On some local properties of sequences of big Galois representations

TL;DR

This paper investigates the local properties of convergent sequences of big Galois representations, showing that ramification occurs at a density zero set of places and establishing a strong multiplicity one theorem.

Contribution

It proves that ramification in such sequences is sparse and extends existing results, including a strong multiplicity one theorem, to these sequences.

Findings

Ramification occurs at a density zero set of places.

Extension of Das--Rajan's result to convergent sequences.

Establishment of a strong multiplicity one theorem for big Galois representations.

Abstract

In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field $F$ with coefficients in a domain finite over a power series ring over a $p$-adic integer ring, the set of places of $F$ where some of the representations ramifies has density zero. Using this, we extend a result of Das--Rajan to such convergent sequences. We also establish a strong multiplicity one theorem for big Galois representations.