A local field approach to the Riemann Hypothesis

Abstract

Since the seminal work of Wan, Poonen, and Sheats in the 1990's, we have been searching for the correct general statement of the Riemann Hypothesis ("RH") which appears implicit in their results. Recently, upon viewing the extension $\C/\R$ in light of results derived for the Carlitz module, we were led to view the RH as a statement about ramification which we explore in this short work. We shall see that, combined with some ideas flowing from the proofs of Wan, Poonen, and Sheats, this ramification idea has a good deal of explanatory power in finite characteristic. Indeed, unramified extensions of nonarchimedean local fields are cyclotomic in nature and this fits perfectly with the best possible extension of the result of Wan and Sheats. The notion that the zeroes "lie on a line" seems to be the beginning of the story in finite characteristic, and we show how this fits with having the zeroes be unramified.