Period relations and critical values of L-functions

Abstract

I.1. In this paper we present a general conjecture concerning the arithmetic of critical values of the L-functions of algebraic automorphic forms. While individual critical values seem almost always transcendental, the evidence of Shimura (c.f., esp. [Sh1], [Sh2]) shows that interesting relations between values at dierent critical integers, and between values of L-functions related by \twisting, do exist. Furthermore, a general recipe due to Deligne ([D]) enables one to predict many such relations. This recipe further allows one to derive reciprocity laws which certain conjectually algebraic numbers, formed essentially as ratios of critical values, ought to obey. To give an example, let K be a quadratic imaginary extension of Q,