The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $\Lambda \subset \mathbb{C}$. 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 $\Lambda$ (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: $$ G_n (\Lambda) = \sum_{\omega \in \Lambda, \omega \neq 0} \frac{1}{\omega^n} $$ 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?

(Sorry if this is a silly question. This is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006. Ultimately I would like a parametrization of lattices where one can "see" the Hopf fibration $S^3 \rightarrow S^2$. The usual parameterization $\Lambda \mapsto (G_4(\Lambda), G_6(\Lambda))$ doesn't quite work because if you rotate the lattice via $\Lambda \mapsto e^{i \theta}\Lambda$ then $G_4$ and $G_6$ transform as the fourth and sixth multiple of $\theta$ respectively... so you can't neatly "see" the $S^1$ action on the space of lattices (including degenerate ones) $S^3$. The Dedekind eta function might provide a more useful parametrization of lattices in this regard.)