  In the first part of the article, we consider the conjecture of K. Buzzard
and T. Gee proposing that every C-algebraic automorphic representation is
C-arithmetic, and we show that it can be reduced to the the analogous statement
for cuspidal C-algebraic automorphic representations. We also show that if
every cuspidal C-algebraic automorphic representations is C-arithmetic, then
every L-algebraic automorphic representation is L-arithmetic. In the second
part, we restrict to the simpler setting of anisotropic groups. We prove a
criterion which shows that, under certain conditions, the trace formula can be
used to establish that an automorphic representation is C-arithmetic. This
criterion does not require the existence of a geometric model for the
non-Archimedean part of the representation: it only depends on the possibility
of isolating a finite family of automorphic representations containing the
given one, and on a certain arithmetic condition on the Archimedean orbital
integrals. As an application, we give a new proof that regular discrete series
are C-arithmetic.
