Let $H$ be a hyperellipitic curve of genus $g$ defined over $\mathbb{F}_q$. The Frobenius endomorphism operates on the divisor class group of $H$ and satisfies a characteristic polynomial $P\in\mathbb{Z}[T]$ of degree $2g$. This polynomial $P$ fulfils the equation $$P(T) = T^{2g} L(1/T),$$ i.e. $P$ is the reciprocal polynomial of $L$, where $L(T)$ denotes the numerator of the zeta-function of $H$ over $\mathbb{F}_q$.

Where do I find that statement (for citation)?

FYI: The statement can be found (without a proof) for example, as preprint of T. Lange, page 23, Theorem 2.35. In the case of elliptic curves ($g=1$), it can be found in the book of Koblitz "A course in number theory and cryptography" or Silverman "The Arithmetic of Elliptic Curves".