A $p$-adic Hermitian Maass lift

TL;DR

This paper generalizes the Hermitian Maass lift to include $p$-oldforms and $p$-adic families, connecting Hermitian Jacobi forms with automorphic forms on unitary groups.

Contribution

It introduces a new Hermitian Maass space for general levels and establishes an isomorphism with special Hermitian Jacobi forms, extending the classical lift to $p$-adic families.

Findings

Defined a Hermitian Maass space for general level

Proved the space is isomorphic to Hermitian Jacobi forms

Constructed a $p$-adic family of automorphic forms

Abstract

For $K$ an imaginary quadratic field with discriminant $-D_K$ and associated quadratic Galois character $\chi_K$, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level $D_K$ and nebentypus $\chi_K$ via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group $U(2,2)$ split over $K$. We generalize this (under certain conditions on $K$ and $p$) to the case of $p$-oldforms of level $pD_K$ and character $\chi_K$. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space of special Hermitian Jacobi forms. We then show how to adapt this construction to lift a Hida family of modular forms to a $p$-adic analytic family of automorphic forms in the Maass space of level $p$.