[Math] How to one express the Dedekind eta function as a sum over the lattice

Question

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors lies along the real axis, then pick another basis vector $\tau$ in the upper half plane, then compute the usual formula
$$
\eta(\tau) = q^\frac{1}{24} \prod_{n=1}^\infty (1-q^n)
$$
where $q=e^{2 \pi i \tau}$. It would be nice if there were a more "canonical" way to compute it directly from the lattice $L$ (i.e. without this rotate-and-pick-a-basis-vector story which breaks symmetry). I'm looking for a formula similar to that of the Eisenstein series where one sums over all points of the lattice:
$$
G_n (L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^n}
$$
We have the theorem of Jacobi that the 24th power of $\eta$ computes as the discriminant of the lattice,
$$
(2\pi)^{12} \eta(\tau)^{24} = 20G_4(\mathbb{Z} + \tau \mathbb{Z})^3 -49 G_6 (\mathbb{Z} + \tau \mathbb{Z})^2,
$$
which is great, since it shows that the 24th power of $\eta$ can be defined canonically via a sum over the lattice points… but how about $\eta$ itself?

Answer 1

The functions $G_k$ and $\Delta = \eta^{24}$ can be regarded as functions on the set of lattices because they're modular of level 1. The $\eta$ function isn't modular of level 1 (its level is 24) so there's no natural way to regard it as a function on lattices -- it's a function on lattices with additional "level structure".

Similarly, in order to regularise $G_2$ to get something with good convergence, you end up having to make it depend on some level structure as well.