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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 $G_6$?

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 $G_6$?

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 $G_6$?

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 $G_6$?