## September 11, 2019

### The Riemann Hypothesis (Part 3)

#### Posted by John Baez Now I’ll say a little about the Weil Conjectures and Grothendieck’s theory of ‘motives’. I will continue trying to avoid all the technical details, to convey some general flavor of the subject without assuming much knowledge of algebraic geometry.

I will start using terms like ‘variety’, but not much more. If you don’t know what that means, imagine it’s a shape described by a bunch of polynomial equations… with some points at infinity tacked on if it’s a ‘projective variety’. Also, you should know that a ‘curve’ is a 1-dimensional variety, but if we’re using the complex numbers it’ll look 2-dimensional to ordinary mortal’s eyes, like this: This guy is an example of a ‘curve of genus 2’.

Okay, maybe now you know enough algebraic geometry for this post.

Last time I explained Hasse’s theorem saying how to count points on elliptic curves over finite fields: the number of points is a pretty obvious guess plus two ‘correction terms’ that grow and oscillate as the number of elements in your finite field increases. The obvious guess grows like $q+1$, the number of elements in your field together with a point at infinity. The correction terms grow like $\sqrt{q}$ — and that fact is a baby version of the Riemann Hypothesis!

To be precise: for any elliptic curve defined over the integers, its number of $\mathbb{F}_q$-points where $q = p^n$ is

$p^n - \alpha^n - \beta^n + 1$

where $|\alpha| = |\beta| = \sqrt{p}$.

The brilliant algebraic geometer André Weil knew about this, and he guessed how it should generalize to other kinds of curves. He announced his generalization seven months after Germany invaded Poland, starting World War II. At the time, he was confined to a French military prison for failing to report for duty — he was a pacifist.

He announced his result before he’d fully proved it. He just had an outline of a proof. Later he wrote:

In other circumstances, publication would have seemed very premature. But in April 1940, who could be sure of a tomorrow? It seemed to me that my ideas contained enough substance to merit not being in danger of being lost.

How could you guess this generalization? If we work over the field $\mathbb{C}$, an elliptic curve looks like an ordinary torus: the surface of a 1-holed doughnut. Other curves, look like the surfaces of doughnuts with other numbers of holes. We call the number of holes the ‘genus’ of the curve.

There’s only one kind of curve with genus 0, namely the projective line, and its number of $\mathbb{F}_q$ points is

$p^n + 1$

since we can chop it into two pieces: the ordinary line $\mathbb{F}_q$ and a point at infinity.

Based on this extremely limited data, if I had to guess the number of $\mathbb{F}_q$-points of a curve of genus $g$, I would guess

$p^n - \alpha_1^n - \cdots - \alpha_{2g}^n + 1$

And that’s what Weil guessed, and eventually proved!

Weil’s Theorem. Given a smooth algebraic curve of genus $g$ defined over $\mathbb{F}_p$, its number of $\mathbb{F}_q$-points where $q = p^n$ is

$p^n - \alpha_1^n - \cdots - \alpha_{2g}^n + 1$

where all the $\alpha_i \in \mathbb{C}$ have $|\alpha_i| = \sqrt{p}$.

So, instead of just two ‘correction terms’, there are $2g$ of them.

It took a lot of work for Weil to prove his theorem: even after getting the basic strategy, he had to redo the foundations of algebraic geometry, generalizing a lot of stuff from the complex numbers to finite fields. As he later wrote:

I could see that, to ensure the validity of the Italian methods in characteristic $p$, all the foundations would have to be redone, but the work of van der Waerden, together with that of the topologists, allowed me to believe that it would not be beyond my strength.

I don’t want to talk about his proof yet. For that I highly recommend this paper, which is where I’m borrowing the quotes from:

Instead, today I just want to give a rough intuitive idea of what’s going on. This will be vague, verging on mystical. I can come back later and firm it up.

I’ll illustrate what I want to say with an elliptic curve, but the same idea works for a curve of any genus.

