The purpose of this semi-expository article is to give another proof of a
classical theorem of Shimura on the critical values of the standard L-function
attached to a Hilbert modular form. Our proof is along the lines of previous
work of Harder and Hida (independently). What is different is an organizational
principle based on the period relations proved by Raghuram and Shahidi for
periods attached to regular algebraic cuspidal automorphic representations. The
point of view taken in this article is that one need only prove an algebraicity
theorem for the most interesting L-value, namely, the central critical value of
the L-function of a sufficiently general type of a cuspidal automorphic
representation. The period relations mentioned above then gives us a result for
all critical values. To transcribe such a result into a more classical context
we also discuss the arithmetic properties of the dictionary between holomorphic
Hilbert modular forms and automorphic representations of GL(2) over a totally
real number field F.Comment: To appear in the Journal of the Ramanujan Mathematical Societ
