Bounding exponential sum of the form $\sum_{\mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n } \chi_1(x_1)\cdots \chi_n (x_n) e(a F(\mathbf{x})/q)$

I have encountered the following exponential sum and I would like to obtain a non-trivial upper bound for it. I am not quite sure where to start, and I would greatly appreciate any suggestions on how I can go about to obtain an upper bound or where to look.

Let $q \in \mathbb{N}$, and $\chi_i$ be a Dirichlet character modulo $q$ (not necessarily primitive), and $F(x_1, \ldots, x_n)$ is a homogeneous polynomial of degree $d$ with integer coefficients which defines a smooth hypersurface over $\mathbb{P}^{n-1}$.

The exponential sum in question is $$\sum_{\mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n } \chi_1(x_1)\cdots \chi_n(x_n) \ e \left(F(\mathbf{x}) \frac{a}{q} \right),$$ where $(a,q)=1$.

I would greatly appreciate any suggestions or comments. Thank you very much for your time.

• By Chinese remainder theorem you probably can reduce to the case where q is a prime power. Then using a twisted version of Lefschetz trace formula and Weil conjectures, the problem is reduced to bounding dimensions of some cohomology groups and in particular if you care about $q$ aspect, you should be able to get roughly square root cancellation. The keyword here is "trace functions", and you probably want to look at recent works of P. Michel, E.Kowalski et al. See for example, Kowalski's Pisa survey: people.math.ethz.ch/~kowalski/trace-functions-pisa.pdf – user31415 Aug 4 '17 at 18:18
• @user31415 Thank you very much for this. As I am not too familiar with Lefschetz trace formula and Weil conjectures, I was wondering after CRT does the argument still work when $q = p^t$ with $t>1$ (ie when it's not over a field)? – Johnny T. Aug 8 '17 at 1:33