Irreducibility of Newton strata in Picard modular surfaces and split local Galois representations

TL;DR

This paper establishes a link between the existence of companion forms for Picard modular forms and the splitting behavior of associated local Galois representations, using monodromy and Newton stratification analysis.

Contribution

It demonstrates the equivalence between companion forms and Galois representation splitting via monodromy and Newton stratum irreducibility in Picard modular surfaces.

Findings

Companion forms correspond to split local Galois representations.

Irreducibility of the non-ordinary Newton stratum is established.

Monodromy group computations support the main results.

Abstract

We show that for a Picard modular form, the existence of companion forms is equivalent to the splitting properties of the associated local Galois representation. This result is obtained by using the computation of the monodromy group and the irreducibility for the closure of the non-ordinary Newton stratum in the special fiber of the Picard modular surface at a split prime.