Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Haocheng Fan

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.

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

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.

Paper Structure

This paper contains 7 sections, 24 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Picard modular surface
  3. Automorphic bundles and dual BGG complexes
  4. Connectedness and monodromy groups of the non-basic Newton strata
  5. Rigid cohomology
  6. Hecke actions and partial Frobenius
  7. Main results

Key Result

Theorem 1.1

The local representation $\sigma_p$ is split if and only if there exists an overconvergent modular form $g\in S^{\dagger}_{2-k}(\Gamma_1(N))$ such that $\tilde{f}:=f(z)-\beta_pf(pz)=\theta^{k-1}(g)$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.2
  • Definition 2.3
  • ...and 42 more