3
votes
0answers
18 views

Modular forms on the theta group

The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
2
votes
0answers
13 views

Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
3
votes
1answer
134 views

Monstrous Moonshine for $M_{24}$?

This is connected to my MO post "Monstrous Moonshine for $M_{24}$ and K3?". In page 44 of this paper, eqn(7.16) and (7.19) yield, ...
1
vote
1answer
89 views

Coefficients of powers of the theta function

Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$ Now, I shall show that the powers of $\theta$ are given by $$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$ where $S_r(n)$ ...
2
votes
0answers
144 views

A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+… = 0$

Given the Dedekind eta function $\eta(\tau)$. Define, $y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$ Prove the multi-grade identity [1], $y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + ...
4
votes
1answer
106 views

Reference request: theta functions for lattices which are not unimodular or even

In A Course in Arithmetic, Serre works out some of the theory of theta functions for even, self-dual lattices, e.g. such theta functions are modular forms of weight $n/2$, where $n$ is the dimension ...