## Tuesday, June 12, 2012

### The Riemann Hypothesis as a statement about ramification

When I was learning algebraic number theory long ago, it was remarked that one did not know how far the analogy between Archimedean local fields and nonArchimedean ones could be pushed. In particular, one did not know whether one should view the extension ${\mathbf C}/{\mathbf R}$ as being ramified or not. For example, the value groups (of the usual absolute value) of both $\mathbf R$ and $\mathbf C$ are the same, which  is the hallmark of a unramified extension of nonArchimedean local fields....

On the other hand, the theory of the Carlitz module seems to mandate viewing ${\mathbf C}/{\mathbf R$ as being (totally) ramified. Indeed, one obviously has ${\mathbf C}={\mathbf R}(2\pi i)$ where $2\pi i$ is the period of the exponential function $\exp(x)$. Now let $A:={\mathbf F}_q[\theta]$ be the  polynomial ring over the finite field ${\mathbf F}_q$ with $q$ elements.The usual dictionary has $A$ being the analog of $\mathbf Z$ and $K={\mathbf F}_q((1/\theta))$ being the analog of $\mathbf R$. And associated to A one has the Carltiz  exponential $\exp_C(x)$ which plays a role extremely similar to the one played by $\exp(x)$ in classical arithmetic. The function $\exp_C(x)$ too has a period which we denote $\tilde{\xi}$ and we define $K_1:=K(\tilde \xi)$.

The key point is simply that $K_1/K$ is a totally ramified abelian extension, with Galois group
isomorphic to $A^\ast$ (just as the the Galois group of ${\mathbf C}/{\mathbf R}$ is isomorphic to
$\mathbf Z^\ast$). So let's agree to view ${\mathbf C}/{\mathbf R}$ as totally ramified also.

Now let $\Xi(s):=1/2 \pi^{-s/2}s(s-1) \Gamma(s/2)\zeta(s)$ be the usual completion of the Riemann
zeta function $\zeta(s)$ obtained by multiplying it with the Euler factor at infinity. Of course everybody knows that $\Xi(s)=\Xi(1-s)$ and that the Riemann Hypothesis is the statement that all the zeroes of $\Xi(s)$ have real part equal to $1/2$. Following Riemann, we put $s=1/2+it$ and $\tilde{\Xi}(t):=\Xi(s)$, so that the RH  becomes the statement that the zeroes of $\tilde{\Xi}(t)$ are real. Or, following the above discussion, the RH is the statement that the zeroes of $\tilde{\Xi}(t)$ are unramified.

So far this is simply definitional.  However, viewing the zeroes as being unramified has very serious explanatory power in finite characteristic. Indeed, in the mid 1990's Daqing Wan, B.\ Poonen, D.\ Thakur and Jeff Sheats, showed that the zeroes of the characteristic $p$ zeta function lie in $K$ (and are also simple!). However, it was very confusing to come up with a statement that would be something like a Generalized Riemann Hypothesis. Indeed, the fact that we are using nonArchimedean analysis forces the existence in general of  zeroes not lying in $K$. But, using the above restatement of the classical RH, I recently realized that the best one could hope for in terms of generalizing Wan/Poonen/Thakur/Sheats is that the zeroes are always unramified and that this fits with classical theory too!

I just put a small note on the arXiv, http://arxiv.org/abs/1206.2040, discussing all this in a bit more
detail.

#### 1 comment:

AC said...

Dear David

Rather than the "value group" one can use the module of the local fields (ie the range of the map from the multiplicative group to the positive reals, which measures the way the Haar measure for the additive group is scaled). Let Mod(K) be the module of the local field K. Then following Weil (basic number theory Chapter 1 section 4), a finite extension K' of K is totally ramified if and only if one has Mod(K')=Mod(K). Thus with this natural definition the extension C of R is totally ramified which fits with your conclusion (but not with the "value group" approach).