  We construct infinite families of closed hyperbolic surfaces that are local
maxima for the systole function on their respective moduli spaces. The systole
takes values along a linearly divergent sequence $(L_n)_{n\geq 1}$ at these
local maxima. The only surface corresponding to $L_1\approx 3.057$ is the Bolza
surface in genus $2$. For every genus $g\geq 13$, we obtain either one or two
local maxima in $\mathcal{M}_g$ whose systoles have length $L_2\approx 5.909$.
For each $n\geq 3$, there is an arithmetic sequence of genera $(g_k)_{k\geq 1}$
such that the number of local maxima of the systole function in
$\mathcal{M}_{g_k}$ at height $L_n$ grows super-exponentially in $g_k$. In
particular, level sets of the systole function can have an arbitrarily large
number of connected components. Many of the surfaces we construct have trivial
automorphism group, and are the first examples of local maxima with this
property.
