Preimages for the Shimura map on Hilbert modular forms

TL;DR

This paper introduces a method to explicitly construct preimages for the Shimura correspondence on Hilbert modular forms of odd, square-free level, extending ideas from the rational case.

Contribution

It provides a new explicit construction method for preimages in the Hilbert modular forms setting, building on prior rational case techniques.

Findings

Constructed explicit preimages for the Shimura map

Method applicable to odd, square-free levels

Fourier coefficients of preimages can be computed explicitly

Abstract

In this article we give a method to construct preimages for the Shimura correspondence on Hilbert modular forms of odd and square-free level. The method relies in the ideas presented for the rational case by Pacetti and Tornar\'ia, and is such that the Fourier coefficients of the preimages constructed can be computed explicitly.