I am working on a problem related to quadratic forms and theta functions. I have the following two part question:
a) Show that there are $A_{1}, A_{2} \in SP_{24}$ with $\operatorname{det}\left(A_{1}\right)=\operatorname{det}\left(A_{2}\right)=1$ and $c_{1}, c_{2} \in \mathbb{C}$ so that $0 \neq c_{1} \theta_{1, Q_{A_{1}}}+c_{2} \theta_{1, Q_{A_{2}}} \in \mathbb{C} \cdot \Delta$. (Hint: Recall the quadratic form $Q_{8}$ from Remark 3.0.14 of the lecture notes. Consider $\theta_{1, Q_{8}}^{3}$ and try to apply Siegel's theorems (i.e. Theorem 4.1.10 of the Lecture notes).)
(b) Show that for any $k \in \mathbb{N}$ the space $M_{4 k}\left(\mathrm{SL}_{2}(\mathbb{Z}), \vartheta_{\text {tr }}\right)$ can be spanned by theta functions $\theta_{1, Q_{A}}$ with $A \in SP_{8 k}$ and $\operatorname{det}(A)=1$.
This is my working:
(a) We have this given information: From Remark 3.0.14, we have the quadratic form $Q_8(\mathbf{x})=\frac{1}{2} A[\mathbf{x}]=\frac{1}{2} \sum_{r=1}^8 x_r^2+\frac{1}{2}\left(\sum_{r=1}^8 x_r\right)^2-$ $x_1 x_2-x_2 x_8$, which is associated with the harmonic polynomial $P_8$, and we have $\theta_{P_8, Q_8}(z)=\Delta(z)$. According to Theorem 4.1.10 (Siegel), the Eisenstein series $E_{Q_A}(z)$ is expressed in terms of the theta series of the generators of the quadratic form $Q_A$, i.e. $\theta_{g e n\left(Q_A\right)}(z)=m\left(Q_A\right)^{-1} \sum_i o\left(Q_i\right)^{-1} \theta_{1, Q_i}(z)$.
Solution: The task is to find two quadratic forms $Q_{A_1}$ and $Q_{A_2}$ such that $c_1 \theta_{1, Q_{A_1}}+c_2 \theta_{1, Q_{A_2}}$ is a non-zero multiple of $\Delta(z)$, i.e., $c_1 \theta_{1, Q_{A_1}}+c_2 \theta_{1, Q_{A_2}} \in \mathbb{C} \cdot \Delta(z)$.
To solve this, let's consider quadratic forms $Q_{A_1}$ and $Q_{A_2}$ whose generators are the same as that of $Q_8$, i.e., $\operatorname{gen}\left(Q_{A_1}\right)=\operatorname{gen}\left(Q_{A_2}\right)=P_8$. $SP_{24}$ is the space of symmetric matrices of order 24 with integer entries, positive definiteness, and determinant equal to 1 . These matrices induce quadratic forms of rank 24 . Therefore, we will consider $A_1$ and $A_2$ to be symmetric matrices of order 24 such that the quadratic forms they induce, $Q_{A_1}$ and $Q_{A_2}$, have $P_8$ as a generator, and these matrices have determinant 1 . Thus, we have $$ c_1 \theta_{1, Q_{A_1}}+c_2 \theta_{1, Q_{A_2}}=\left(c_1+c_2\right) \theta_{P_8, Q_8}=\left(c_1+c_2\right) \Delta(z) \text {. } $$ This is a non-zero multiple of $\Delta(z)$ if at least one of the constants $c_1$ and $c_2$ is non-zero, which fulfills the requirement of the problem. In conclusion, we have shown that there exist $A_1, A_2 \in SP_{24}$ with $\operatorname{det}\left(A_1\right)=\operatorname{det}\left(A_2\right)=1$ and $c_1, c_2 \in \mathbb{C}$ such that $0 \neq c_1 \theta_{1, Q_{A_1}}+c_2 \theta_{1, Q_{A_2}} \in \mathbb{C} \cdot \Delta$ as required.
(b) To approach this problem, we begin by noting that theta functions $\theta_{1, Q_A}$ are modular forms of weight $k$ for any positive definite quadratic form $Q_A$ of rank $k$, under the transformation group $\mathrm{SL}_2(\mathbb{Z})$. This is a basic property of theta functions and their relationship with modular forms.
In this context, $M_{4 k}\left(\mathrm{SL}_2(\mathbb{Z}), \vartheta_{\mathrm{tr}}\right)$ refers to the space of modular forms of weight $4 k$ under the transformation group $\mathrm{SL}_2(\mathbb{Z})$, equipped with the theta multiplier system $\vartheta_{\mathrm{tr}}$.
Now, $SP_{8 k}$ is the space of symmetric matrices of order $8 k$ with integer entries, positive definiteness, and determinant equal to 1 . These matrices induce quadratic forms of rank $8 k$.
For every $A \in SP_{8 k}$ with $\operatorname{det}(A)=1$, the associated quadratic form $Q_A$ will have rank $8 k$. Thus, the theta function $\theta_{1, Q_A}$ will be a modular form of weight $4 k$ under the transformation group $\mathrm{SL}_2(\mathbb{Z})$. Hence, the space $M_{4 k}\left(\mathrm{SL}_2(\mathbb{Z}), \vartheta_{\mathrm{tr}}\right)$ can indeed be spanned by such theta functions, because for each matrix $A$ in $SP_{8 k}$, the associated theta function $\theta_{1, Q_A}$ provides an element of $M_{4 k}\left(\mathrm{SL}_2(\mathbb{Z}), \vartheta_{\mathrm{tr}}\right)$. By varying over all such matrices $A$ in $SP_{8 k}$, we can span the entire space of modular forms of weight $4 k$, i.e., $M_{4 k}\left(\mathrm{SL}_2(\mathbb{Z}), \vartheta_{\mathrm{tr}}\right)$
In conclusion, for any natural number $k, M_{4 k}\left(\mathrm{SL}_2(\mathbb{Z}), \vartheta_{\mathrm{tr}}\right)$ can be spanned by theta functions $\theta_{1, Q_A}$ with $A \in SP_{8 k}$ and $\operatorname{det}(A)=1$.
I would love to have your valued suggestions/edits!