Recent Progress and Open Problems in Function Field Arithmetic — The Influence of John Tate’s Work

The goal of this paper is to give a quick survey of some important recent results and open problems in the area of function field arithmetic, which studies geometric analogs of arithmetic questions. We will sketch related developments and try to trace the multiple influences of works of John Tate in this context. keywords: Tate, function field, Drinfeld modules, elliptic curves, zeta values. This paper is dedicated to my teacher John Tate. I am glad and honored to be invited to contribute to this special volume celebrating his 80th birthday. The goal of this paper is to give a quick survey of some important recent results and open problems in the area of function field arithmetic, which studies geometric analogs of arithmetic questions. We will sketch related developments and try to trace the multiple influences of works of John Tate in this context. We mainly, but not fully, limit ourselves to topics where these are clearly visible. Also, we focus mainly on results simple to state, and leave variants or generalizations to the references. General references for background on recent results in function field arithmetic are [Ros02, Gos96, G+92, G+97, Tha04]. Received November 13, 2006. ∗Supported in part by NSF grant.