L - Series and Transcendental Numbers

This paper is a survey of some recent work of mine done jointly with V. Kumar Murty, N. Saradha, S. Gun and P. Rath. In all of these works, the motivating question is the following. Given an automorphic representation π , we consider the L-series L(s, π) attached to π according to the Langlands formalism. We are interested in the possible transcendence of special values L(k, π) when k is a positive integer. Sometimes, we will be interested in algebraic linear combinations of such values. There are many conjectures in the literature regarding the arithmetic significance of L(k, π) for certain classes of π , but in this paper, we are concerned with its transcendence. In these investigations, a central role is played by the celebrated theorem of Alan Baker, proved in 1966. This theorem states the following. If α1, . . . , αn are non-zero algebraic numbers and β1, . . . , βn are algebraic numbers, then β1 logα1 + · · · + βn logαn is either zero or transcendental. The latter case arises if logα1, . . . , logαn are linearly independent over Q and β1, . . . , βn are not all zero. The latter case also arises if β1, . . . , βn are linearly independent over Q (see for example, [20]).