Conjugation of Hilbert modular forms and trace formula

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 $\mathbb{C}$ 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