Purity for families of Galois representations

TL;DR

This paper introduces a notion of purity for big Galois representations and pseudorepresentations, showing how it influences the structure and variation of automorphic Galois families and improves aspects of the local Langlands correspondence.

Contribution

It formulates a new concept of purity for p-adic Galois families and demonstrates its applications in understanding automorphic families and refining the local Langlands correspondence.

Findings

Purity ensures monodromy powers remain maximal under specializations.

Purity clarifies the variation of local Euler factors and automorphic types.

Improves the local Langlands correspondence for GL_n in families.

Abstract

We formulate a notion of purity for $p$-adic big Galois representations and pseudorepresentations of Weil groups of $\ell$-adic number fields for $\ell\neq p$. This is obtained by showing that all powers of the monodromy of any big Galois representation stay "as large as possible" under pure specializations. The role of purity for families in the study of the variation of local Euler factors, local automorphic types along irreducible components, the intersection points of irreducible components of $p$-adic families of automorphic Galois representations is illustrated using the examples of Hida families and eigenvarieties. Moreover, using purity for families, we improve a part of the local Langlands correspondence for $\mathrm{GL}_n$ in families formulated by Emerton and Helm.