An explicit formula for the generic number of dormant indigenous bundles

TL;DR

This paper proves Joshi's conjecture by deriving an explicit formula for the degree of the moduli stack of dormant indigenous bundles over the moduli stack of curves, advancing understanding in p-adic Teichmuller theory.

Contribution

It provides the first explicit formula for the degree of the moduli stack of dormant indigenous bundles, confirming Joshi's conjecture.

Findings

Explicit formula for the degree of the moduli stack

Verification of Joshi's conjecture

Advancement in p-adic Teichmuller theory

Abstract

A dormant indigenous bundle is an integrable projective line bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in the $p$-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 of Joshi.