find $\rm{dim} (\Theta_k)$ :
$\mathbb H=\{x+iy|y>0;x,y\in \mathbb R\}$is the upper half plane. $z=x+iy$ and $k$ can be any positive real number
If $f:\mathbb H\to\mathbb C$ ,which satisfies:
1.$f(z)=f(z+2)$
2.$f(-1/z)=(-iz)^kf(z)$
3.$f(z)=O(y^{-k})+O(1)$
Then we called $f\in \Theta_k$ , We can easily see that $\Theta_k$ is a linear space
Maybe this is a very basic conclusion, but I know very little about the automorphic form. apparently, $\rm{dim}(\Theta_k)\ge1$ , because $\theta^{2k}(z)\in \Theta_k$ ,which $\theta(z)=\sum_{n\in\mathbb z}e^{\pi\rm {i} n^2 z}$
I'm asking this question because I'm trying to figure out what other functions have properties similar to theta functions.And could we find the generators like $G_4$,$G_6$ as in the case of modular forms?
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2$\begingroup$ Please don't delete and and repost your question instead of editing: mathoverflow.net/questions/497824/… (for one, comments get lost) $\endgroup$Kimball– Kimball2025-07-17 03:14:04 +00:00Commented Jul 17 at 3:14
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$\begingroup$ @Kimball Sorry, I'm not very familiar with the rules of this site. $\endgroup$8451543498– 84515434982025-07-17 06:58:26 +00:00Commented Jul 17 at 6:58
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1 Answer
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You are asking for the generators of the ring of modular forms for the theta subgroup of the modular group. I think that $$M_*(\Gamma_\theta)=\mathbb{C}[\theta_2^4,\theta_3^4, \theta_4^4]/(\theta_3^4-\theta_4^4-\theta_2^4)$$ in terms of the standard theta constants. The generators $$G_4,G_6$$ of the ring of modular forms for the full modular group are 8th and 12th order polynomials respectively in the theta constants which can be worked out by demanding the correct modular transformations.
