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The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. 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 $L$ (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: $$ G_n (L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^n} $$ 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 $S^3 \rightarrow S^2$!. Let me explain.

The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}^2$ (including the degenerate ones) and $\mathbb{C}^2 - \{0\}$. By scaling the lattice $L \mapsto t L$ we can arrange that these two numbers satisfy $|z_1|^2 + |z_2|^2 = 1$; in other words we have that

$$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3. $$

Now $S^3$ carries a beautiful action of $S^1$ given by sending $(z_1, z_2) \mapsto (e^{i \theta}z_1, e^{i \theta}z_2)$, the quotient being $S^2$; this is the Hopf fibration $S^3 \rightarrow S^2$.

Since the collection of lattices up to rescaling identifies with $S^3$ in such a nice way, it is tempting to try and see this action of $S^1$ at the level of lattices. It would be nice if it were to correspond to rotation of the lattice! But alas, it does not. Firstly, it can't, because the action of $S^1$ on $S^3$ is free, while rotating a lattice might `click' it back into itself before one has rotated a full rotation. In fact we see that if we rotate the lattice via $L\mapsto e^{i\theta} L$, we find that the invariants change as $$ (G_4(L), G_6(L)) \mapsto (e^{-4i\theta}G_4(L), e^{-6i\theta}G_6(L)) $$ which is not the behaviour we are looking for.

So what does the action of $S^1$ on $S^3$ correspond to in the space of lattices up to rescaling?

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

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 $S^3 \rightarrow S^2$!. Let me explain.

The map $\Lambda \mapsto (G_4(\Lambda), G_6(\Lambda))$ gives a bijection between all lattices $\Lambda \subset \mathbb{C}^2$ (including the degenerate ones) and $\mathbb{C}^2 - \{0\}$. By scaling the lattice $\Lambda \mapsto t \Lambda$ we can arrange that these two numbers satisfy $|z_1|^2 + |z_2|^2 = 1$; in other words we have that

$$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3. $$

Now $S^3$ carries a beautiful action of $S^1$ given by sending $(z_1, z_2) \mapsto (e^{i \theta}z_1, e^{i \theta}z_2)$, the quotient being $S^2$; this is the Hopf fibration $S^3 \rightarrow S^2$.

Since the collection of lattices up to rescaling identifies with $S^3$ in such a nice way, it is tempting to try and see this action of $S^1$ at the level of lattices. It would be nice if it were to correspond to rotation of the lattice! But alas, it does not, because $G_4$ and $G_6$ don't quite scale correctly - if we rotate the lattice via $\Lambda \mapsto e^{i\theta} \Lambda$, we find that the invariants change as $$ (G_4(L), G_6(L)) \mapsto (e^{-4i\theta}G_4(L), e^{-6i\theta}G_6(L)) $$ which is not the behaviour we are looking for.

So: I'm looking for an identification $$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3 $$ in such a way that rotation of the lattice corresponds to sending $(z_1, z_2) \mapsto (e^{i\theta}z_1, e^{i\theta}z_2)$.

I had thought that the Dedekind eta function (or at least its inverse square) might help in this regard since if you rotate the lattice $L \mapsto e^{i\theta} L$ then, by Jacobi's result above, the inverse square of the $\eta$ function should rotate correctly, namely $\eta(L)^{-2} \mapsto e^{i \theta} \eta(L)^{-2}$.

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. 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 $L$ (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: $$ G_n (L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^n} $$ 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 $S^3 \rightarrow S^2$!. Let me explain.

The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}^2$ (including the degenerate ones) and $\mathbb{C}^2 - \{0\}$. By scaling the lattice $L \mapsto t L$ we can arrange that these two numbers satisfy $|z_1|^2 + |z_2|^2 = 1$; in other words we have that

$$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3. $$

Now $S^3$ carries a beautiful action of $S^1$ given by sending $(z_1, z_2) \mapsto (e^{i \theta}z_1, e^{i \theta}z_2)$, the quotient being $S^2$; this is the Hopf fibration $S^3 \rightarrow S^2$.

