The n-Category Café

No one else likes to state it this way, but it seems to me that post-Weil, it has become increasingly clear that it’s better to think of a four-part picture:

1. Arithmetic of algebraic number fields

2. Arithmetic of function fields of curves over finite fields

3. Geometry of function fields of curves over finite fields

4. Geometry of Riemann surfaces

The typical case in translating a problem from one of these settings to another is that you have to go through all of these discrete steps, but that each step is straightforward. On the other hand if you set up the picture without step 3 then the jump between steps 2 and 4 often seems magical and arbitrary.

Of course the downside of this perspective is that 3 requires us to define etale cohomology to even start talking about it.

The whole business with Deligne’s proof of the Weil conjecture was deeply ironic because indeed the nut Deligne found was only very slightly unripe. One can imagine him borrowing a small wooden hammer from Rankin or Kazhdan and Margulis and hitting the nut, watching that open. But in every other respect Deligne’s work was a complete vindication of the ideas in SGA (defining etale cohomology, treating sheaves rather than cohomology of varieties as the most important object, using sheaves appropriately to perform inductions on the dimension and otherwise reduce to the simplest case)

This nut has indeed posthumously gotten riper over time, as Laumon developed the tools to simplify Deligne’s proof which perhaps “should have been” the background material.

Grothendieck’s idea of motives, on the other hand, has been tremendously helpful in too many problems in algebraic geometry and number theory to count, despite leading to very little progress on the standard conjectures themselves and no progress at all on the Weil conjectures. I want to imagine Grothendieck snatching the nut out of the water of etale cohomology, just before it was about to crack, and encasing it in a sphere of solid gold, which has remained impervious after decades in the ocean, but from which mathematicians have been able to chip off and distribute tiny bits of gold dust.

Thanks for all that helpful information/philosophy!

Yeah, when I used to read about Grothendieck getting the sulks when Deligne proved the Riemann Hypothesis part of the Weil Conjectures using a “trick”, I got the impression that this “trick” came out of left field and had little to do with Grothendieck’s plan. But as I slowly read Milne’s paper The Riemann Hypothesis over Finite Fields: from Weil to the present day, it’s becoming clear to me that Deligne really combined Grothendieck’s strategy with some other ideas.

By the way, this paper is the perfect mix of history and mathematical exposition for someone like me: it starts with Tim Gowers interviewing Deligne about how he got the idea for this proof method, and then it explains it.

I think this stuff Deligne says in that interview makes him sound like the anti-Grothendieck:

In part because of Serre, and also from listening to lectures of Godement, I had some interest in automorphic forms. Serre understood that the $p^{11/2}$ in the Ramanujan conjecture should have a relation with the Weil conjecture itself. A lot of work had been done by Eichler and Shimura, and by Verdier, and so I understood the connection between the two. Then I read about some work of Rankin, which proved, not the estimate one wanted, but something which was a $1/4$ off — the easy results were $1/2$ off from what one wanted to have.

I can’t imagine Grothendieck talking about an estimate that was $1/4$ off what he wanted.

Frenkel proposed to add a fourth language to Weil’s three – ‘Quantum physics’ – in Gauge Theory and Langlands Duality, p. 11.

I posed a question on what makes for such a language on MO here. Nearly had it closed down.

I should point out that from the point of view of the Weil conjectures, which at least in their original form deal only with smooth projective varieties, it is more enlightening to think in terms of Grothendieck’s categories of pure motives with respect to various equivalence relations. That’s what these categories were invented for, after all! Mixed motives, like Deligne 1-motives, are involved in understanding cohomological phenomena for singular and/or open varieties.

The motivic story for smooth projective curves is completely understood in the following sense. Over a field $k$ and for $X$ a smooth projective curve over $k$, the Chow motive $h(X)$ in Grothendieck’s category $M_{\sim}(k,\mathbb{Q})$ of motives with respect to an adequate equivalence relation (for instance rational equivalence) decomposes as

where as you can guess $h^i(X)$ realises to $H^i(X)$ for all (Weil) cohomology theories, for instance $\ell$-adic cohomology for any $\ell$ prime to the characteristic of $k$. This kind of so-called Chow-Künneth decomposition is expected to exist for all smooth projective varieties, but this is known in very few cases besides curves and surfaces.