An elliptic curve over $\mathbb{C}$ can always be gotten by taking a parallelogram in the complex plane and identifying opposite edges. For simplicity let’s consider a square. When we identify its opposite edges, the resulting torus is the disjoint union of a point, two open intervals, and an open square: So, if we were trying to count the points in the torus — a futile quest, since there are uncountably many — we might guess the answer is

$|\mathbb{R}^2| + |\mathbb{R}^1| + |\mathbb{R}^1| + |\mathbb{R}^0 |$

and with luck this might equal

$|\mathbb{R}|^2 + |\mathbb{R}| + |\mathbb{R}| + 1$

Now there actually is a reasonable choice of the value for $|\mathbb{R}|$. Namely, -1. That may seem very weird, but it’s the right choice in Schanuel’s approach to Euler characteristic. In particular, this choice gives

$(-1)^2 + (-1) + (-1) + 1 = 0$

which is the correct Euler characteristic of the torus. In general if you take a nice enough compact space and chop it into disjoint ‘open cells’ that are copies of $\mathbb{R}^n$, this method gives the Euler characteristic. You can find lots more details here:

Now, that may seem weird enough, but another weird thing is that we’re doing complex algebraic geometry, so you might think the real numbers shouldn’t be in the story at all! But somehow they are lurking there. For example, algebraic geometers call an elliptic curve $E$ over the complex numbers a curve, because it’s 1-dimensional as a complex variety, but they still like to use its cohomology groups:

$\begin{array}{ccl} H^2(E, \mathbb{Q}) &=& \mathbb{Q} \\ H^1(E, \mathbb{Q}) &=& \mathbb{Q}^2 \\ H^0(E, \mathbb{Q}) &=& \mathbb{Q} \end{array}$

which admit that topologically it’s a 2-dimensional thing that can be built out of a 2-dimensional cell, two 1-dimensional cells and one 0-dimensional cell.

Of course everyone in algebraic geometry is used to this. But to emphasize the weirdness, we could draw our elliptic curve like this: showing that $\mathbb{R}$ appears as the ‘square root’ of $\mathbb{C}$.

The reason I’m making such a big deal out of this is that Weil’s job was to figure out how all this works with $\mathbb{F}_q$ replacing $\mathbb{C}$. There’s really no field that deserves to be called $\mathbb{F}_q^{1/2}$, in general (although you could argue there is when $\mathbb{F}_q$ is a quadratic extension of some other field). But we could still try to guess the number of points in an elliptic curve over $\mathbb{F}_q$ based on these wacky ideas, like this: $q + \sqrt{q} + \sqrt{q} + 1$

or if $q = p^n$,

$p^n + p^{n/2} + p^{n/2} + 1$

But this is wrong.

It’s not surprising that this stupid guess is wrong; what’s surprising is that it’s not completely off the mark! The right answer is

$p^n - \alpha^n - \beta^n + 1$

for some $\alpha, \beta$ with $|\alpha| = |\beta| = p^{1/2}$.

So, we have some numbers $\alpha$ and $\beta$ saying that the 1-dimensional pieces of our elliptic curve have more ‘personality’ than the stupid guess predicts. These numbers have the same absolute value as the stupid guess predicts, but they also have a complex phase depending on our curve.

We also have some minus signs that aren’t in the stupid guess, familiar from Schanuel’s theory of Euler characteristic. Somehow the 1-dimensional pieces of our curve don’t add points, they subtract them!

All this may sound too wacky to be worth thinking about, but in fact some ideas like this turn out to work not just for curves, but also for higher-dimensional algebraic varieties. That’s what the full-fledged Weil Conjectures say!

Namely, suppose we have a smooth $d$-dimensional projective algebraic variety $V$ defined by polynomials with integer coefficients, and we want to count its points over $\mathbb{F}_q$, where $q = p^n$. Then we can chop it into very abstract ‘pieces’ of ‘homological dimension’ $0, 1, \dots , 2d$. Note the dimension doubling here: for example, when $d = 1$ we have a curve, but this has a piece of homological dimension two, as shown in the pictures above.

In general, each piece of $V$ having homological dimension $k$ will contribute

$\pm \alpha^n$

to the number of points, where $|\alpha| = p^{k/2}$ and the sign depends on whether $k$ is even or odd. To get the total number of points of $V$, we just add up these contributions.

