## Question

Up to equivalence, there are two positive-definite even unimodular lattices in $16$ dimensions: $D_{8}^{+}\oplus D_{8}^{+}$ and $D_{16}^{+}$. As observed by Witt in 1941, the theory of modular forms implies the "strange and interesting" fact that these inequivalent lattices have identical theta functions. He comments that while modular forms make this fact obvious, geometrically it remains opaque, seemingly a pure coincidence.

Is there any known geometric explanation for this fact? For example, is it possible to construct a (geometrically meaningful) explicit length-preserving bijection between $D_{8}^{+}\oplus D_{8}^{+}$ and $D_{16}^{+}$?

**Reference**: Witt, Ernst. "Eine Identität zwischen Modulformen zweiten Grades." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Vol. 14. No. 1. Springer-Verlag, 1941.

## Background

### Definitions

By *lattice* I mean a free abelian group $L$ of finite rank $n$ equipped with a positive-definite inner product. Two lattices are *equivalent* if they are isomorphic as groups by an isomorphism which preserves the inner product.

The *theta function* of a lattice $L$ is the generating function $\theta_{L}(q)$ counting the number of lattice points of a given squared-length in $L$:
$$\theta_{L}(q):=\sum_{x\in L}q^{x\cdot x}=\sum_{\alpha\in\left\{ x\cdot x\mid x\in L\right\} }N_{\alpha}q^{\alpha},\quad N_{\alpha}:=\#\left\{ x\in L\bigm|x\cdot x=\alpha\right\}.$$
(Commonly there is a factor of $\tfrac{1}{2}$ in the exponent.)

The $D_{n}$ root lattice is the sublattice of $\mathbb{Z}^{n}$ consisting of all vectors whose components sum to an even number. Let $c_{n}\in\mathbb{Z}^{n}$ denote the characteristic vector $(1,1,\ldots,1)$. Then $D_{n} =\left\{ x\in\mathbb{Z}^{n}\bigm|c_{n}\cdot x\equiv0\pmod{2}\right\}$, and we define

$$ D_{n}^{+}:=D_{n}\cup\left(D_{n}+\tfrac{1}{2}c_{n}\right). $$

Note that $D_{8}^{+}$ is the usual $E_{8}$ lattice.

## Collected results

#### The lattices are even.

$D_{n}^{+}$ is a lattice when $n\equiv0\pmod{2}$. If additionally $n\equiv0\pmod{4}$, then $D_{n}^{+}$ is integral: $\left\{ x\cdot x\mid x\in L\right\} \subset\mathbb{Z}$. If moreover $n\equiv0\pmod{8}$, then $D_{n}^{+}$ is even: $\left\{ x\cdot x\mid x\in L\right\} \subset2\mathbb{Z}$.

#### The lattices are unimodular.

$D_{n}$ has index $2$ in both $\mathbb{Z}^{n}$ and $D_{n}^{+}$. Since $\det\mathbb{Z}^{n}=1$ and $\det(L')/\det(L)=[L:L']^2$, it follows that $\det D_{n}=4$ and $\det D_{n}^{+}=1$.

#### The lattices have the same theta functions.

As a consequence of Poisson summation, if $L$ is even and unimodular with dimension $n$, then $\theta_{L}$ is a modular form of level $1$ and weight $n/2$.

The unital ring of modular forms of level $1$ is graded by weight and is freely generated by the Eisenstein series

$$ \begin{align} E_{4}(q) &=1+240q^{2}+2160q^{4}+\cdots, \\ E_{6}(q) &=1-504q^{2}-16632q^{4}+\cdots. \end{align} $$

Thus the modular forms of weight $8$ are spanned by $E_{4}(q)^{2}=1+480q^{2}+61920q^{4}+\cdots$, and any positive-definite even unimodular lattice of dimension $16$ must have $E_{4}(q)^{2}$ as its theta series.

#### The lattices are inequivalent.

The 480 vectors of length $\sqrt{2}$ in $D_{16}^{+}$ generate a proper sublattice, while those in $D_{8}^{+}\oplus D_{8}^{+}$ generate the full lattice. Concretely, the 480 vectors of length $\sqrt{2}$ in $D_{16}^{+}$ are the $2^{2}\cdot\binom{16}{2}$ vectors of the form $\left\{ \pm e_{i}\pm e_{j}\mid1\leq i<j\leq16\right\}$. The vector $\tfrac{1}{2}c_{16}$ is excluded since it has length $\sqrt{4}$. The included vectors span $D_{16}\subset D_{16}^{+}$. On the other hand, the 240 vectors with length $\sqrt{2}$ in $D_{8}^{+}$ are $2^{2}\cdot\binom{8}{2}+2^{7}$, where $2^{7}$ represents all vectors of the form $\left(\pm\tfrac{1}{2},\cdots\pm\tfrac{1}{2}\right)$ with an even number of minus signs. Since $\tfrac{1}{2}c_8$ is included, the span is all of $D_8^+$.

## Motivation

In Four-Dimensional Lattices With the Same Theta Series, Conway and Sloane construct an explicit length-preserving bijection between isospectral lattices (changing the sign of the first coordinate which is divisible by 3). I'm curious if a similar construction is possible in the above case.