Exponential sums over finite fields with even characteristic

Question

I am looking for an elementary evaluation (if one exists) of the exponential sum

$$ G_r(a,b) = \sum_{x \in \mathbb{F}_{2^r}} \psi(ax^2 + bx), $$

where $a,b \in \mathbb{F}_{2^r}^*$ are both units, $\psi(x) = e(Tr(x)/2)$ and $Tr : \mathbb{F}_{2^r} \to \mathbb{F}_2$ is the usual field Trace map

$$ Tr(x) = \sum_{i=0}^{r-1} x^{2^i}. $$

It should be noted that

$$ G_r(a,0) = G_r(0,a) = 0, $$

since the map $x \mapsto x^2$ permutes the elements of $\mathbb{F}_{2^r}$.

I feel that such a sum must have surely been studied before, but I am having trouble both evaluating the sum and finding references for it. Short of an explicit formula for the sum, any information (or any reference to where this sum might be studied) would be appreciated. I found no information on this sum in the usual suspects: Ireland-Rosen, Iwaniec-Kowalski and "Gauss and Jacobi Sums," by Berndt, Evans, and Williams.

Answer 1

Trace is additive, and ${\rm Tr}(u)={\rm Tr}(u^2)$ for all $u$, so $ax^2+bx$ has the same trace as $(a+b^2)x^2$. Therefore the sum is $2^r$ if $a=b^2$ and zero otherwise.

In general, for a polynomial $P(x)$ over the field of $2^r$ elements, the sum of $\psi(P(x))$ is $2^r$ less than the number of affine points on the "hyperelliptic" curve $y^2+y=P(x)$. Here $P(x) = ax^2+bx$, so (for much the same reason I gave above: polynomials $\eta^2+\eta$ can be absorbed into $y^2+y$) the curve is rational, with $2^r$ points, unless $a=b^2$ when it is the union of two disjoint lines and has $2^{r+1}$ points.