In particular, there will be one piece of the top dimension, and I’m pretty sure this always contributes $p^d$ to the number of points. Then there are oscillating ‘correction terms’ that grow like $p^{d - \frac{1}{2}}, p^{d-1},$ and so on, coming from the lower-dimensional pieces.

These ‘pieces’ usually aren’t so vividly geometrical as I’ve tried to made them seem in the elliptic curve example. To get your hands on them you need to use cohomology theory. Grothendieck became obsessed with trying to understand them, and he called them ‘motives’. They’re quite amazing entities, since as we’ve seen they have dimensions that are half-integer multiples of the usual dimensions that varieties can have, and their ‘number of points’ can be a complex number — although when put together to form a variety, these complex numbers always add up to give a natural number. They remind me a bit of quantum mechanics, where you often compute real-valued observable quantities as a sum of complex numbers, and we have lots of oscillating ‘phases’. This analogy could be fleshed out in quite a bit more detail.

Grothendieck never fully succeeded in clarifying his thoughts on motives, and much remains mysterious about them — though also a lot is known. In particular, he never finished proving the Weil Conjectures using his line of thinking! He proved a lot, but he never showed these complex numbers $\alpha$ have the absolute values that the Weil Conjectures predict. So he never really worked out the growth rates of these ‘correction terms’ I keep talking about. He reduced this problem to what are now called the ‘Standard Conjectures’. These can be seen as conjectures about motives. They are absolutely fundamental to algebraic geometry. But nobody has ever managed to prove them! Deligne finished proving the Weil Conjectures in a different way.

I would like to say more about all this someday. But for now let’s just take stock of where we are. I started talking about the Riemann Hypothesis, but then I switched to a simpler version, the Weil Conjectures. In both the problem is understanding the oscillating ‘correction terms’ to a naive way of counting something. I sketched how in the Weil Conjectures, where we are trying to count points on a variety, these correction terms come from lower-dimensional ‘pieces’ of the variety, called motives. So, we can hope that in the Riemann Hypothesis something similar is going on.

But I have not said anything about how the Riemann Hypothesis is connected to the Weil Conjectures. What sort of ‘points’ on what sort of thing are we trying to count in the Riemann Hypothesis? How might we chop this thing up into generalized ‘motives’ of some sort? There’s a lot known about this, but much more remains unknown. We can hope, though, that this thing — whatever it is — is some sort of space such that integers act like functions on this space. We can also hope that this thing is some sort of generalized ‘curve’, built from a piece of homological dimension 2, a piece of homological dimension 0, and infinitely many pieces of homological dimension 1, which correspond to the nontrivial zeros of the Riemann zeta function. The fact that these pieces have homological dimension 1 should explain why these zeros have real part 1/2.

I also have not said anything about how Weil proved the Weil Conjectures for curves, or how Grothendieck almost proved them for higher-dimensional algebraic varieties. As I mentioned, Weil had to redo the foundations of algebraic geometry to handle the case of curves — and Grothendieck had to redo them over again, in a deeper way, to handle the more general case. People trying to prove the Riemann Hypothesis by continuing the same strategy feel the need to redo the foundations yet again, and that’s where things get really exciting.

Indeed, the more I learn about this, the more in awe I am of how much mathematics has arisen, directly or indirectly, from the Riemann Hypothesis — and how much more could still come out of it.

Posted at September 11, 2019 10:25 AM UTC

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### Re: The Riemann Hypothesis (Part 3)

Perhaps this is a good occasion to mention Weil’s letter from his prison cell to his philosopher sister, Simone, translated here, which invokes an analogy reused by John here:

my work consists in deciphering a trilingual text [cf. the Rosetta Stone]; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance.

Serge Lang was somewhat critical in his response to the publication of this translation.

