Generic Newton polygons for curves of given p-rank

Original · EN

We survey results and open questions about the p-ranks and Newton polygons of Jacobians of curves in positive characteristic p. We prove some geometric results about the p-rank stratification of the moduli space of (hyperelliptic) curves. For example, if 0 ≤ f ≤ g-1, we prove that every component of the p-rank f+1 stratum of Mg contains a component of the p-rank f stratum in its closure. We prove that the p-rank f stratum of Mg is connected. For all primes p and all g ≥ 4, we demonstrate the existence of a Jacobian of a smooth curve, defined over Fₚ, whose Newton polygon has slopes {0, 1/4, 3/4, 1}. We include partial results about the generic Newton polygons of curves of given genus g and p-rank f.