Deuring's mass formula of a Mumford family

TL;DR

This paper investigates the Newton polygon jumping locus of a Mumford family in characteristic p, showing it consists solely of supersingular points with a specific cardinality, using p-adic Hodge theory.

Contribution

It establishes a precise description and count of the supersingular points in the Newton polygon jumping locus for Mumford families in characteristic p.

Findings

Jumping locus contains only supersingular points.

Cardinality of the jumping locus is (p^r - 1)(g - 1).

Technique used is p-adic Hodge theory.

Abstract

We study the Newton polygon jumping locus of a Mumford family in char $p$. Our main result says that, under a mild assumption on $p$, the jumping locus consists of only supersingular points and its cardinality is equal to $(p^r-1)(g-1)$, where $r$ is the degree of the defining field of the base curve of a Mumford family in char $p$ and $g$ is the genus of the curve. The underlying technique is the $p$-adic Hodge theory.