A support theorem for the Hitchin fibration: the case of $S{L}_n$

We prove that the direct image complex for the $D$-twisted $S{L}_n$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $G{L}_n$ is due to P.-H. Chaudouard and G. Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\delta$-regularity results for some auxiliary weak abelian fibrations.