Supersingular curves via the Shimura--Taniyama method
Jeremy Booher, Rachel Pries
TL;DR
The paper classifies primes $p$ for which reductions of genus $g$ curves $X$ that are abelian covers of ${f P}^1$ branched at three points are supersingular, focusing on genera $5\le g\le 10$. It combines Ekedahl’s Ramification/Inertia framework, Shimura--Taniyama’s CM-based Newton polygon method, and Kani--Rosen decompositions, with SageMath enumerations to treat both cyclic and non-cyclic Galois groups. The authors provide exact congruence conditions on $p$ for each genus and compute natural densities $\\delta_g$ of primes yielding supersingular reductions, demonstrating a density higher than naively expected and revealing a non-transversal interaction between Torelli and supersingular loci for larger genus. These results advance explicit constructions of supersingular curves and illuminate density phenomena in the moduli of abelian varieties and Jacobians. All results are expressed with explicit $p$-moduli conditions and decompositions into lower-genus constituents, grounded in the Shimura--Taniyama framework and CM theory.
Abstract
For a curve which admits an abelian cover of the projective line branched at three points, we study when its reduction to positive characteristic is supersingular. Using the method of Shimura and Taniyama, we give a complete classification when the genus of the curve is at most 10. The natural density of the set of primes for which this construction yields a supersingular curve is larger than expected.