Posted by: David Corfield on September 11, 2019 11:57 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Thanks for these interesting references! Weil’s three languages seem to be:

1. algebraic number fields
2. function fields of curves over finite fields
3. fields of meromorphic functions Riemann surfaces

The first two are now called global fields, and I see that an axiomatic presentation of them was given by Artin and Whaples around 1945-1946. Trying to unify the second and third led Weil to generalize cohomological ideas from Riemann surfaces to function fields of curves over finite fields; I tried to very tersely sketch how this is crucial to the Weil conjectures, by giving us way to think of these curves as 2-dimensional, with the tricky stuff happening in half the top dimension. Later Grothendieck built these ideas into étale cohomology in order to prove the full-fledged Weil Conjectures.

Often people cite this saying of Grothendieck as just a general platitude on how to go about mathematics wisely:

I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

However, I feel more and more that this approach was specially suited to the Weil Conjectures, because Grothendieck felt sure they would “open like a perfectly ripened avocado” when the analogy between items 2 and 3 was sufficiently developed.

This seems to be true, with the curious problem that completing the analogy leads to the Standard Conjectures, which have remained open since the 1960s. Some even doubt their truth. Milne’s paper is a great account of all this.

So as it happened, Deligne went ahead and cracked the slightly unripe nut.

Posted by: John Baez on September 11, 2019 4:06 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.

Posted by: Will Sawin on September 11, 2019 7:28 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.

Posted by: John Baez on September 12, 2019 7:00 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.

Posted by: David Corfield on September 12, 2019 8:54 AM | Permalink | Reply to this

### Will

None of these analogies will likely be as close as the number fields / functions fields / curves / curves. Usually these are models of a few phenomena in number fields, or vice versa, rather than the vast quantities of number-theoretic phenomena that are translatable into geometric language.

Of course that’s fine if you’re OK with a weaker link…

Posted by: Will Sawin on September 13, 2019 2:57 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

I’ve sometimes thought that the topologists got it wrong when they decided to number homology groups by integers: a physicist would instead use half-integers. Indeed, the Koszul sign rules say that the odd degree terms are fermionic, and fermions have half-integral spin.

Remembering that they are fermions does a few things. First, it emphasizes that algebraic curves are curves: their homological dimension is 1 for the “physics numbering”. Second, it emphasizes that the fermionic degrees of freedom should count negatively. Finally, it emphasize that the differential can be understood as a type of supersymmetry operator.

Posted by: Theo Johnson-Freyd on September 11, 2019 7:51 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Right!!! I want to get deeper into this in my story. My ‘half-integer’ remark was supposed to make people think of fermions versus bosons. And the connection is real: to count points in a variety we take the trace of some power of the Frobenius operator acting on cohomology, using the Lefschetz fixed point theorem. But this is really a supertrace, with minus signs thrown in for the odd dimensions. So yeah, these dimensions are acting fermionic.

Sorry if I’m “talking down” to you: I’m really trying to explain this to all the intimidated onlookers.

What I’d really like to know is this: if we take elliptic curves over $\mathbb{Z}$ and chop them up into motives, what motives do we get? Since I’m just a beginner I might like to use pure effective Chow motives or pure Chow motives, to keep my feet on the ground. I guess these things are hard to understand in general which is why people massage them in sophisticated ways. But I’m hoping that if we stick to the world of elliptic curves, the motives of cohomological dimension 1 that show up could be somewhat manageable in themselves. They are somehow “1-dimensional fermionic gadgets” that gives rise to the correction terms $-\alpha^n - \overline{\alpha}^n$ that I keep talking about, where $|\alpha| = \sqrt{p}$ and (though I didn’t say it) $\alpha$ is an algebraic integer. I would like to understand these gadgets better!

Posted by: John Baez on September 12, 2019 6:30 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

I wrote:

But I’m hoping that if we stick to the world of elliptic curves, the motives of cohomological dimension 1 that show up could be somewhat manageable in themselves.

It turns out Simon Pepin Lehalleur gave some useful hints in a comment to Part 1!

Posted by: John Baez on September 12, 2019 8:24 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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

(1)$h(X) = h^0(X) \oplus h^1(X) \oplus h^2(X)$

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.

