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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

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

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

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 $G_6$?

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"Then G4 and G6 can be recovered as certain period integrals along the fundamental cycles of the elliptic curve. Is that right?". I don't think so. G_4 and G_6 are the numbers showing up in the polynomial equation relating the Weierstrass P-function to its derivative. See any book on elliptic curves over C, or Serre's "Course in arithmetic" (last chapter) for discussions on this sort of thing. –  Kevin Buzzard Feb 16 '10 at 15:53
    
Note that a major obstruction to a purely geometric description of these numbers is that they aren't invariant under rotation, which rules out any interpretation in terms of angles, areas, and any other geometric invariants of a lattice which are invariant under rotation. –  Qiaochu Yuan Oct 28 at 19:25

2 Answers 2

up vote 7 down vote accepted

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.

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Thanks, this helps, especially the part about the differential being included in the data, i.e. Lambda |-> (projective curve, differential). I still feel a bit short-changed though. It would be nice if there was a lattice-theoretic description of the numbers G_4(L) and G_6(L), something geometric that you could explain to someone who just knew about R^2 and had never heard of complex numbers before, let alone projective curves. –  Bruce Bartlett Feb 16 '10 at 19:31

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 $2 \le k \le 5$.

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