Robert's Mistakes

Diagram 13: Mumford's treasure map / the arithmetic surface

Dec 06, 2013

David Mumford first came up with this diagram in his 'red book'. Lieven Lebruyn writes: "It was believed to be the first depiction of one of Grothendieck's prime spectra [...], and as such was influential for generations of arithmetic geometers".

Also called the arithmetic surface, it's a natural habitat for F_1, the contradictory field with one element. It is studied nevertheless, evading the problematic ontology by euphemizing it as "virtual"... Let me first explain what we see in the picture:

The vertical lines correspond to the prime fields. That's clockwork arithmetic for prime number sized clock faces. Non-prime number sized clockwork algebras suffer zero divisors. For example,

24:00 is zero'th hour,
so 2*12 is zero,
that means 2 divides zero
(as would 12 and any other factor of 24).

Ordinary numbers don't behave like that, also it complicates things in a way that the prime fields help to understand... A story for another time. Anyway, that's why a vertical line is drawn only for clockwork algebras of prime number size (i'll call them F_p, even if labeled V((p)) in the picture).

Then there's a horizontal curve for every integral polynomial (one with integer coefficients), that happens to be a prime ideal . Remember, a prime ideal of Z[x] is a set of integral polynomials that together do the same job as a single prime number does for ordinary numbers: Every integral polynomial can be factored into a product of elements of prime ideals , just as every number can be written as a product of prime numbers. I've written a bit about prime ideals in diagram 11.

Mumford's treasure map shows the prime ideals for F_p[x]. It is a complete map of the prime-number-like structure of the polynomials in any finite field! But why do some curves visit the same point multiple times?

Take the curve for (x^2+1). In F_5 it isn't a prime ideal! But we'll put something else on the same curve, since in F_5,

(x+2)*(x+3) = x^2 + 5x + 6 = x^2 + 1.

But there's more! To the top there's the zero polynomial, there's one for every F_p and each is contained in all the other ideals below - hence the vertical doilies.

To the right we find the prime ideals for all the integral polynomials Z[x] interpreted as F_infinity.

There's even more: Yuri I. Manin added some limit lines: On the bottom he adds the arithmetic axis , another horizontal line, corresponding to the prime numbers and "explaining" the integers. Just like the curves above do for the polynomials.

He also added the geometric axis: A vertical line left to the one for F_2, which would correspond to the elusive F_1 - the field that cannot exist... But that might be used to place other well known mathematical structures in the diagram. A task that has not yielded a satisfying result, yet.

I'm indebted to Lieven Lebruyn, the picture and most of what i know i got from him - but all the errors are mine. Here's the blog:

His "absolute geometry" is the same content, prepped into a pdf. I warmly suggest you try this excellent and entertaining paper, full of insights and anecdotes. Starting with our picture it'll lead you to F_1, Mazur's knotted dictionary and other fun stuff:

There should be newer finds in the more recent preprint "Absolute geometry and the Habiro topology" , but i haven't really gotten to it. Also, it's less of an expository leisure trip than the link above:

I probably should tell you here that the affine scheme is also known as the spectrum of a ring. And that i think, using this diagram, one should be able to explain Zariski topology and what Alexandre Grothendieck has done to it. But right now i'm not sure i can.

And then, wikipedia says about the spectrum of a ring:

It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.