Pointless hyperelliptic curves

In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over Fq but has no rational points over Fq for various pairs (g,q). As an example of such a result, we show that if p is a prime such that is also prime then there will be pointless hyperelliptic curves over Fp of every genus g⩾q−1.