Onp-adic ArtinL-functions

In this paper we will discuss p -adic Artin L -functions. The existence of these functions is a simple consequence of a theorem of Deligne and Ribet [4]. One can formulate a “ p -adic Artin conjecture” for these functions. Our primary purpose here is to relate this conjecture to the “main conjecture” discussed by Coates in [3]. We will describe the precise formulations of these conjectures that we will use later. Our main result will be that in fact the main conjecture implies the p -adic Artin conjecture.