Posted by: Simon Pepin Lehalleur on September 12, 2019 12:43 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.

Posted by: John Baez on September 13, 2019 7:11 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Since $E$ is an elliptic curve (smooth projective connected of genus $1$ with a distinguished rational point), $\mathrm{Jac}(E)$ is $E$ itself, but this time considered as a $1$-dimensional abelian variety (smooth commutative complete algebraic group).

The notation $\mathrm{Jac}(E)\otimes\mathbb{Q}$ refers to the fact that we are formally inverting isogenies (finite surjective algebraic group homomorphisms). A priori this seems like a bad start; how could we hope to recover the number of $\mathbb{F}_{p^n}$ points if we do this? Isogenies tend to kill torsion points, and over a finite field every rational point of an abelian is a torsion point!

The point is that from $\mathrm{Jac}(E)\otimes\mathbb{Q}$ you can still recover the rational $\ell$-adic Tate module of $E$ (for every prime $\ell$ prime to the characteristic)

(1)$T_\ell(E)\otimes\mathbb{Q} = \left(\mathrm{lim}_n \mathrm{Ker}([\ell^n]:E(\overline{\mathrm{F}}_p)\to E(\overline{\mathrm{F}}_p))\right)\otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell.$

Now a computation using the Weil pairing on the integral Tate module shows that the characteristic polynomial of Frobenius acting on $T_\ell(E)\otimes\mathbb{Q}$ has coefficients in $\mathbb{Z}$ (rather than just in $\mathbb{Z}_\ell$) and has constant term $p$. The complex roots of this polynomial are exactly $\alpha$ and $\bar{\alpha}$. This is explained for instance in Silverman’s Arithmetic of Elliptic Curves, Section 5.2.

There is a way to reformulate Weil’s proof of the Weil conjectures for more general smooth projective curves which generalizes the argument above for elliptic curves:

• relate the point count for the curve to intersection theory on the square of curve (rational points occurring in the transverse intersection of the diagonal and the graph of Froebenius),

• relate the intersection theory on the square of the curve with the endomorphisms of its Jacobian (this is the “motivic” part I mentioned in my previous answer),

• relate the resulting questions about endomorphisms of the Jacobians to endomorphisms of its Tate module (can be thought of “taking the $\ell$-adic cohomology of the motive $h^1(X)$”),

• solve the problem on the Tate module.

This — minus the motivic reformulation of the second step — is all due to Weil and explained in its historical context in Section 1 of Milne’s survey “The Riemann hypothesis over finite fields”.

The last two steps can be done for an arbitrary abelian variety and, combined with the $\ell$-adic approach to the Weil conjectures (i.e. the interpretation via $\ell$-adic cohomology and the Lefschetz trace formula) and the simple structure of $\ell$-adic cohomology of abelian varieties, they also give you the Weil conjectures for the abelian varieties themselves.

Posted by: Simon Pepin Lehalleur on September 14, 2019 10:42 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Thanks yet again. I am now trying to understand Tate modules. For people trying to follow along at home, this explanation on Math Stackexchange may be helpful.

Posted by: John Baez on September 15, 2019 7:14 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Is there a take-away slightly-inaccurate slogan along the lines of “The Riemann Hypothesis predicts that Spec(Z) behaves like a space with such-and-such homology?” Perhaps you are building up to something along those lines.

Posted by: Theo Johnson-Freyd on September 11, 2019 7:53 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Is there a take-away slightly-inaccurate slogan along the lines of “The Riemann Hypothesis predicts that $Spec(\mathbb{Z})$ behaves like a space with such-and-such homology?”

Yes! A slightly better version is to use some kind of “completion” or “compactification” of $Spec(\mathbb{Z})$ to some thing $\overline{Spec(\mathbb{Z})}$, just like we took the solutions of a cubic equation and added a point at infinity to get an elliptic curve whose homology is nicer to think about.

What I was trying to hint at in my post is that this $\overline{Spec(\mathbb{Z})}$ thing, whatever it is, acts kinda like a space with rational homology $H^0 = H^2 = \mathbb{Q}$ but $H^1 = \mathbb{Q}^\infty$, with one 1-cycle for each zero of the Riemann zeta function.

