Test Schwartz functions at $p$ for theta lifts of Hida families

TL;DR

This paper develops a method to construct Hida families of theta lifts using specially chosen Schwartz functions at places dividing p, enhancing the understanding of p-adic families in automorphic forms.

Contribution

It introduces a novel selection of Schwartz functions with equivariance properties for p-adic theta lifts, advancing the construction of Hida families for orthogonal and unitary groups.

Findings

Successful construction of Hida families of theta lifts

Introduction of Schwartz functions with equivariance properties

Enhanced understanding of p-adic automorphic forms

Abstract

We construct Hida families of theta lifts from definite orthogonal and unitary groups. A major ingredient of the construction is the choice of test Schwartz functions at places dividing $p$. We select a special type of Schwartz functions endowed with the equivariance property for the action of $\mathbb{U}_p$-operators.