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ABC conjecture rumor June 12, 2012

Posted by Scott Carnahan in Algebraic Geometry, Number theory.
20 comments

There is a rumor circulating here in Japan, to the effect that S. Mochizuki has proved the ABC conjecture. My understanding is that blogs are for spreading such things.

Apparently, he had predicted some years ago that he would finish a proof in 2012, so I suppose this is an “on-time delivery”. It is certainly no secret that his research program has been aiming at the conjecture for several years.

Our very own Noah Snyder did some original work on the function field version of this conjecture, when he was a high-school student.

Update (Sept 4, 2012): This rumor seems to be true! You can find the four “Inter-universal Teichmuller Theory” papers on the very bottom of his papers page.

Fun with y^2=x^p-x May 3, 2012

Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.
3 comments

Here’s a basic example that comes up if you work with elliptic curves: Let $p$ be a prime which is $3 \mod 4$ . Let $E$ be the elliptic curve $v^2=u^3-u$ over a field of characteristic $p$ . Then $E$ has an endomorphism $F(u,v) = (u^p, v^p)$ . It turns out that, in the group law on $E$ , we have $F^2 = [-p]$ . That is to say, $F(F(u,v))$ plus $p$ copies of $(u,v)$ is trivial.

I remember when I learned this trying to check it by hand, and being astonished at how out of reach the computation was. There are nice proofs using higher theory, but shouldn’t you just be able to write down an equation which had a pole at $F(F(u,v))$ and vanished to order $p$ at $(u,v)$ ?

There is a nice way to check the prime $3$ by hand. I’ll use $\equiv$ for equivalence in the group law of $E$ . Remember that the group law on $E$ has $-(u,v) \equiv (u,-v)$ and has $(u_1,v_1)+(u_2,v_2)+(u_3,v_3) \equiv 0$ whenever $(u_1, v_1)$ , $(u_2, v_2)$ and $(u_3, v_3)$ are collinear.

We first show that

$\displaystyle{ F(u,v) \equiv (u-1, v) - (u+1, v) \quad (\dagger)}$

Proof of $(\dagger)$ : We want to show that $F(u,v)$ , $(u+1,-v)$ and $(u-1,v)$ add up to zero in the group law of $E$ . In other words, we want to show that these points are collinear. We just check:

$\displaystyle{ \det \begin{pmatrix} 1 & u^3 & v^3 \\ 1 & u-1 & -v \\ 1 & u+1 & v \end{pmatrix} = 2 v (v^2-u^3+u) = 0}$

as desired. $\square$ .

Use of $(\dagger)$ : Let $(u_0, v_0)$ be a point on $E$ . Applying $F$ twice, we get

$\displaystyle{ F^2(u_0,v_0) \equiv F \left( (u_0-1,v_0) - (u_0+1,v_0) \right)}$

$\displaystyle{ \equiv (u_0-2,v_0) - 2 (u_0,v_0) + (u_0+2,v_0)}$ .

Now, the horizontal line $v=v_0$ crosses $E$ at three points: $(u_0, v_0)$ , $(u_0-2, v_0)$ and $(u_0+2, v_0)$ . (Of course, $u_0 -2 =u_0+1$ , since we are in characteristic three.) So $(u_0-2, v_0) + (u_0, v_0) + (u_0+1, v_0) \equiv 0$ and we have

$\displaystyle{F^2(u_0, v_0) \equiv -3 (u_0, v_0)}$

as desired. $\square$ .

I was reminded of this last year when Jared Weinstein visited Michigan and told me a stronger statement: In the Jacobian of $y^2 = x^p-x$ , we have $F^2 = [(-1)^{(p-1)/2} p]$ , where $F$ is once again the automorphism $F(x,y) = (x^p, y^p)$ .

Let me first note why this is related to the discussion of the elliptic curve above. (Please don’t run away just because that sentence contained the word Jacobian! It’s really a very concrete thing. I’ll explain more below.) Letting $C$ be the curve $y^2 = x^p-x$ , and letting $p$ be $3 \mod 4$ , we have a map $C \to E$ sending $(x,y) \mapsto (y x^{(p-3)/4}, x^{(p-1)/2})$ , and this map commutes with $F$ . I’m going to gloss over why checking $F^2 = [(-1)^{(p-1)/2} p]$ on $C$ will also check it on $E$ , because I want to get on to playing with the curve $C$ , but it does.

So, after talking to Jared, I was really curious why $F^2$ acted so nicely on the Jacobian of $C$ . There are some nice conceptual proofs but, again, I wanted to actually see it. Now I do.

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Rationality of the zeta function mod p December 12, 2011

Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.
5 comments

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 $X$ be a smooth hypersurface in $\mathbb{P}^{n}$ , over the field $\mathbb{F}_p$ with $p$ elements. The Weil conjectures are conjectures about the number of points of $X$ over $\mathbb{F}_{p^k}$ . Specifically, they say that there should be some matrix $A$ such that

$\displaystyle{ \# X(\mathbb{F}_{p^k}) = 1+p^k+p^{2k} + \cdots + p^{(n-1)k} + (-1)^{n-1} \mathrm{Tr} (A^k),}$

and that the eigenvalues of $A$ should be algebraic integers of norm $p^{(n-1)/2}$ .
Here I am using the Lefschetz hyperplane theorem to know what $H_{et}^i(X)$ is for $i \neq n-1$ .

