Strongly branched coverings of closed Riemann surfaces

Let $b:{W_1} \to {W_2}$ be a $B$-sheeted covering of closed Riemann surfaces of genera ${p_1}$ and ${p_2}$ respectively. $b$ is said to be strongly branched if ${p_1} > {B^2}{p_2} + {(B - 1)^2}$. If ${M_2}$ is the function field on ${W_1}$ obtained by lifting the field from ${W_2}$ to ${W_1}$, then ${M_2}$ is said to be a emphstrongly branched subfield if the same condition holds. If ${M_1}$ admits a strongly branched subfield, then there is a unique maximal one. If ${M_2}$ is this unique one and $f$ is a function in ${M_1}$ so that \[ (B - 1)o(f) < ({p_1} - B{p_2}) + (B - 1)\] then $f \in {M_2}$, where $o(f)$ is the order of $f$. (This is a generalization of the hyperelliptic situation.) These results are applied to groups of automorphisms of ${W_1}$ to obtain another generalization of the hyperelliptic case.