TITLE: Jacobi sums on tori
QUESTION [6 upvotes]: The Jacobi sum of $n$ multiplicative character $\chi_1,\dots,\chi_n$ on a finite field
$\mathbb  F_q$ is defined as 
$$J(\chi_1,\dots,\chi_n) = \sum_{x_1,\dots,x_n \in \mathbb F_q, x_1+\dots+x_n=1} \chi_1(x_1) \chi_2(x_2) \dots \chi_n(x_n).$$ One also considers the variant:
$$J_0 (\chi_1,\dots,\chi_n) = \sum_{x_1,\dots,x_n \in \mathbb F_q, x_1+\dots+x_n=0} \chi_1(x_1) \chi_2(x_2) \dots \chi_n(x_n)$$
It is well-known (see e.g. Ireland and Rosen) that one can compute the complex modulus
of $J(\chi_1,\dots,\chi_n)$ and $J_0(\chi_1,\dots,\chi_n)$ (when the $\chi_i$ are "general" in some precise sense, $|J(\chi_1,\dots,\chi_n)|=q^{n-3/2}$, and the other cases are not hard to determine as well).
The definition of the Jacobi sum can be rewritten in a more compact way using the maximal diagonal torus $T$ of $Gl_n$: for $\chi=(\chi_1,\dots,\chi_n)$ a character of the maximal torus,
$$J(\chi) = \sum_{x \in T(\mathbb F_q), \ tr\  x = 1} \chi(x)$$
$$J_0(\chi)=  \sum_{x \in T(\mathbb F_q), \ tr\  x = 0} \chi(x)$$
Now it is clear that the definition above of $J(\chi)$ makes sense when $T$ is replaced by any
subtorus of $Gl_n$ defined over $\mathbb F_q$, not necessarily maximal or split. My question is

Is it possible to determine, or at least estimate, $J(\chi)$ for general character of
  $T$ when $T$ is a general torus of $Gl_n$ defined over $\mathbb F_q$? 

In  particular, by how much (if anything) is it possible to improve on the trivial bound
$|J(\chi)| \leq \sum_{x \in T(\mathbb F_q), \ tr\  x = 1} 1$?
If the general question is too hard, here is one case I am especially interested: $T$ is the torus of $Gl_4$ of diagonal matrices $(x,y,y^{-1},x^{-1})$. So in this case, $T$ is still 
a split torus, but is not maximal (actually it it is the maximal torus of the symplectic group $Sp_4$ seen as a torus of $Gl_4$ through the natural inclusion). So in this case, in down-to-earth terms, $J_0(\chi)=\sum_{x,y \in \mathbb F_q, x + y + x^{-1} + y^{-1} =0} \chi_1(x) \chi_2(y)$.

REPLY [3 votes]: For general tori this problem gets really weird. For instance, given any polynomial equation $f(x_1,..,x_n)$ defined over $\mathbb F_p$, I can encode the point-counting problem $|\{x_1,\dots,x_n  \in \mathbb F_q | x_1,\dots x_n \neq 0, f(x_1,\dots,x_n)=0 \}|$ as one of these. This point-counting problem is usually at least as hard as counting solutions to $f$ which is a tough problem that we can usually only approximate the answer to.
Indeed, just consider a diagonal matrix where each monomial in $x_1,\dots,x_n$ occurs with multiplicity congruent modulo $p$ to its coefficient in $f$. As $x_1$ and $x_n$ vary, this defines a torus. Choose a trivial character of this torus, and you recover the point-counting problem.
Or choose a nontrivial, generic one - it shouldn't make the problem any easier.
Of course, in this specific case, Felipe Voloch has the solution.