On the smallest non-trivial quotients of mapping class groups

We prove that the smallest non-trivial quotient of the mapping class group of
a connected orientable surface of genus at least 3 without punctures is
$\mathrm{Sp}_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the
process, we generalise Korkmaz's results on $\mathbb{C}$-linear representations
of mapping class groups to projective representations over any field.