Here’s a neat argument about counting points that you could present at the end of a second course in number theory. I’m sure it’s not original, but, hey, that’s what blogs are for!
Let be a smooth hypersurface in , over the field with elements. The Weil conjectures are conjectures about the number of points of over . Specifically, they say that there should be some matrix such that
and that the eigenvalues of should be algebraic integers of norm .
Here I am using the Lefschetz hyperplane theorem to know what is for .
This is, of course, a famously hard theorem. The claim about the eigenvalues is the hardest part, but simply the existence of a matrix for which this formula holds is already quite hard; the first proof was due to Dwork.
What I am going to show you is that there is a much easier proof of the above formula modulo ; a proof of the sort that could be appear in Ireland and Rosen. Many of the terms above disappear mod , so our goal is just to show that there is some matrix such that
Some polyhedral notation
Let have degree . Let be the simplex in . We will use the standard shorthand to mean the monomial , for . For any polytope , we’ll write for the lattice points in .
Let be the defining equation of , so is of the form . Let
The rows and columns of will be indexed by the lattice points in the interior of . We’ll write for the interior of .
Remark: I’m going to stick to the case of hypersurfaces in projective space, but this argument generalizes to hypersurfaces in any toric variety, and those of you who are used to toric varieties will recognize that I am choosing my notation accordingly.
The Chevalley-Warning trick
We start with a trick which may be familiar from the proof of the Chevalley-Warning theorem.
Notice that, for a nonnegative integer, we have
Let be any polynomial , where ranges through some finite subset of . We deduce that
Let be the hypersurface in affine space defined by the polynomial . For , we have
We now compute in two ways and get:
So the above formula is
The second equality is just thinking about which exponents of the form could occur in .
Finally, we shift from affine space to projective space. We have . So
Let’s look at the polynomial over . The polytope is a line segment of length , so there are two interior lattice points, namely and . We have, in part,
(All coefficients are reported modulo .)
So the sum in is . and we deduce that the number of roots of in is . Sure enough, has no roots in .
So, as before, we deduce that the number of roots of in is also and, indeed, the polynomial has no roots in that field either.
So the number of roots of in is and, indeed, all three roots of the polynomial are in this field.
If you compute for higher and higher values, you’ll see that the coefficients of and cycle through , and with period .
So we are to show that there is some matrix over whose powers have traces , and , repeating cyclically. Certainly, it’s true in this case (a diagonal matrix with entries and works). But why is it true in general?
Let’s not just look at the coefficients of , for . Let’s look at the coefficient of , for and . For, example, continuing the previous example, we’ll be looking at the coefficients of , , and . We’ll organize them into a matrix, with rows indexed by and columns by . Call this matrix .
In the above example, , so . We also get that , , and the values repeat from there. (I highly recommend taking a computer algebra system and having it work out these powers for you. It’s really fun to watch them go!)
It is now obvious what we should prove. We should show that . Then, taking , we will have and , as desired.
Finishing the proof
I feel guilty spelling out the proof. It is so much more fun for you to find it yourselves. Really, once you know what you should be proving, there are only a few reasonable things to try. We adopt the convenient notation for the coefficient of in the polynomial .
Okay, here it is. Set . Let . So we have . Since the coefficients of our polynomials are in , we have , or .
So, for any and , we have
We just have to think through what ranges over in this sum.
Well, had better be a lattice point, or there will be no term in . Also, has to be in , as it is to be an exponent of . Set . Then . So lies on the interior of the line segment between , which is in , and , which is in . So is in . We conclude that the only nonzero terms in the above sum come from , and
This is exactly the equation for multiplying matrices. QED.
Some concluding thoughts
I first learned about the Weil conjectures from the introduction to Freitag and Kiehl. This made it seem like an amazing, and thoroughly unmotivated insight, that there should be some cohomology groups around such that the traces of the Frobenius action give the point counts. Looking at examples like this makes the idea seem much more natural. After all, what is the equation but a statement that we have a representation of the group here? And, in our example, the matrices repeat with period three — and the Frobenius for the cubic in our above example has order three! Once you see the matrices, it is hard for the modern mind not to look for the group representation.
Of course, this is very anachronistic of me; the modern mathematical mind looks for the group action BECAUSE in part of the success of that method in proving the Weil conjectures. But, to my mind, every bit of demystification helps.
Those who are familiar enough with the theory may be bothered that the size of my matrices is the number of lattice points in the interior of , which is , not . This is because the argument I am giving here is the low-tech version of Fulton’s fixed point formula, not of the Lefschetz fixed point formula. Unfortunately, Fulton’s theorem only works modulo — if you want to count points modulo higher powers of , you’ll need to work with larger matrices.
Which brings me to a suggestion for someone who really knows this -adic material, and wants to turn out an awesome blog post. It is my vague understanding that Dwork’s great accomplishment was to figure out how to generalize this argument to higher powers of . If someone wanted to write up how this works to count points modulo , in the same sort of elementary way, I’d love to read it. UPDATE I have since realized that rationality of the zeta function modulo is not a good approximation to Dwork’s proof. See comments below.