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 $p$ ; a proof of the sort that could be appear in Ireland and Rosen. Many of the terms above disappear mod $p$ , so our goal is just to show that there is some matrix $B$ such that

$\displaystyle{ \# X(\mathbb{F}_{p^k}) \equiv 1 + (-1)^{n-1} \mathrm{Tr} (B^k) \mod p.}$

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Farey fractions, Ford circles, and SL_2. October 18, 2011

Posted by Scott Carnahan in group theory, Number theory.
12 comments

The topic of this post came up during a conversation with some physicists about the fractional quantum Hall effect (which is quite fascinating, but I don’t feel particularly qualified to discuss). I have decided to set it down here in the hope that, as long as I have an internet-capable device with me, I won’t have to rederive it in front of people again. Some of this material appears in Apostol’s Modular functions and Dirichlet series in number theory and Conway’s The sensual form. I’d be happy to hear about other good treatments.

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The Brauer Groupoid August 11, 2010

Posted by Noah Snyder in fusion categories, groupoids, Number theory, quantum algebra.
3 comments

Recall that the Brauer group of a field k consists of central simple algebras over k up to Morita equivalence with the group operation given by tensor product. For example, the Brauer group of the real numbers is Z/2 because the only central simple algebras are matrix algebras over $\mathbb{R}$ or matrix algebras over the quaternions $\mathbb{H}$ , and $\mathbb{H} \otimes \mathbb{H} \cong M_4(\mathbb{R})$ . It is a well-known and fundamental fact that the Brauer group is isomorphic to the second Galois cohomology $H^2(\text{Gal}(\bar{k}/k), \bar{k}^*)$ where $\bar{k}$ is the seperable closure of k.

What I’d like to explain in this post is a follow-your-nose proof of this isomorphism which comes from thinking about fusion categories. Namely, attached to any fusion category there is a very natural object called Brauer-Picard groupoid (introduced by Etingof-Nikshych-Ostrik). For the special case of the fusion category of vector spaces over k the Brauer-Picard groupoid has a point for every seperable extension of k and the group of automorphisms of the point k gives exactly the Brauer group. However, one can also look at the group of automorphisms of other points, in particular the point $\bar{k}$ . The group of automorphisms of that point is instead naturally isomorphic to the Galois cohomology $H^2(\text{Gal}(\bar{k}/k), \bar{k}^*)$ . Since the groupoid is connected we see that the Brauer group coincides with the Galois cohomology. In fact, there’s a natural choice of arrow from $k$ to $\bar{k}$ and so a natural choice of isomorphism between the two groups.

This example came up in work in progress with Pinhas Grossman where we compute the Brauer-Picard groupoid of the fusion categories which come from the Haagerup subfactor. As we’ll see the automorphism group of a point in the Brauer-Picard groupoid has a subgroup consisting of certain “outer automorphisms.” I wanted to have a good example in hand where the outer automorphism group of different points were different in order to test certain lemmas. The example in this post is as extreme as things can get, for $k$ there are no nontrivial outer automorphisms, while for $\bar{k}$ the whole group consists of outer automorphisms.
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Divergent sums and the class number formula May 28, 2010

Posted by David Speyer in Number theory.
4 comments

Over on MathOverflow, we’ve had a bunch of discussions about the class number formula. In particular, Keith Conrad pointed out a paper of Orde which gives a beautiful nonsense proof of the class number formula. This post is my attempt to understand why Orde’s argument works. Specifically, I am going to use an idea which I learned from Terry Tao’s blog: Arguments about divergent sums are often really arguments about the constant term in asymptotic expressions for smoothed sums.

This post assumes familiarity with the concepts and notations of a first course in algebraic number theory.
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Lattices and their invariants May 14, 2010

Posted by Scott Carnahan in linear algebra, Number theory.
1 comment so far

This post started out as an exposition on the monster Lie algebra, but it grew out of control, so I’m hacking off a chunk. Here, I’ll describe the basics of integer lattices.

Lattices show up in many mathematical contexts, some of which may be unexpected to the uninitiated. These contexts include the study of optimal periodic sphere-packings, the topology of 4-manifolds (where lattices give a full classification in the simply connected case), algebraic number theory, finite group theory, and theoretical high-energy physics. I will say almost nothing about these applications, though.

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The Weil Conjectures: The approach via the Standard Conjectures May 3, 2010

Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.
7 comments

The aim of this post is to outline a strategy for proving the Weil conjectures, proposed by Grothendieck and others. This strategy is incomplete; at various stages, we will need to assume conjectures which are still open today.

