\mathbb{Z}^n$, the points with integer coordinates. We write $\langle x,y\rangle$ for the usual inner product and call $\langle x,x\rangle$, the squared length, the norm of $x$.

Up to rotations and reflections, a lattice is determined by the inner products of its basis vectors. They form the Gram matrix

$$G=\bigl(\langle b_i,b_j\rangle\bigr)_{1\le i,j\le n}.$$

If a vector of $\Lambda$ is written as $x=c_1b_1+\dots+c_nb_n$, its norm is $cGc^{\mathsf T}$ with $c=(c_1,\dots,c_n)$.

The basis of a lattice is not unique. Changing the basis replaces $G$ by $UGU^{\mathsf T}$ for an integer matrix $U$ with $\det U=\pm1$, so the determinant $\det G$ does not depend on the choice of basis. We call it the determinant $\det\Lambda$ of the lattice. Its square root is the volume of the parallelepiped $\{t_1b_1+\dots+t_nb_n : 0\le t_i<1\}$, called a fundamental cell. Translating the fundamental cell by all vectors of the lattice covers $\mathbb{R}^n$ without overlaps.

All lattices on this website are given by Gram matrices with integer entries (the lattices studied here have integer inner products, as the next section explains). The figure shows the Gram matrix of a lattice in $\mathbb{R}^8$ called $E_8$. Among the even unimodular lattices defined in the next section, $E_8$ is the one of smallest dimension, and it appears throughout this website.

A lattice is integral if all inner products $\langle x,y\rangle$ of its vectors are integers, that is, if its Gram matrix has integer entries. Among integral lattices, those with the following properties have names of their own.

• An integral lattice is even if every norm $\langle x,x\rangle$ is even. This holds exactly when the diagonal entries of $G$ are even.
• An integral lattice is unimodular if $\det G=1$. Equivalently, the lattice coincides with its dual lattice $$\Lambda^{\#}=\{\,y\in\mathbb{R}^n : \langle x,y\rangle\in\mathbb{Z}\ \text{for all}\ x\in\Lambda\,\}.$$

Even unimodular lattices exist only when $n$ is a multiple of $8$ [Ser73, Ch. V] (the reason is explained in the next section). The lattice $\mathbb{Z}^n$ is unimodular but not even, since its basis vectors have norm $1$. On the other hand, the Gram matrix of $E_8$ in the figure above has $2$ on the diagonal and determinant $1$, so $E_8$ is even unimodular.

If we identify two lattices when a rotation or reflection carries one onto the other, there are only finitely many even unimodular lattices in each dimension. In dimension $8$ the only one is $E_8$. In dimension $16$ there are two: $E_8\perp E_8$ and a lattice called $D_{16}^{+}$. Here $\perp$ denotes the orthogonal sum: the two copies of $E_8$ lie in mutually orthogonal $8$-dimensional subspaces of $\mathbb{R}^{16}$.

In dimension $24$ there are $24$ of them. They are called the Niemeier lattices, after the mathematician who classified them [Nie73]. In dimension $32$ there are more than ten million, even counting only those without vectors of norm $2$ [Kin03, Cor. 5], and the exact number is not known. The mass formula page explains how such lower bounds are obtained.

Among the $24$ Niemeier lattices, only one has no vectors of norm $2$. This is the Leech lattice $\Lambda_{24}$, published by Leech in 1967 [Lee67]. Conway proved that every even unimodular lattice of dimension $24$ without vectors of norm $2$ is the Leech lattice [Con69]. The nonzero vectors of $\Lambda_{24}$ have norm at least $4$, and there are $196\,560$ vectors of norm $4$. The figure shows a Gram matrix of $\Lambda_{24}$.

For a lattice $\Lambda\subset\mathbb{R}^n$, the theta series is the function

$$\theta_\Lambda(\tau)=\sum_{x\in\Lambda}e^{\pi i\tau\langle x,x\rangle},\qquad \tau\in\mathbb{H}=\{\tau\in\mathbb{C} : \operatorname{Im}\tau>0\}.$$

Collecting the vectors of equal norm, we can write it as $\theta_\Lambda(\tau)=\sum_{k\ge0}N_k\,e^{\pi i\tau k}$, where $N_k$ is the number of vectors of norm $k$. For an even lattice only even norms occur, so with $q=e^{2\pi i\tau}$ the theta series is a power series in $q$ whose coefficients count vectors:

$$\theta_\Lambda(\tau)=1+N_2\,q+N_4\,q^2+N_6\,q^3+\cdots .$$

For $E_8$ and the Leech lattice it begins

$$\begin{aligned} \theta_{E_8}(\tau)&=1+240\,q+2160\,q^2+6720\,q^3+\cdots,\\ \theta_{\Lambda_{24}}(\tau)&=1+0\cdot q+196\,560\,q^2+16\,773\,120\,q^3+\cdots . \end{aligned}$$

