Algebraicity of Hilbert Modular Forms

The conjugate $\sigma(f)$ of a Hilbert modular form $f$ by an automorphism
$\sigma$ of the field of complex numbers is obtained by applying $\sigma$
to the Fourier coefficients of $f$. A classical Theorem of Shimura states
that the conjugate of a Hilbert modular form $f$ of weight $k$ again is a
Hilbert modular form of conjugate weight $\sigma(k)$. We want to describe
an approach to Shimura's Theorem, which deduces the Theorem from a
comparison of trace formulas.