the number of rational points of a curve modulo 2 Consider the language $L=\{f, q\}$ - the number of solutions of equations $f(x,y)=0$ in $\mathbb{F}_q^2$ is equal to zero modulo $2$, where $q = 2^m$. Does $L$ belong to $P$? to $NP$? ($f$ is written as $\sum a_{ij} x^i y^j. $)
This language belongs to P. See Corollary 7.4 at 
"Algorithmic theory of zeta functions
over finite fields"
by Daqing Wan. (There $n$ is the number of variables of a hypersurface, $d$ is the total degree of it, and  $q$ is the cardinality of the corresponding finite field.)