Uniqueness of the maximal function field with the third largest genus in characteristic $3$

For a prime power $q$, a function field of genus $g$ over $\mathbb{F}_{q^2}$ is said to be maximal if it attains the Hasse--Weil upper bound, i.e., if it has exactly $q^2+1+2gq$ places of degree one. It is well known that the largest possible genus for a maximal function field is $g_1 := q(q-1)/2$, and that the Hermitian function field is the unique maximal function field of this genus up to isomorphism. The second and third largest possible genera are known to be $g_2 := \lfloor(q-1)^2/4\rfloor$ and $g_3:= \lfloor (q^2-q+4)/6\rfloor$, and uniqueness is also known for $g_2$. 

In this talk, we will present a recent result establishing that there is also just one maximal function field of genus $g_3$ in the case where $q$ is a power of three. The talk is based on joint work with Peter Beelen and Maria Montanucci.