On subfields of the second generalization of the GK maximal function field

The second generalized GK function fields $K_n$ are a recently found family of maximal function fields over the finite field with $q^{2n}$ elements, where $q$ is a prime power and $n \ge 1$ an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of $K_n$. In case $\gcd(q+1,n)=1$ and either $q$ is even or $q \equiv 1 \pmod{4},$ we find a complete list of Galois subfields of $K_n.$ Our construction adds several previously unknown genera to the genus spectrum of maximal curves.