Why does the Gamma function satisfy a functional equation?

Question

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from the perspective of Tate's thesis, which I don't really have the background to appreciate yet, so I'm asking for another perspective.

The perspective I want goes something like this: the Riemann zeta function is a product of the local zeta functions of a point over every finite prime $p$, and the Gamma function should therefore be the "local zeta function of a point at the infinite prime."

Question 1: Can this intuition be made precise without the machinery of Tate's thesis? (It's okay if you think the answer is "no" as long as you convince me why I should try to understand Tate's thesis!)

Multiplying the local zeta functions for the finite and infinite primes together, we get the Xi function, which has the nice functional equation. Now, as I learned from Andreas Holmstrom's excellent answer to my question about functional equations, for the local zeta functions at finite primes the functional equation

$$\zeta(X,n-s) = \pm q^{ \frac{nE}{2} - Es} \zeta(X, s)$$

(notation explained at the Wikipedia article), which for a point is just the statement $\frac{1}{1 - p^s} = -p^{-s} \frac{1}{1 - p^{-s}}$, reflects Poincare duality in etale cohomology, and the hope is that the functional equation for the Xi function reflects Poincare duality in some conjectural "arithmetic cohomology theory" for schemes over $\mathbb{Z}$ (or do I mean $\mathbb{F}_1$?).

Question 2: Can the reflection formula for the Gamma function be interpreted as "Poincare duality" for some cohomology theory of a point "at the infinite prime"? (Is this question as difficult to answer as the more general one about arithmetic cohomology?)

Answer 1

Your questions are a part of what Deninger has been writing about for 20 years. He's proposed a point of view that sort of explains a lot of things about zeta functions. It's important to say that this explanation is more in a theoretical physics way than in a mathematical way, in that, as I understand it, he's predicted lots of new things which he and other people have then gone on to prove using actual mathematics. I guess it's kind of like the yoga surrounding the Weil conjectures before Dwork and Grothendieck made actual cohomology theories that had a chance to do the job (and eventually did). It's pretty clear to me that he's put his finger on something, but we just don't know what yet.

Let me try to say a few things. But I should also say that I never worried too much about the details, because the details he has are about a made up picture, not the real thing. (If he had the real thing, everyone would be out of a job.) So my understanding of the actual mathematics in his papers is pretty limited.

Question 1: He gives some evidence that Euler factors at both finite and infinite places should be seen as zeta-regularized characteristic polynomials. For the usual Gamma function, see (2.1) in [1]. For the Gamma factors of general motives, see (4.1) in [1]. For the Euler factors at the finite places, see (2.3)-(2.7) in [2]. He gives a description that works simultaneously at the finite and infinite places in (0.1) of [2]. Beware that some of this is based on an artificial cohomology theory that is designed to make things uniform over the finite and infinite places. (Indeed, at the risk of speaking for him, probably the whole point was to see what such a uniform cohomology theory would look like, so maybe one day we'll be able to find the real thing.)

Question 2: He expects his cohomology theory to have a Poincare duality which is "compatible with respect to the functional equation". See the remarks and references in [3] between propositions 3.1 and 3.2.

I'd recommend having a look at [3]. It's mainly expository. Also, I remember [4] being a good exposition, but I don't have it in front of me now, so I can't say much. He also reviews things in section 2 of his recent Archive paper [5].

[1] "On the Gamma-factors attached to motives", Invent. Math. 104, pp 245-261

[2] "Local L-factors of motives and regularized determinants", Invent. Math. 107, pp 135-150

[3] "Some analogies between number theory and dynamical systems on foliated spaces", Proceedings of the ICM, Vol. I (Berlin, 1998), pp 163-186

[4] "Evidence for a cohomological approach to analytic number theory", First ECM, Vol. I (Paris, 1992), pp 491-510

[5] "The Hilbert-Polya strategy and height pairings", arxiv.org

Answer 2

I am no expert, but let me give you some guesses as to the answers.

Q1) I am going to go for "no". I think it's precisely Tate's thesis that shows that the gamma factor for Riemann zeta can be interpreted as a local factor analogous to the usual local factors at the finite primes. However let me absolutely stress that you do not need to read all the technical details of Tate's thesis to understand the analogue completely. Any function on a local field (including $\mathbf{Q}_p$ and $\mathbf{R}$) has a Fourier transform, which is another such function. If you normalise things in a sane way, then the characteristic function of $\mathbf{Z}_p$ is its own Fourier transform, and there's a standard function on $\mathbf{R}$ that is its own Fourier transform. Now do a certain explicit integral to these functions---the same sort in both cases---this is in Tate's thesis. On the $p$-adic side you get $(1-p^{-s})^{-1}$ and on the real side you get the correct Gamma factor. None of this is a mystery really and is really just scratching the surface of Tate's thesis (the meat of which is the functional equation, not the definition!). In fact, let's do it now.

So $k$ is either $\mathbf{Q}_p$ or $\mathbf{R}$, and $\mu^*$ is a Haar measure on $k^*$. Now if $f$ a function on $k$, let's define $$\zeta(f,s)=\int_{k^\times}f(t)|t|^s d\mu^*$$ (the integral should be over $k^\times$) Let's now compute this integral for some choices of $f$, $k$. If $k=\mathbf{Q}_p$ and $f$ is the characteristic function of $\mathbf{Z}_p$ (which turns out to be its own Fourier transform) and $\mu^*$ is normalised to make $\mathbf{Z}_p^*$ have measure 1, then the integral is (breaking up $\mathbf{Z}_p\backslash\{0\}$ into a sum of $p^m\mathbf{Z}_p^\times$ for $m\geq0$) $$\zeta(f,s)=\sum_{m\geq0}p^{-ms}=(1-p^{-s})^{-1}.$$

Now let's try another example: let's let $f$ be $e^{-\pi x^2}$, which is its own Fourier transform, let's let Haar measure on $\mathbf{R}^\times$ be $dx/|x|$, and let's compute the integral. It's done on p317 of Cassels-Froehlich---Tate's thesis: we need to compute $$\int_{x\in\mathbf{R}}e^{-\pi x^2}|x|^{s-1} dx$$ which is readily checked to be $\pi^{-s/2}\Gamma(s/2)$.

Q2) I think there is a misunderstanding here (perhaps mine, perhaps yours). The global functional equation reflects Poincare duality in the function field case. I don't really see a local functional equation for $(1-p^{-s})^{-1}$ so I don't really know what you're asking.