What do the numbers G_4 and G_6 of a lattice actually measure?

Question

If you have a lattice $L \subset \mathbb{C}$, you can compute the following numbers:

$ G_4(L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^4}, \quad G_6(L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^6}. $

By a 'lattice', I just mean a closed discrete additive non-trivial subgroup of $\mathbb{C}$ (so I'm allowing degenerate lattices like $\mathbb{Z}$). Anyhow, these numbers are important invariants of the lattice, because they set up a bijection

$\{ \mbox{Lattices in }\mathbb{C}\}\rightarrow \mathbb{C}^2 \setminus \{0\}$

$L \mapsto (G_4(L), G_6(L))$.

But what do these numbers actually measure about the lattice, geometrically? Some kind of combination of angles? Some area? I'm confused. We have these numbers that get used over and over, but what do they actually measure?

I guess one possible answer is: consider the Riemann surface (torus) $\mathbb{C} / \Lambda$. Then $G_4$ and $G_6$ can be recovered as certain period integrals along the fundamental cycles of the elliptic curve. Is that right? Is there a more direct geometric understanding of $G_4$ and $G_6$?

Answer 1

As far as I know $G_4$ and $G_6$ don't have a direct geometric interpretation of the type you are looking for. Rather, they appear as coefficients in the algebraic equation for $\mathbb C/\Lambda.$ More precisely, the pair $(\mathbb C/\Lambda, dz)$ consisting of the complex torus $\mathbb C/\Lambda$ and the everywhere-holomorphic differential form $dz$ is isomorphic to the pair $(E,dx/y),$ where $E$ is the smooth complex projective curve cut out by the (homogeneous equation associated to) the equation $y^2 = 4 x^3 - 60 G_4(L) x - 140 G_6(L).$ (Here the letter $E$ is for "elliptic".)

As you probably know, this is more or less the content of the theory of Weierstrass's elliptic functions.

In summary: lattices in $\mathbb C$ are the same thing as elliptic curves over ${\mathbb C}$ equipped with a choice of non-zero holomorphic differential (via $\Lambda \mapsto (\mathbb C/\Lambda, dz)$, and the quantities $G_4$ and $G_6$ give an explicit formula for this correspondence, by describing the coefficients of the algebraic equation for the corresponding elliptic curve.

Answer 2

I will try to complement Emerton's answer. Let $M_{2k}$ and $S_{2k}$ denote, respectively, the vector spaces of modular and cusp forms of weight $2k$; from the definition of cusp forms, it follows that $S_{2k}$ is a subspace of codimension at most 1 in $M_{2k}$. Since $G_{2k}$ is modular but not a cusp form, we have the decomposition (for $k \ge 2$) $M_{2k} = S_{2k} \oplus \mathbb{C}G_{2k}$. In particular, $M_{2k} = \mathbb{C}G_{2k}$ for $2 \le k \le 5$.