I'm currently trying to "compute" the zeta function of some hyperelliptic curves over a finite field (of odd characteristic). Precisely, let $b\in\mathbb{F}_q^\times$ be a non zero element (I also assume that $b\neq \pm 1$), consider the hyperelliptic curves $C_d$ given by $$y^2=x^{2d}+2bx^d +1.$$ Note that $d$ is any positive integer prime to $q$ (I don't want to restrict $d$ to a bounded interval): the conditions on $b$ and $d$ ensure that $C_d$ is smooth. Now, to write down the zeta function of $C_d$, one tries to compute the number of solutions of such an equation : one sees that $C_d$ has two points at infinity (who are rational) and $$\#C_d(\mathbb{F}_q) = q+2 +\sum_{x\in\mathbb{F}_q}\mu(x^{2d}+2bx^d+1),$$ where $\mu$ is the quadratic character of $\mathbb{F}_q^\times$.
One can "change variables in the sum" setting $z=x^d$ and noting that $$ \#\{x\in\mathbb{F}_q : y=x^d \}=\sum_{\chi^d=1} \chi(y),$$ the sum being over all multiplicative characters whose $d$-th power is the trivial character. This gives $$\#C_d(\mathbb{F}_q) = q+1 +\sum_{\substack{\chi^d = 1 \\ \chi\neq 1}}\left(\sum_{z\in\mathbb{F}_q}\chi(z)\mu(z^{2}+2bz+1)\right).$$
That's where I'm stuck. The curve $C_d$ is of genus $d-1$, so one would expect each of the $d-1$ sums $$S_b(\chi) := \sum_{z\in\mathbb{F}_q}\chi(z)\mu(z^{2}+2bz+1) $$ to "split" in two parts, probably ressembling Jacobi sums, giving $2(d-1)$ parts hopefully of complex modulus $\sqrt{q}$. Note that $S_b(\chi)$ is a real number.
My intuition comes from the computation made by Weil in his paper about number of solutions of equation over finite fields : he considers instead curves like $$C'_d: y^2=x^{2d}+1,$$ (ie. put $b=0$). This curve is of genus $d-1$ and the number of points over $\mathbb{F}_q$ can be written $$\#C'_d(\mathbb{F}_q) = q+1 - \sum_{\substack{\chi^{2d} = 1 \\ \chi\neq 1,\mu}}J(\chi, \mu).$$ In this case, one has a sum of $2(d-1)$ Jacobi sums $J(\chi, \mu)$ which directly give (via the Hasse-Davenport relation) an expression for the zeta-function $$ Z(C'_d, T) = \frac{\prod_{a}(1-J_aT^{n_a})}{(1-T)(1-qT)},$$ where $a$ runs through a set of representatives of the action of $q$ on $\mathbb{Z}/d\mathbb{Z}$, and $J_a$ is a certain Jacobi sums...
That's the kind of expression I'm looking for in the case of the hyperelliptic curves mentionned above. Has anyone done this computation before ? I failed to find a reference in which there's no restriction on $d$ ... Does anyone know how to compute the sums $S_b(\chi)$ in terms of Jacobi sums ?
Thanks :)
