# p-adic Asai transfer

## TL;DR

This paper constructs a rigid analytic map interpolating the Asai transfer from automorphic representations of GL(2) over a real quadratic field to GL(4) over Q, connecting eigenvarieties in a p-adic setting.

## Contribution

It introduces a novel rigid analytic map between eigenvarieties that captures the Asai transfer in a p-adic analytic framework.

## Key findings

- Constructed a rigid analytic map between eigenvarieties.
- Interpolates Asai transfer at classical points.
- Bridges automorphic forms over quadratic fields and Q in p-adic families.

## Abstract

Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In this paper, we construct a rigid analytic map from the universal eigenvariety of $GL_{2}/K$ to the universal eigenvariety of $GL_{4}/Q$, which at nice classical points interpolate this Asai transfer.