Petersson-product of the cusp part of the theta series

Question

Who can help me solving this problem: $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the thetaseries $$\theta_Q(z):=\sum_{m\in\mathbb{Z}^{2k}}q^{Q(m)}\ (q=e^{2\pi i z})$$ is a modular form of weight $k$ on the congruence group $\Gamma_0(N)$ ( for some integer $N$ ) with some character $\chi$ $\bmod$ $N$, i.e. $$\theta_Q \left(\frac{az+b}{cz+d}\right)=\chi(d)(cz+d)^k \theta_Q(z)$$ for all $\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\in \Gamma_0(N)$. Let us denote the space of modular forms of weight $k$ on $\Gamma_0(N)$ with character $\chi$ by M$_k(\Gamma_0(N),\chi)$. Let S$_k(\Gamma_0(N),\chi)$ be the subspace of cusp forms and $\mathcal{E}_k(\Gamma_0(N),\chi)$ be the eisenstein subspace. We know $$\text{M}_k(\Gamma_0(N),\chi)=\mathcal{E}_k(\Gamma_0(N),\chi)\oplus\text{S}_k(\Gamma_0(N),\chi)$$ and this decomposition is orthogonal under the Petersson inner product. Let $S=S(z)\in S_k(\Gamma_0(N),\chi)$ be the Cusp part of $\theta_Q$. My Question is: Is there any explicit bound $b=b(N,k,\chi)$ for $\langle S,S\rangle$, where $\langle \dot,\dot \rangle$ is the Petersson-Skalarproduct.

Answer 1

I don't know that anyone has worked out a completely explicit and completely general bound on the Petersson inner product of the cusp form part of a theta series. There are a number of ways to approach this including the definition, Poincare series, and the approximate functional equation for Rankin-Selberg $L$-functions. Here are some references that may be useful:

$\bullet$ Schulze-Pillot has a paper "On explicit versions of Tartakovski's theorem" that gives an explicit bound in the $2k = 4$ case.

$\bullet$ Duke has a paper that gives some bounds depending only on the discriminant in the ternary case ("On ternary quadratic forms", Journal of Number Theory, 2005).

$\bullet$ Schulze-Pillot's paper was inspired in part by an older paper of Fomenko ("Estimates for scalar squares of cusp forms, and arithmetic applications", Journal of Soviet Mathematics, 1991). You might also take a look at my own paper "Quadratic forms representing all odd positive integers", which will appear next month in the American Journal of Math. In this I study the $2k = 4$ case for quadratic forms of fundamental discriminant, but I stop short of giving a completely explicit bound, even in this case.

$\bullet$ There are some other papers that use analytic methods to study quadratic forms, including papers of G.L. Watson from the 1960s, and the recent paper of Browning and Dietmann ("On the representation of integers by quadratic forms", Proceedings of the London Math Society, 2008).