Timeline for Lattices and stable homotopy groups of spheres
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 13, 2022 at 11:00 | comment | added | pregunton | E.g. if $k=2$, the sum of cusp forms is $A_1\Delta (a E_4^3+b E_6^2) + A_2 \Delta^2$ for $a+b=1$, which can be re-expressed as $A_1 \Delta E_4^3 + A_2' \Delta^2)$ where $A_2' = A_2-1728b$. But since the theta series corresponds to an extremal lattice, one needs to succesively kill the $a_i q^i$ terms for $i=1, \ldots, k$. So an easy induction shows that $A_i = \text{integer} \times \frac{24k}{B_{12k}}$, which implies that all remaining nonconstant coefficients of the theta series are necessarily multiples of the numerator of $\frac{24k}{B_{12k}}$, that is, multiples of $|J(\pi_{24k})|$. | |
| Aug 13, 2022 at 10:58 | comment | added | pregunton | I think I can prove that $|J(\pi_{24k})|$ divides the $\operatorname{gcd}$. The theta series can be decomposed as the sum of an Eisenstein series $E_{12k}=1 + \frac{24k}{B_{12k}} \sum_{n>0}a_nq^n$, $a_n \in \mathbb{Z}$, and the cusp forms $A_1 \Delta \Pi_1, A_2 \Delta^2 \Pi_2, \ldots, A_k \Delta^k$, where $A_i \in \mathbb{Q}$ and $\Pi_i=1+\ldots$ is some linear combination of products of Eisenstein series of appropriate weights. It's easy to see that the cusp forms $\Delta^i \Pi_i$ can be simultaneously chosen such that all coefficients are integers and the first nonzero coefficient is $1$. | |
| Aug 8, 2022 at 15:51 | comment | added | user164898 | A direct connection between $\pi_*^S$ and the theta-series of lattices would be terrific and fascinating, by the way! But it seems much easier to believe that any connection between im J and theta-series of lattices simply occurs due to properties of Bernoulli numbers. | |
| Aug 8, 2022 at 15:42 | comment | added | user164898 | The order of im J in $\pi_{24k-1}^S$ is the denominator of the Bernoulli number $B_{12k}/24k$, and $B_n/n$ shows up as the only term in Eisenstein series that isn't a sums-of-divisors function. You can see from your formula how the theta-series decomposes into coefficients of the Eisenstein $E_{12}$ and the cusp form $\Delta$. The relation between the theta-series coefficients of a lattice and im J just comes about because of how the theta-series decomposes in terms of modular forms, and in particular what multiple of the Eisenstein series in the relevant weight appears in the decomposition. | |
| Aug 8, 2022 at 15:26 | history | edited | Tyrone |
Added Alg. Top. tag.
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| Aug 8, 2022 at 14:42 | history | asked | Adam P. Goucher | CC BY-SA 4.0 |