FINITENESS OF SUPERELLIPTIC CURVES WITH CM JACOBIANS

This paper proves the Coleman conjecture for superelliptic curves: there are, up to isomorphism, at most finitely many superelliptic curves whose Jacobians are CM abelian varieties, as long as these curves are of genus at least 8. Here superelliptic curves are smooth projective curves overC admitting affine equations of the form yn D ζ.x/ with ζ a separable polynomial. The proof is reduced to the geometry of superelliptic Torelli locus T Sg in the Siegel modular variety Ag: we establish the generic exclusion from T Sg of any special subvariety of dimension > 0 in Ag for g ≥ 8, and the stability properties of Higgs bundles associated to surface fibrations play a crucial role in our study.