The $h^0(X)$ and $h^2(X)$ are easily understood in terms of the field of constants of $X$ and Tate twists. The interesting part is the $h^1(X)$. The full subcategory of $M_{\sim}(k,\mathbb{Q})$ consisting of direct summands of $h^1(X)$ with $X$ ranging over all smooth projective curves is equivalent to the category of abelian varieties over $k$ up to isogenies, with the equivalence sending $h^1(X)$ to $\mathrm{Jac}(X)\otimes\mathbb{Q}$. If you unwind the definitions, this is a theorem of Weil about divisors on products of two smooth projective curves which is directly related to Weil’s proof of the Weil conjectures for curves. This is well explained in Scholl’s “Classical Motives”, Section 3.

This result is one of the origins of the philosophy of motives, and essentially the only case where everything is completely understood! If we pretend, of course, that we know everything about abelian varieties… But at least abelian varieties are very amenable to linear algebra methods.

For higher dimensional varieties, we have isolated situations where the geometry is understood well enough to have a sufficient supply of algebraic cycles with controlled cohomological properties (often via ingenious constructions relating back to curves or abelian varieties), and the rest is an ocean of intricately related conjectures.

Thanks, Simon! In Part 1 I was asking about motives for singular cubics, but I think I’m more interested in motives for elliptic curves, so this is very helpful:

The $h^0(X)$ and $h^2(X)$ are easily understood in terms of the field of constants of $X$ and Tate twists. The interesting part is the $h^1(X)$. The full subcategory of $M_{\sim}(k,\mathbb{Q})$ consisting of direct summands of $h^1(X)$ with $X$ ranging over all smooth projective curves is equivalent to the category of abelian varieties over $k$ up to isogenies, with the equivalence sending $h^1(X)$ to $\mathrm{Jac}(X)\otimes\mathbb{Q}$. If you unwind the definitions, this is a theorem of Weil about divisors on products of two smooth projective curves which is directly related to Weil’s proof of the Weil conjectures for curves. This is well explained in Scholl’s “Classical Motives”, Section 3.

Thanks very much! This indeed sounds very ‘classical’, which I like.

So it sounds like if $E$ is an elliptic curve over $\mathbb{F}_p$ and

$$# \{points \; of \; E \; over \; \mathbb{F}_{p^n}\} = p^n - \alpha^n - \overline{\alpha}^n + 1$$

for all $n$, then the number $\alpha$, at least up to complex conjugation, can be read off from $Jac(E) \otimes \mathbb{Q}$ somehow. Right?

I’m just fascinated by the geometrical meaning of this number right now, since it’s a baby version of one of the Riemann zeta zeros.

Thanks for the links, David! I’ve thought about that stuff and written a little about it in This Week’s Finds, but much of it remains elusive and now I really need to understand it.

James Borger writes:

For example, let’s look first at function fields. $Spec\mathbb{C}[z]$ is just the complex line $\mathbb{C}$. As we start inverting elements of $\mathbb{C}[z]$, as we must do to make $\mathbb{C}(z)$, the effect on the spectrum is to remove bigger and bigger finite sets of points. The limit is where we remove all the points and we’re just left with some kind of mesh.

If we had started with a Riemann surface of genus $g$, then we’d be left with a mesh of genus $g$, a surface sewn out of the cloth from which fly screens for windows are made. If we want to recover the original surface from the surface mesh, we just put it out back in the shed for a while and let the mesh fill up with dirt. This is just the familiar fact that a (smooth compact, say) Riemann surface can be recovered from the field of meromorphic functions on it.

The other James, James Dolan, used to talk about this process of poking holes through every point of a Riemann surface in terms of a pincushion or voodoo doll. He joked that the Riemann surface was a kind of “voodoo spectrum” of the field of meromorphic functions.

I see now that I said this before, in that previous conversation. But it’s good for me to revive this set of images.

A real closed field $k$ is one such that its algebraic closure $K$ is a finite extension. In this case, $K/k$ has degree two and $k$ is an ordered field, so has characteristic zero.

So your naive intuition didn’t fail you.

More generally there’s a construction due to Artin, which Lang dubbed ‘digging holes in algebraically-closed fields’.