For lattices $L$ and $M$, the theta series of the orthogonal sum $L\perp M$ is the product of their theta series. A vector of $L\perp M$ is a pair $(x,y)$ with $x\in L$ and $y\in M$, and its norm is $\langle x,x\rangle+\langle y,y\rangle$. So

$$\theta_{L\perp M}(\tau)=\sum_{x\in L}\sum_{y\in M}e^{\pi i\tau(\langle x,x\rangle+\langle y,y\rangle)}=\theta_L(\tau)\,\theta_M(\tau).$$

The theta series of an even unimodular lattice obeys two transformation rules. First, since $\langle x,x\rangle$ is even, $e^{\pi i\langle x,x\rangle}=1$ for every $x$, and therefore

$$\theta_\Lambda(\tau+1)=\sum_{x\in\Lambda}e^{\pi i\tau\langle x,x\rangle}\,e^{\pi i\langle x,x\rangle}=\theta_\Lambda(\tau).$$

Second, the Poisson summation formula gives, for any lattice,

$$\theta_\Lambda\!\left(-\frac1\tau\right)=\left(\frac{\tau}{i}\right)^{n/2}\frac{1}{\sqrt{\det\Lambda}}\;\theta_{\Lambda^{\#}}(\tau),$$

with the dual lattice on the right [Ser73, Ch. VII, §6]. For a unimodular lattice $\Lambda^{\#}=\Lambda$ and $\det\Lambda=1$, so the same function appears on both sides:

$$\theta_\Lambda\!\left(-\frac1\tau\right)=\left(\frac{\tau}{i}\right)^{n/2}\theta_\Lambda(\tau).$$

This is why unimodularity is required: it makes the theta series transform into itself, up to an explicit factor.

These two rules already force $n$ to be a multiple of $8$. Combining them, the map $U(\tau)=-1/(\tau+1)$ acts on the theta series by

$$\theta_\Lambda\bigl(U(\tau)\bigr)=A(\tau)\,\theta_\Lambda(\tau),\qquad A(\tau)=\left(\frac{\tau+1}{i}\right)^{n/2}.$$

Applying $U$ three times returns every point to itself: $U(U(U(\tau)))=\tau$. Hence

$$\theta_\Lambda(\tau)=A\bigl(U(U(\tau))\bigr)\,A\bigl(U(\tau)\bigr)\,A(\tau)\;\theta_\Lambda(\tau),$$

and since $\theta_\Lambda$ is not identically zero and the factors depend continuously on $\tau$, the product of the three factors is $1$ for every $\tau$. Now take $\rho=e^{2\pi i/3}$. Then $U(\rho)=\rho$, and $(\rho+1)/i=e^{-\pi i/6}$, so each factor equals $A(\rho)=e^{-\pi i n/12}$. Therefore $e^{-\pi i n/4}=1$, which means that $n$ is a multiple of $8$.

Once $n$ is a multiple of $8$, the factor $(\tau/i)^{n/2}$ is simply $\tau^{n/2}$. A holomorphic function on $\mathbb{H}$ that is unchanged under $\tau\mapsto\tau+1$, is multiplied by $\tau^{k}$ under $\tau\mapsto-1/\tau$, and stays bounded as $\operatorname{Im}\tau\to\infty$ is called a modular form of weight $k$. So the theta series of an even unimodular lattice of dimension $n$ is a modular form of weight $n/2$ [Ser73, Ch. VII, §6]. This matters because, for a fixed weight, the modular forms of that weight form a vector space of small dimension. As the next section shows, this leaves little freedom for the numbers $N_2,N_4,\dots$.

The minimum $\min\Lambda$ of a lattice is the smallest norm of a nonzero vector. For an even lattice it is an even number. How large can the minimum of an even unimodular lattice be?

A basis of the modular forms of weight $n/2$ can be built from the two lattices we have met. The orthogonal sum of $a$ copies of $E_8$ and $b$ copies of $\Lambda_{24}$ is an even unimodular lattice of dimension $8a+24b$, and its theta series is $\theta_{E_8}^{\,a}\,\theta_{\Lambda_{24}}^{\,b}$. When $n$ is a multiple of $8$, the products with $8a+24b=n$, one for each $b=0,1,\dots,\lfloor n/24\rfloor$, form a basis of the modular forms of weight $n/2$ [Ser73, Ch. VII, §3]. Consequently, the theta series of every even unimodular lattice of dimension $n$ is a linear combination of these $\lfloor n/24\rfloor+1$ products.

Take $n=48$, where there are three products. The table lists their coefficients from $q^0$ to $q^4$, together with those of one particular linear combination of them.

The combination was chosen to make the coefficients of $q$ and $q^2$ vanish. With three products we can impose three conditions: the constant term is $1$, and the coefficients of $q$ and $q^2$ are $0$. These conditions determine the combination, and with it the coefficient of $q^3$, which is $52\,416\,000$.

