Mass formula for non-ordinary curves in one dimensional families

TL;DR

This paper develops a mass formula for counting non-ordinary curves in one-dimensional families of cyclic covers of the projective line over positive characteristic fields, generalizing classical results and applying to new cases.

Contribution

It introduces two new equations for the mass formula using intersection theory and $a$-numbers, extending classical formulas to broader families of curves.

Findings

Derived mass formulas for various families of curves.

Generalized the Eichler--Deuring mass formula.

Applied results to hyperelliptic and cyclic cover families.

Abstract

This paper is about one dimensional families of cyclic covers of the projective line in positive characteristic. For each such family, we study the mass formula for the number of non-ordinary curves in the family. We prove two equations for the mass formula: the first relies on tautological intersection theory; and the second relies on the $a$-numbers of non-ordinary curves in the family. Our results generalize the Eichler--Deuring mass formula for supersingular elliptic curves; they also generalize some theorems of Ibukiyama, Katsura, and Oort about supersingular curves of genus $2$ that have an automorphism of order $3$ or order $4$. We determine the mass formula in many new cases, including linearized families of hyperelliptic curves of every genus and all families of cyclic covers of the projective line branched at four points. keywords: curve, hyperelliptic curve, cyclic cover, Jacobian, mass formula, cycle class, tautological ring, Hodge bundle, intersection theory, Frobenius, non-ordinary, $p$-rank, $a$-number.