Conjugation of Hilbert modular forms and trace formula

TL;DR

This paper provides a new proof of Shimura's theorem showing that conjugation by automorphisms preserves the Hilbert modular form structure, using a comparison of trace formulas in a representation theoretic framework.

Contribution

It introduces a novel, representation theoretic approach to prove Shimura's theorem on the conjugation of Hilbert modular forms, simplifying the existing proof.

Findings

Conjugation by automorphisms preserves the level and weight of Hilbert modular forms.

A new proof of Shimura's theorem using trace formula comparison.

The approach clarifies the structure of conjugate forms in the automorphic setting.

Abstract

We describe (in a representation theoretic setting) a simple comparison of trace formulas, which implies that the conjugate of a Hilbert modular form $f$ by an automorphism of ${\Bbb C}$ again is a Hilbert modular form of the same level and conjugate weight as $f$. This is a Theorem of Shimura for which we obtain a new proof (cf. Theorem 3.3 and Corollary 3.4