An Explicit Formula for the Generic Number of Dormant Indigenous Bundles

Abstract

A dormant indigenous bundle is an integrable ${\mathbb{P}}^1$-bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in $p$-adic Teichmuller theory developed by S. Mochizuki. Kirti Joshi proposed a conjecture concerning an explicit formula for the degree over the moduli stack of curves of the moduli stack classifying dormant indigenous bundles. In this paper, we give a proof for this conjecture.