Notes on the arithmetic of Hilbert modular forms

TL;DR

This paper offers a new proof of Shimura's theorem on the critical values of the standard L-function for Hilbert modular forms, utilizing period relations and algebraicity results for automorphic representations.

Contribution

It introduces an organizational approach based on period relations to prove algebraicity of critical L-values, simplifying and unifying previous methods.

Findings

Provides a new proof of Shimura's theorem using period relations.

Establishes algebraicity of the central critical L-value for certain automorphic representations.

Discusses the arithmetic correspondence between Hilbert modular forms and automorphic representations.

Abstract

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.