Our aim is to prove:

Theorem:
Let $X$ be a smooth projective variety, over a field of any characteristic. Let $H^*$ be a “reasonable” cohomology theory. Let $\omega \in H^2(X)$ be the hyperplane class for a projective embedding of $X$ . Let $F$ be an automorphism of $X$ , such that $F^* \omega = q \omega$ . Then the eigenvalues of $F^*: H^r(X) \to H^r(X)$ are algebraic numbers and, when interpreted as elements of $\mathbb{C}$ , have norm $q^{r/2}$ .

In previous posts, we gave for a proof in characteristic zero and a proof in the case that $X$ is a curve. I also explained why I need to say how I am embedding these eigenvalues into $\mathbb{C}$ . Our proof requires all the ideas of these previous posts, plus some new ones.
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Cyclotomic integers, fusion categories, and subfactors (March) April 15, 2010

Posted by Noah Snyder in fusion categories, Number theory, Paper Advertisement, quantum algebra, subfactors.
1 comment so far

Frank Calegari, Scott Morrison, and I recently uploaded to the arxiv our paper Cyclotomic integers, fusion categories, and subfactors. In this paper we give two applications of cyclotomic number theory to quantum algebra.

A complete list of possible Frobenius-Perron dimensions in the interval (2, 76/33) for an object in a fusion category.
Given a family of graphs G_n obtained from a graph G by attaching a chain of n edges to a chosen vertex, an effective bound on the greatest n so that G_n can be the principal graph of a subfactor.

Neither of these results look like they involve number theory. The connection comes from a result of Etingof, Nikshych, and Ostrik which says that the dimension of every object in a fusion category is a cyclotomic integer.

A possible subtitle to this paper is

What’s so special about $(\sqrt{3} + \sqrt{7})/2$ ?

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Christol’s theorem and the Cartier operator February 11, 2010

Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.
3 comments

Let’s suppose that we want to compute $2^n \mod p$ , and we have already been given $n$ written out in base $p$ as $n = \sum n_i p^i$ . Here $p$ is a small prime and we want to do this conversation repeatedly for many $n$ ‘s.

Remember that $2^p \equiv 2 \mod p$ and thus $2^n \equiv 2^{n_0+n_1 p + n_2 p^2 + \cdots + n_{\ell} p^{\ell}} \equiv 2^{n_0} 2^{n_1} \cdots 2^{n_{\ell}} \mod p$ . So, start with $1$ , multiply it by $2^{n_0}$ , then by $2^{n_1}$ , then by $2^{n_2}$ and so forth. When you get to the end, read off $2^n$ .

We can precompute the effect on $\mathbb{F}_p$ of multiplying by $2^k$ , for $0 \leq k \leq p-1$ . Then we can compute $2^n$ just by scanning across the base $p$ representation of $n$ and applying these precomputed maps to the finite set $\mathbb{F}_p$ .

The precise way to say that this is a simple, one-pass, process is that it is a computation which can be done by a finite-state automaton. Here is the definition: let $I$ , $S$ and $O$ be finite sets (input, states and output), and $s \in S$ (the start). For each $i \in I$ , let $A(i)$ be a map $S \to S$ . We also have a map $r:S \to O$ (readout). Given a string $(i_0, i_1, \ldots, i_{\ell})$ in $I^{\ell}$ , we compute $r( A(i_{\ell}) \circ \cdots A(i_1) \circ A(i_0) (s))$ . So our input is a string of characters from $I$ , and our output is in $O$ .

We can say that our example above shows that $(n_{\ell}, \ldots, n_1, n_0) \mapsto 2^{\sum n_i p^i} \mod p$ is computable by a finite-state automaton. (In our example, the sets $I$ , $S$ and $O$ all have cardinality $p$ , but I do not want to identify them.)

This is a special case of an amazing result of Christol et al: Let $f_n$ be a sequence of elements of $\mathbb{F}_p$ . Then $(n_{\ell}, \ldots, n_1, n_0) \mapsto f_{\sum n_i p^i}$ can be computed by a finite-state automaton if and only if the generating function $\sum f_n x^n$ is algebraic over $\mathbb{F}_p(x)$ !

We have just explained the case $\sum f_n x^n = 1/(1-2x)$ . The reader might enjoy working out the cases $1/(1-x-x^2)$ (the Fibonacci numbers) and $(1 - \sqrt{1-4x})/2x$ (the Catalans).

In this post, I will use Christol’s theorem as an excuse to promote the Cartier operator, an amazing tool for working with differential forms in characteristic $p$ .

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ABC conjecture rumor June 12, 2012

Fun with y^2=x^p-x May 3, 2012

Rationality of the zeta function mod p December 12, 2011

Farey fractions, Ford circles, and SL_2. October 18, 2011

The Brauer Groupoid August 11, 2010

Divergent sums and the class number formula May 28, 2010

Lattices and their invariants May 14, 2010

The Weil Conjectures: The approach via the Standard Conjectures May 3, 2010

Cyclotomic integers, fusion categories, and subfactors (March) April 15, 2010

Christol’s theorem and the Cartier operator February 11, 2010

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