Supersingular curves on Picard modular surfaces modulo an inert prime

TL;DR

This paper investigates supersingular curves on Picard modular surfaces over inert primes, analyzing automorphic vector bundles in characteristic p and deriving a formula linking supersingular locus components to the surface's second Chern class.

Contribution

It provides a new formula connecting the number of supersingular locus components to the second Chern class for Picard modular surfaces at inert primes.

Findings

Derived a formula relating supersingular components to second Chern class

Analyzed automorphic vector bundles in characteristic p

Characterized supersingular curves on Picard modular surfaces

Abstract

We study the supersingular curves on Picard modular surfaces modulo a prime $p$ which is inert in the underlying quadratic imaginary field. We analyze the automorphic vector bundles in characteristic $p$, and as an application derive a formula relating the number of irreducible components in the supersingular locus to the second Chern class of the surface.