If you want to “peek at the back of the book” and see an expert’s speculations on this, try:

See the bottom half of page 123, for example.

Posted by: John Baez on September 12, 2019 6:23 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

We’ve had some 3-dimensional images of $Spec(\mathbb{Q})$ and $Spec(\mathbb{Z})$ described to us in the past at the Café and later.

We have some material gathered on the 3d issue at nLab: arithmetic topology.

Posted by: David Corfield on September 12, 2019 9:10 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.

Posted by: John Baez on September 12, 2019 4:15 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

I am really enjoying this series; thanks so much for writing it! But here I feel like something got slipped under the rug, or maybe I just missed seeing when it went under. I can see why a “1-dimensional” complex variety is 2-dimensional as a real space, and I can imagine that there is some way of defining its cohomology groups that will detect this. (You didn’t say what cohomology theory you’re talking about, but I assume it’s one of those algebro-geometric things involving Grothendieck topologies and words like “etale” or “Nisnevich” or “fpqc”.)

But did you say anything about why it happens that a “1-dimensional” variety over a finite field is also “2-dimensional” in some sense, and moreover some sense detected “algebraically” by its (co)homology? Why do finite fields behave more like $\mathbb{C}$ than like $\mathbb{R}$? I can’t think offhand of any property shared by $\mathbb{C}$ and $\mathbb{F}_q$ but not by $\mathbb{R}$ that might explain why they are alike in this way.

Maybe another way of asking the question is, what if you try to do something similar for a variety over $\mathbb{R}$ (or some other field like $\mathbb{Q}$)? Does such a variety also have an Euler characteristic that can be computed in a similar way? Is there a “$\mathbb{R}^{1/2}$” (or any formula that behaves as if there is)?

Posted by: Mike Shulman on September 11, 2019 9:51 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Mike wrote:

I am really enjoying this series; thanks so much for writing it!

Thanks! It’s tons of fun to write, since it makes me learn stuff. It’s a bit scary talking about material I don’t understand very well, but luckily I’ve had lots of practice at that.

(You didn’t say what cohomology theory you’re talking about, but I assume it’s one of those algebro-geometric things involving Grothendieck topologies and words like “etale” or “Nisnevich” or “fpqc”.)

I’m just trying to tell the classic tale of the Weil conjectures, so it’s étale cohomology that I was vaguely alluding to.

But did you say anything about why it happens that a “1-dimensional” variety over a finite field is also “2-dimensional” in some sense, and moreover some sense detected “algebraically” by its (co)homology? Why do finite fields behave more like $\mathbb{C}$ than like $\mathbb{R}$? I can’t think offhand of any property shared by $\mathbb{C}$ and $\mathbb{F}_q$ but not by $\mathbb{R}$ that might explain why they are alike in this way.

That’s a great point, and it reveals a defect in my exposition, though probably not one I want to fix. When counting points on a elliptic curve over a finite field $\mathbb{F}_q$ using cohomological methods, we take that curve and turn it into a curve $E$ over the algebraic closure $\overline{\mathbb{F}}_p$, where $q = p^n$. The $\mathbb{F}_q$-points are then the points fixed by the $n$th power of a map $f \colon E \to E$.

So its not that $\mathbb{F}_q$ is more like $\mathbb{C}$ than $\mathbb{R}$. It’s that $\overline{\mathbb{F}}_p$ is more like $\mathbb{C}$ than $\mathbb{R}$, because they’re both algebraically closed.

Now you may think this completely destroys my whole story… but my story was so hand-wavy that it’s hard to destroy. In particular, since

$\displaystyle{ colim \mathbb{F}_{p^n} = \overline{\mathbb{F}}_p }$

we can think of $\mathbb{F}_q$ as a kind of stand-in for the real deal, $\overline{\mathbb{F}}_p$. I can blame my sloppiness on my desire to put off talking about $\overline{\mathbb{F}}_p$ and the Frobenius.