Take $K$ algebraically-closed, an element $x\in K$, and $k\subset K$ a subfield maximal with respect to not containing $x$. If $k$ is perfect, then either it is real closed, or else $K/k$ is Galois with Galois group the $p$-adic integers under addition, for some prime $p$.

(If $k$ is not perfect, then $K/k$ is purely inseparable.)

Thanks; I figured it might be something like that. But there’s still a hole in the analogy between $\mathbb{F}_q$ and $\mathbb{C}$, of course, because in the latter case there is no passage to a “stand-in” and hence no need to restrict to Frobenius$^n$-fixed-points. And while there’s no Frobenius in the characteristic-0 case, I don’t see why I can’t think of $\mathbb{R}$ as a similar sort of stand-in for $\mathbb{C}$ — or, perhaps, the field of real algebraic numbers as a stand-in for the algebraic closure of $\mathbb{Q}$ — since in both cases the former field just happens to be missing some roots of polynomial equations. In fact, $\mathbb{R}$ seems like an even better stand-in for $\mathbb{C}$ than $\mathbb{F}_q$ is for $\overline{\mathbb{F}_p}$, since it’s only missing a root for one polynomial equation. And $\mathbb{R}$ is also, of course, the fixed-points of some endomorphism of $\mathbb{C}$, namely complex conjugation.

Thanks, Andrew! I’d heard of real closed fields — I’ve thought a bit about them from a logic perspective:

• The logic of real and complex numbers.

There are some pretty cool examples.

But I never knew that any field whose algebraic closure is a finite extension is a real closed field! That’s quite nice.

The rest of your remarks are pushing the limits of my feeble knowledge of algebra, but that’s good. I never took a course on fields, Galois theory or commutative algebra in school since I was into physics back then… the only fields I liked were quantum, and the only algebras I liked were C*, Banach and Lie. Most of my commutative algebra was picked up on the street.

Thank you for the marvellous answer! If I may venture a further niaive question, is it your expectation the the solution, once found, is likely to answer the question ‘what are numbers’ in any deep way… and in the event that humanity must wait another century then is there a layman’s view that could be drawn in the meantime if one was to simply assume that ‘integers act as a function on a space which is made up of an infinite number of pieces of homological dimension 1’? Or is it more likely that the revelation is going to be bound up within the specific detail of the exact kind of space and the exact kind of function?
In any case, I hope we may see a ‘Part 4’ to your Riemann series at some stage!!

It’s risky to speculate on the broad implications of a discovery that hasn’t been made yet. And the Riemann Hypothesis is one of those cases where if we knew what general kind of thing we were going to discover, we’d be much closer to discovering it than we actually are. (So, it’s not like discovering the Higgs boson, where we were pretty sure of exactly what we’d find, and it turned out we found exactly that — basically boring.)

I have a guess as to roughly how someone will prove the Riemann Hypothesis. It’s a rather commonly held guess, though also a lot of mathematicians are skeptical of this guess. I sketched at the end of this post, though there’s a lot more detail I’d like to add someday.

If this guess is right, our usual view of ‘number systems’ as commutative rings is just the tip of some iceberg, where the more general number systems include things like the (as-yet-not-understood) ‘field with one element’. There’s a race going on now to understand these more general things, with various competing proposals. Naturally I hope that the winning answer will not be interesting not merely to number theorists, but mathematicians more generally. I’d like it to have some exciting philosophical significance!

As evidence that this is not a completely crazy hope, Grothendieck’s approach to the Riemann Hypothesis did unearth the concept of ‘topos’, which is important far beyond number theory. It also half-unearthed the concept of ‘motive’, which I feel has not yet been fully clarified.

By the way: if people really understood motives, perhaps they’d know how to prove the standard conjectures — the conjectures that Grothendieck succeeded in reducing the last Weil Conjecture to, before Deligne stepped in and took a different approach to proving that last conjecture.

So, in response to your earlier question about “further intermediate waypoints”, I partially take back my earlier reply! The standard conjectures could be a step toward more deeply understanding why the last Weil Conjecture is true. I’ve never heard anyone claim the standard conjectures would help with the Riemann Hypothesis, so they’re probably not exactly an “intermediate waypoint” — but they’d certainly shed a lot of light on algebraic geometry, and they might be easier than the Riemann Hypothesis.

Might be.