The space of 2D lattices in $ \Bbb R^2$ can be represented with the two Eisenstein series $ G_4$ and $ G_6$ . Each lattice uniquely maps to a point in $ \Bbb C^2$ using these two invariants, and the points $ (1,0)$ and $ (0,1)$ map to the unit square and triangular lattices.

This representation requires us to treat $ \Bbb R^2$ as being isomorphic to $ \Bbb C$ , so that we can evaluate, for some lattice $ \Lambda(\omega_1, \omega_2)$ generated by $ \omega_1$ and $ \omega_2$

$ $ G_k(\Lambda) = \Bbb \sum_{0 \neq \omega \in \Lambda(\omega_1, \omega_2)} \frac{1}{\omega^k} $ $

where each element $ \omega \in \Lambda$ is then treated as having the algebra structure from $ \Bbb C$ .

The above is often treated in the setting of modular forms, where it is assumed that everything is in $ \Bbb C$ from the start, and that one of the lattice vectors is $ 1$ . However, it is also given fairly often as an equivalent treatment in terms of real lattices.

So the basic question is, how do you develop a similar theory for $ \Bbb R^n$ ? Is there a similar algebra structure one can place on $ \Bbb R^n$ taking the place of $ \Bbb C$ ?

**My big-picture questions**:

- What is a direct formula to represent the space of lattices in $ \Bbb R^n$ , so that every lattice corresponds to a unique point?
- If the above is too complicated, is there an easier representation if we allow for things like “signed lattices,” or allow different automorphisms of the same lattice to be represented with different elements of the same space, etc? (As an example, in $ \Bbb R^2$ , suppose we differentiated between two 90 degree rotations of the square lattice, or three 60 degree rotations of the triangular lattice, while equivocating between any 180 degree rotation of the same lattice.)

Assuming some generalization of the Eisenstein series is relevant, then the follow-up questions would be

- How do you generalize the Eisenstein series to lattices in $ \Bbb R^3$ , or $ \Bbb R^n$ in general?
- Assuming you manage to do that, the space of lattices for $ \Bbb R^n$ should have dimension $ n^2$ . Which Eisenstein invariants are needed to uniquely represent this space of lattices?
- The Eisenstein/Weierstrass invariants can be viewed as coefficients from the Laurent expansion of the Weierstrass elliptic function. Is the above simplified from looking at the Laurent expansions of higher-dimensional elliptic functions?

I have been pointed in a few interesting directions – Eisenstein Series on Real, Complex, and Quaternionic Half-Spaces, Eisenstein Series in Complexified Clifford Analysis, Generalization of Weierstrass Elliptic functions to $ \Bbb R^n$ , On a New Type of Eisenstein Series in Clifford Analysis, Clifford analysis with generalized elliptic and quasi elliptic functions. People have suggested the theory of automorphic forms and the Langlands’ Eisenstein series.

However, I have somehow been unable to unpack all this to get a simple definition of the space of n-dimensional real lattices. I would assume that some generalized Eisenstein series would be necessary, but I don’t know how to build them for arbitrary $ \Bbb R^n$ . Some of the suggestions for replacing $ \Bbb C$ with an arbitrary Clifford algebra can sometimes equivocate between different lattices. For instance, even in the simplest case of $ \Bbb R^4$ , where we can treat the entire thing as the quaternion algebra, we tend to have that the lattices $ \Lambda(1,2\mathbf e_1,\mathbf e_2,\mathbf e_3)$ and $ \Lambda(1,\mathbf e_1,2\mathbf e_2,\mathbf e_3)$ produce the same Eisenstein series, if we naively use the formula $ \sum_{0 \neq \omega \in \Lambda} \frac{1}{\omega^k}$ .

This was originally posted and bountied at MSE: https://math.stackexchange.com/questions/3260577/using-eisenstein-series-to-represent-the-space-of-lattices-generalization-to

I am hoping for some basic explication of how to do this that doesn’t require extremely deep knowledge of elliptic curve geometry and etc – I just want a basic representation for the space of n-d lattices.