But now you’re making me wonder: is $\overline{\mathbb{F}}_p$ a quadratic extension of some field? If so, could that play a role like $\mathbb{R}$ in my story?

(This question shows how little I know about algebraic closures of finite fields. Are they finite extensions of other fields??? My super-naive instinct says no.)

Posted by: John Baez on September 12, 2019 7:29 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.)

Posted by: andrew hubery on September 12, 2019 8:50 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.

Posted by: Mike Shulman on September 12, 2019 11:38 AM | Permalink | Reply to this

### Will

There will certainly be a fixed-point formula relating the Euler characteristic of the real points of a variety defined over R to the trace of complex conjugation acting on the cohomology of its complex points. But because this is just an Euler characteristic formula, there is no need to raise R to any powers, as we can’t distinguish R^2 from 0.

I don’t think the stand-in perspective is the right one. An algebraic variety is a first-class geometric object, and they are quite stable under base change, so an algebraic variety over a smaller field is essentially just an algebraic variety over a larger field with some extra structure. An algebraic variety has a set of rational points, which is a second-class object, derived from the original one.

Over a finite field, this is a discrete set, so the only invariant is the count, which we have a nice formula for in terms of the original geometry.

Over the reals, this is a manifold, so there are many invariants, but the only one I know a formula for is an Euler characteristic.

But in either case we should think of this as a shadow cast off by the variety.

Posted by: Will on September 13, 2019 3:08 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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

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. Posted by: John Baez on September 12, 2019 3:41 PM | Permalink | Reply to this

### Jeffery

What do you think of the connection between Jensen polynomials and the Riemann Hypothesis?

https://www.pnas.org/content/116/23/11103

Posted by: Jeffery Winkler on September 23, 2019 5:40 AM | Permalink | Reply to this

### Re: Jeffery

I don’t understand it well enough to have an opinion. All I can say is that 1) anything Zagier does is worth paying attention to, and 2) the connection to the Gaussian unitary ensemble is interesting, because that’s another mysterious aspect of the Riemann zeta zeros… one first noticed when the physicist Freeman Dyson and the number theorist Daman Chowla had tea at the Institute for Advanced Studies.

Posted by: John Baez on September 23, 2019 4:20 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Brilliant series, thanks for explaining at a level such that a non-mathematician can get a good feel for the matter. Even though great strides have clearly been made, on the face of it would seem that nearly a century ago the Weil conjectures offered greater promise as a strategy for resolution than exists at all today… unless there is another somewhat well-defined ‘intermediate waypoint’ to aim for which you haven’t mentioned?

Thanks

Posted by: bertie on December 3, 2019 12:03 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

I don’t know other good “intermediate waypoints” en route to the proof of the Riemann Hypothesis. This is probably one reason the hypothesis is considered difficult.

I’ve read that every hard unsolved conjecture has a gap in the proof that nobody knows how to fill, but with the Riemann Hypothesis it comes right at the start. Meaning, I guess, that we’re completely clueless. But I don’t think that’s quite right, because people do have the beginnings of a strategy. I sketched it very vaguely at the end of this post, but someday I’d like to sketch it in a bit more detail.

Still, the sketch is vague enough that people think it’s very hard to fill it in.

Posted by: John Baez on December 5, 2019 7:42 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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!!

Posted by: bertie on December 7, 2019 9:35 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

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.

Posted by: John Baez on December 7, 2019 11:12 PM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Magnificent, thank you for taking the time to share such clear insights and reasons for optimism!! As you allude to, a slightly sobering note amongst all this is that the Standard Conjectures themselves have already celebrated their half-centenial!

Posted by: bertie on December 8, 2019 12:57 AM | Permalink | Reply to this

### Re: The Riemann Hypothesis (Part 3)

Yes — this suggests that there are some fundamental things we don’t understand about this stuff. (Here “we” means the human race. I understand almost nothing compared to the real experts.)

It’s too bad Grothendieck didn’t keep working on this subject. It may take someone with comparable creativity to make serious progress.

Posted by: John Baez on December 8, 2019 1:42 AM | Permalink | Reply to this

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