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.