Now take any even unimodular lattice of dimension $48$. If it has a nonzero vector of norm $2$ or $4$, its minimum is at most $4$. If it has none, its theta series begins $1+0\cdot q+0\cdot q^2$, so it must be this combination, and the lattice has $52\,416\,000$ vectors of norm $6$. In both cases its minimum is at most $6$.

The same reasoning works in every dimension. Write $m=\lfloor n/24\rfloor$. There are $m+1$ products, so we can require the constant term to be $1$ and the coefficients of $q,\dots,q^m$ to vanish. These conditions determine the combination, and with it all coefficients. A little more thought shows that the minimum can never exceed $2m+2$. What is needed is that the coefficient of $q^{m+1}$ in the combination so determined is always positive, and Siegel proved that it is [Sie69].

Theorem (Mallows–Odlyzko–Sloane [MOS75]). An even unimodular lattice of dimension $n$ satisfies

$$\min\Lambda\le 2\left\lfloor\frac{n}{24}\right\rfloor+2 .$$

An even unimodular lattice for which equality holds is called an extremal even unimodular lattice, or simply an extremal lattice. The bound rises by $2$ each time the dimension reaches a multiple of $24$: it is $2$ in dimensions $8$ and $16$, $4$ in dimensions $24$ to $40$, $6$ in dimensions $48$ to $64$, and so on. $E_8$ and the Leech lattice are extremal.

The theta series of an extremal lattice is the combination determined above. The number of vectors of each norm is therefore known before the lattice itself is found. The second table lists the number of vectors of minimal norm.

The coefficients of this combination are not always positive. Since a number of vectors cannot be negative, a negative coefficient rules out an extremal lattice in that dimension. Mallows, Odlyzko and Sloane showed that a negative coefficient appears once the dimension is large enough. Jenkins and Rouse then showed that one appears in every dimension greater than $163\,264$, so there are no extremal lattices in those dimensions [JR11].

In dimensions up to $163\,264$, on the other hand, whether an extremal lattice exists has to be decided dimension by dimension. The history page lists the dimensions in which examples are known.

This section is a side topic. The bound of the previous section is a statement about theta series, but extremal lattices are also of interest for geometric reasons, and we mention two of them here.

The first is sphere packing. Place a ball of radius $\tfrac12\sqrt{\min\Lambda}$ at every point of a lattice. The balls do not overlap, and there is one ball per fundamental cell, whose volume is $\sqrt{\det\Lambda}$. The proportion of space that the balls fill is therefore

$$\frac{\text{volume of a ball of radius }\tfrac12\sqrt{\min\Lambda}}{\sqrt{\det\Lambda}} .$$

For unimodular lattices the denominator is $1$. Among even unimodular lattices of a given dimension, a larger minimum therefore means a denser packing, and an extremal lattice gives the densest packing that the bound allows.

Each ball touches as many others as there are vectors of minimal norm. This number is called the kissing number of the packing; for extremal lattices it is the last column of the second table above. The figure shows the same construction in the plane, where the balls become disks: each disk touches six others, so the kissing number of this packing is $6$.

In three dimensions the largest possible kissing number is $12$. Whether a thirteenth ball can also touch the central one was debated in the seventeenth century, and the answer, no, was proved by Schütte and van der Waerden [SvdW53]. For $E_8$ the kissing number is $240$, and for the Leech lattice it is $196\,560$. These are the largest possible values in dimensions $8$ and $24$ [OS79].

An extremal lattice is not necessarily the densest packing of its dimension. The comparison above is only among even unimodular lattices of that dimension; other lattices, and packings that do not come from lattices, are not included. Finding the densest packing is hard even in three dimensions. Kepler conjectured in 1611 that no packing of equal balls is denser than the face-centred cubic arrangement (the way oranges are often stacked), which fills $\pi/\sqrt{18}\approx 74.05\%$ of space. Hales announced a proof in 1998, published in 2005 [Hal05].

In dimensions $8$ and $24$, by contrast, the extremal lattices $E_8$ and $\Lambda_{24}$ give the densest sphere packings of all. Viazovska proved this in dimension $8$ [Via17], and Cohn, Kumar, Miller, Radchenko and Viazovska in dimension $24$ [CKMRV17].

The second reason concerns how the vectors of a fixed norm are distributed: in an extremal lattice, they are spread evenly over the sphere on which they lie. This is made precise as follows. A finite set $X$ on a sphere is called a spherical $t$-design if, for every polynomial $f$ of degree at most $t$, the average of $f$ over $X$ equals its average over the whole sphere.

Venkov showed the following. In an extremal lattice of dimension $n$, for every nonzero norm, the vectors of that norm form a spherical $11$-design, $7$-design or $3$-design according as $n$ leaves remainder $0$, $8$ or $16$ on division by $24$ [BV01, Cor. 4.1].