Conductors in $p$-adic families

TL;DR

This paper proves that pure specializations of Weil-Deligne representations in p-adic families share the same conductor, extending to collections of pure representations with lifts parametrized by pseudorepresentations.

Contribution

It establishes the invariance of conductors in p-adic families of Weil-Deligne representations and generalizes to collections with lifts over larger domains.

Findings

Pure specializations have identical conductors.

Conductors are constant in families with pseudorepresentation parametrization.

Results apply to collections of pure representations with lifts.

Abstract

Given a Weil-Deligne representation of the Weil group of an $\ell$-adic number field with coefficients in a domain $\mathscr{O}$, we show that its pure specializations have the same conductor. More generally, we prove that the conductors of a collection of pure representations are equal if they lift to Weil-Deligne representations over domains containing $\mathscr{O}$ and the traces of these lifts are parametrized by a pseudorepresentation over $\mathscr{O}$.