# Stability of Asai local factors for $GL(2)$

## TL;DR

This paper proves the stability of Asai local factors for $GL(2)$ over quadratic extensions of non-archimedean fields, using Mellin transforms and Bessel functions, under highly ramified twists.

## Contribution

It establishes the stability of local factors for $GL(2,E)$ representations, including Asai and Rankin-Selberg factors, a result not previously known.

## Key findings

- Stability of Asai local factors under ramified twists
- Expression of gamma factors as Mellin transforms with Bessel functions
- Applicability to pairs of representations

## Abstract

Let $F$ be a non-archimedean local field of characteristic not equal to $2$ and let $E/F$ be a quadratic algebra. We prove the stability of local factors attached to (complex) irreducible admissible representations of $GL(2,E)$ via the Rankin-Selberg method under highly ramified twists. This includes both the Asai as well as the Rankin-Selberg local factors attached to pairs. Our method relies on expressing the gamma factor as a Mellin transform using Bessel functions.