Since the collection of lattices up to rescaling identifies with $S^3$ in such a nice way, it is tempting to try and see this action of $S^1$ at the level of lattices. It would be nice if it were to correspond to rotation of the lattice! But alas, it does not. Firstly, it can't, because the action of $S^1$ on $S^3$ is free, while rotating a lattice might `click' it back into itself before one has rotated a full rotation. In fact we see that if we rotate the lattice via $L\mapsto e^{i\theta} L$, we find that the invariants change as $$ (G_4(L), G_6(L)) \mapsto (e^{-4i\theta}G_4(L), e^{-6i\theta}G_6(L)) $$ which is not the behaviour we are looking for.

So what does the action of $S^1$ on $S^3$ correspond to in the space of lattices up to rescaling?

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 $S^3 \rightarrow S^2$!. Let me explain.

The map $\Lambda \mapsto (G_4(\Lambda), G_6(\Lambda))$ gives a bijection between all lattices $\Lambda \subset \mathbb{C}^2$ (including the degenerate ones) and $\mathbb{C}^2 - \{0\}$. By scaling the lattice $\Lambda \mapsto t \Lambda$ we can arrange that these two numbers satisfy $|z_1|^2 + |z_2|^2 = 1$; in other words we have that

$$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3. $$

Now $S^3$ carries a beautiful action of $S^1$ given by sending $(z_1, z_2) \mapsto (e^{i \theta}z_1, e^{i \theta}z_2)$, the quotient being $S^2$; this is the Hopf fibration $S^3 \rightarrow S^2$.

Since the collection of lattices up to rescaling identifies with $S^3$ in such a nice way, it is tempting to try and see this action of $S^1$ at the level of lattices. It would be nice if it were to correspond to rotation of the lattice! But alas, it does not, because $G_4$ and $G_6$ don't quite scale correctly - if we rotate the lattice via $\Lambda \mapsto e^{i\theta} \Lambda$, we find that the invariants change as $$ (G_4(L), G_6(L)) \mapsto (e^{-4i\theta}G_4(L), e^{-6i\theta}G_6(L) $$ which is not the behaviour we are looking for.

So: I'm looking for an identification $$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3 $$ in such a way that rotation of the lattice corresponds to sending $(z_1, z_2) \mapsto (e^{i\theta}z^1, e^{i\theta}z^2)$.

I had thought that the Dedekind eta function (or at least its square) might help in this regard since if you rotate the lattice $L \mapsto e^{i\theta} L$ then, by Jacobi's result above, the square of the $eta$ function should rotate correctly, namely $\eta(L)^2 \mapsto e^{i \theta} \eta(L)^2$.

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 $S^3 \rightarrow S^2$!. Let me explain.

The map $\Lambda \mapsto (G_4(\Lambda), G_6(\Lambda))$ gives a bijection between all lattices $\Lambda \subset \mathbb{C}^2$ (including the degenerate ones) and $\mathbb{C}^2 - \{0\}$. By scaling the lattice $\Lambda \mapsto t \Lambda$ we can arrange that these two numbers satisfy $|z_1|^2 + |z_2|^2 = 1$; in other words we have that

$$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3. $$

Now $S^3$ carries a beautiful action of $S^1$ given by sending $(z_1, z_2) \mapsto (e^{i \theta}z_1, e^{i \theta}z_2)$, the quotient being $S^2$; this is the Hopf fibration $S^3 \rightarrow S^2$.

Since the collection of lattices up to rescaling identifies with $S^3$ in such a nice way, it is tempting to try and see this action of $S^1$ at the level of lattices. It would be nice if it were to correspond to rotation of the lattice! But alas, it does not, because $G_4$ and $G_6$ don't quite scale correctly - if we rotate the lattice via $\Lambda \mapsto e^{i\theta} \Lambda$, we find that the invariants change as $$ (G_4(L), G_6(L)) \mapsto (e^{-4i\theta}G_4(L), e^{-6i\theta}G_6(L)) $$ which is not the behaviour we are looking for.

So: I'm looking for an identification $$ \{\mbox{lattices in } \mathbb{C}^2 \mbox{ up to rescaling}\} \cong S^3 $$ in such a way that rotation of the lattice corresponds to sending $(z_1, z_2) \mapsto (e^{i\theta}z_1, e^{i\theta}z_2)$.

I had thought that the Dedekind eta function (or at least its inverse square) might help in this regard since if you rotate the lattice $L \mapsto e^{i\theta} L$ then, by Jacobi's result above, the inverse square of the $\eta$ function should rotate correctly, namely $\eta(L)^{-2} \mapsto e^{i \theta} \eta(L)^{-2}$.