Hello fellow mathematicians,
I have the following problem: Let $u_1X^3 + u_2Y^3 + u_3Z^3 - cXYZ = 0$ be a homogeneous cubic equation, where the coefficients $u_1$, $u_2$, $u_3$ and $c$ are in $\mathbb{F}_q[t]$. When looking for solutions we can assume that $X$, $Y$ and $Z$ are in $\mathbb{F}_q[t]$ and that $\operatorname{gcd}(X,Y,Z) = 1$. I am trying to find a condition on the solubility of this cubic equation. Or more specifically, the non-solubility of the equation. What we can do is the following: consider a suitable chosen prime ideal $P$ and reduce the equation modulo this prime ideal. My questions are: is there a way to determine which prime ideal to take? And, how would we determine solvability of the equation modulo $P$. We can ofcourse brute force all the possibilities as there are only finitely many of them, but that quickly gets out of hand. To illustrate the problem, consider
\begin{equation} t^2X^3 + Y^3 - 2t^2Z^3 - 2tXYZ = 0 \end{equation} over $\mathbb{F}_7[t]$. Reducing this modulo $(t+3)$ yields:
\begin{equation} 2X^3 + Y^3 - 4Z^3 + 6XYZ = 0. \end{equation} Going through all $7^3$ possibilities we see that this has no solution and hence the original equation has no solution.
Any help/reference would extremely helpful!
EDIT: I am following the paper: ELEMENTARY 3-DESCENT WITH A 3-ISOGENY by HENRI COHEN AND FABIEN PAZUKI. I try to find function-field analogues for the statements in chapter 5 of this paper.