Geometric explanation for coincidence in lengths of 16-dimensional even unimodular lattices?

Question

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.

Answer 1

Outline

Allowing for rotations and reflections of cosets, it's straightforward to see that the cosets of the respective lattices are congruent upon breaking up the lattices into 4D chunks, primarily thanks to the existence of 4D rotations taking $D_4^+$ to $\mathbb{Z}^4$. The resulting chunks can be expressed in terms of $D_4$ and its translate. The decompositions of $D_{16}^+$ and $D_8^+\oplus D_8^+$ are easily computed and identical, establishing the desired result.

Details

For notational convenience, I set up a semiring in which two lattices are equivalent iff they can be split along cosets which can be rotated/reflected into each other. Then I show constructively that $D_{16}^{+}\sim D_{8}^{+}\oplus D_{8}^{+}$ in this semiring. By fixing certain choices and tracing through the steps, it is straightforward (though rather tedious) to construct an explicit length-preserving bijection between the two lattices.

The semiring consists of formal $\mathbb{N}$-linear combinations of lattice cosets under the equivalence relation $\sim$ stated above. Multiplication is orthogonal direct sum, which is clearly commutative, and is graded by dimension. Substituting the cosets with their theta functions induces a semiring homomorphism to formal power series with nonnegative integer coefficients and exponents in $\mathbb{R}_{\geq0}$.

(Unfortunately there is some notational ambiguity between scale factors and coset multiplicities. To resolve this, I give precedence to multiplicities, so that $6\mathbb{Z}$ denotes the integer lattice with multiplicity $6$, and not the integer multiples of $6$.)

Denote $\tilde{D}_{n}:=\mathbb{Z}^{n}\backslash D_{n}=\left\{ x\in\mathbb{Z}^{n}\bigm|c_{n}\cdot x\equiv1\pmod{2}\right\}$. Some straightforward relations in this semiring are:

\begin{align} D_{m+n} & \sim D_{m}D_{n}+\tilde{D}_{m}\tilde{D}_{n},\\ \tilde{D}_{m+n} & \sim\tilde{D}_{m}D_{n}+D_{m}\tilde{D}_{n},\\ D_{n}+\tfrac{1}{2}c_{n} & \sim\tilde{D}_{n}+\tfrac{1}{2}c_{n},\\ D_{m+n}+\tfrac{1}{2}c_{m+n} & \sim2(D_{m}+\tfrac{1}{2}c_{m})(D_{n}+\tfrac{1}{2}c_{n}),\\ D_{4}+\tfrac{1}{2}c_{4} & \sim\tilde{D}_{4}. \end{align}

The first and second relations are parity counting. For the third, reflecting in any coordinate is equivalent to shifting by the unit vector of that coordinate. The fourth relation is parity counting plus an application of the third relation. Finally, the fifth relation uses the same rotations taking $D_4^+$ to $\mathbb{Z}^4$.

Exploiting the above relations, \begin{align*} D_{4m}+\tfrac{1}{2}c_{4m} & \sim2^{m-1}\tilde{D}_{4}^{m},\\ D_{8} & \sim D_{4}^{2}+\tilde{D}_{4}^{2},\\ D_{16} & \sim D_{4}^{4}+6D_{4}^{2}\tilde{D}_{4}^{2}+\tilde{D}_{4}^{4}. \end{align*}

Thus

$$ D_{16}^{+}\sim D_{16}+\left(D_{16}+\tfrac{1}{2}c_{16}\right)\sim D_{4}^{4}+6D_{4}^{2}\tilde{D}_{4}^{2}+9\tilde{D}_{4}^{4}\sim\left(D_{4}^{2}+3\tilde{D}_{4}^{2}\right)^{2}\sim(D_{8}^{+})^{2}. $$

I'd be very interested to know whether or not the theta homomorphism is injective (i.e. if all coincidences between theta functions of lattice cosets are explainable by rotations and reflections).