Hida family of theta lift from U(1) to definite U(2)

TL;DR

This paper constructs Hida families interpolating theta lifts of algebraic Hecke characters to a definite unitary group U(2) over a CM extension, analyzing their properties under certain non-vanishing conditions.

Contribution

It introduces a novel method to interpolate theta lifts into Hida families for unitary groups over CM fields, linking L-value non-vanishing to primitivity.

Findings

Hida families constructed for theta lifts from U(1) to U(2)

Primitivity of Hida families under specific L-value conditions

Connection between non-vanishing L-values and family properties

Abstract

Let K/F be a CM extension satisfying the ordinary assumption for an odd prime p. In this article, we construct Hida families that interpolate theta lifts of algebraic Hecke characters to a definite unitary group U(2) defined from skew-Hermitian spaces over K, and show that the Hida family is primitive when the central L-value of the branch character of the family satisfies certain non-vanishing modulo p conditions.