Monthly Archives: August 2009

In which I like a bit of hip-hop vernacular as much as the next man

IMG_0148Home from the UK.  I like reading the British newspaper for the pleasure of encountering sentences that read, to my American ear, like something an American would write in an attempt to sound droll and British.  Two from yesterday’s Evening Standard:

“Olympia Haralambous, 16, scored 10 A* grades despite being hit in the head with a rounders bat the day before her first exam.”

“I like a bit of hip-hop vernacular as much as the next man.”

These are funniest when read aloud in a plummy British newsreader accent.

I was also going to compliment the United Kingdom on the ready availability of steak-flavored potato chips, except that as it turns out these tasted nothing at all like steak.

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Anabelian puzzle 4: What is the probability that a set of n points has no 3 collinear?

OK, this isn’t really an anabelian puzzle, but it was presented to me at the anabelian conference by Alexei Skorobogatov.

Let X_n be the moduli space of n-tuples of points in A^2 such that no three are collinear.  The comment section of this blog computed the number of components of X_n(R) back in January.  Skorobogatov asked what I could say about the cohomology of X_n(C).  Well, not a lot!  But if I were going to make a good guess, I’d start by trying to estimate the number of points on X_n over a finite field F_q.

So here’s a question:  can you estimate the number of degree-n 0-dimensional subschemes S of A^2/F_q which have no three points collinear?  It seems very likely to me that the answer is of the form

P(1/q) q^{2n} + o(q^{2n})

for some power series P.

One way to start, based on the strategy in Poonen’s Bertini paper:  given a line L, work out the probability P_L that S doesn’t have three points on L.  Now your first instinct might be to take the product of P_L over all lines in A^2; this will be some version of a special value of the zeta function of the dual P^2.  But it’s not totally clear to me that “having three points on L_1″ and “having three points on L_2″ are independent.

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Anabelian puzzle 3: why are there virtual sections?

Suppose X is a scheme over a field K, and write Xbar for the basechange of X to Kbar, so that as usual we have an exact sequence.

1 \rightarrow \pi_1(\bar{X}) \rightarrow \pi_1(X) \rightarrow G_K \rightarrow 1.

Now there may be no section from G_K back to \pi_1(X).  But certainly X has a rational point over some finite extension L/K, which means that there is definitely a section from the finite-index subgroup G_L to \pi_1(X).  This is so easy that I can’t help wondering:  is there a way to see the existence of such a “virtual section” from group theory alone?  My intuition is to say no.  But I just thought I’d mention it, while we’re puzzling anabelianly.

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Anabelian puzzle 2: birational sections in the least non-abelian case

(Note:  this is being posted from an airport shortly before boarding, so this is less edited than usual.  I have some more concrete remarks on the puzzle below but will save them for later.)

Let X/K be a variety over a number field; then we have an exact sequence of etale fundamental groups

\pi_1(X/\bar{K}) \rightarrow \pi_1(X/K) \rightarrow G_K

and every point of X(K) induces a section from G_K back to \pi_1(X/K).  Grothendieck’s section conjecture asserts that, at least for certain classes of varieties X including projective smooth curves of genus greater than 1, the group-theoretic sections above are in fact in bijection with X(K).

A point P of X(K) is also a point of U(K), where U is any open subscheme of X containing P.  This gives you a section G_K \rightarrow \pi_1(U/K) for any such U; the limit of all these fundamental groups is the Galois group of the maximal extension of the function field K(X) unramified at P.

In fact, you can associate to P \in X(K) a section from G_K to the whole absolute Galois group of K(X) (or, better, a “bouquet” of related sections.)  The choice of P determines a decomposition group in G_{K(X)}, isomorphic to a semidirect product of its inertia group by a copy of G_K; now a section of this semidirect product is a section from G_K to G_{K(X)}.

So a weaker version of the section conjecture, the birational section conjecture, asserts that all of these sections from G_K to G_{K(X)} come from X(K).  Koenigsmann proved a few years ago that the birational section conjecture holds for K = Q_p.  A recent paper of Florian Pop proves something much stronger; that the birational section conjecture holds over finite extensions of Q_p even when G_{K(X)} is replaced with a really puny quotient.  Namely:  let K’ be the maximal elementary 2-abelian extension of Kbar(X), and let K” be the maximal elementary 2-abelian extension of K”.  Then you have a group Gamma described by an exact sequence

1 \rightarrow G(K''/\bar{K}(X)) \rightarrow \Gamma \rightarrow G_K

and what Pop proves is that this group Gamma, which you might call the “geometrically metabelian mod-2 fundamental group of X,” “remembers” enough about the curve X that already the sections from G_K to Gamma are all given by points of X.  Pop calls this a “minimalist birational section conjecture.”

One then wonders:  just how minimal can one get?  Abelianizing Gal(Kbar(X)) is too brutal; the resulting “geometrically abelian” fundamental group has lots of “extra” sections coming from points of Jac(X).  (Note:  just noticed this paper of Esnault and Wittenberg about exactly this, haven’t read it yet.)

Now here’s the puzzle — suppose we let K’ be the maximal elementary 2-abelian extension of Kbar(X) (i.e. the compositum of all quadratic extensions) and K” be the maximal elementary abelian 2-extension of K’ such that Gal(K”/Kbar(X)) has nilpotence class 2.  Then again you have

1 \rightarrow G(K''/\bar{K}(X)) \rightarrow \Gamma' \rightarrow G_K

where \Gamma' could be called the “geometrically 2-nilpotent mod 2 fundamental group.”

So what are the sections from G_K to \Gamma'?  Is this fundamental group so minimalist that there are tons of extra sections, or is it the maximally minimalist context where a birational section conjecture could hold?

Update:  A bit jetlagged, but let me at least add one concrete question to this post.  I claim that a birational section as above would give you a function

f: U(\mathbf{Q}) \rightarrow \mathbf{Q}^*/(\mathbf{Q}^*)^2

for some open subscheme U of P^1, with the property that the Hilbert symbol

(f(a)/(a-b), f(b)/(b-a))

always vanishes.  Can you think of any such functions besides f(a) = a + constant, or f(a) = constant?

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Astonishing, logarithmic

Sometimes I wish people would stop saying “exponential” when they just mean “fast” or “a lot.”  Be careful what you wish for.  From Benjamin F. Carlson in the Atlantic:

A Quantum Leap, marvels Simon Rogers at The Guardian in a post that graphs Bolt’s astonishing, logarithmic rise in speed.

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Anabelian puzzle 1: lifting Galois representations from symplectic groups to mapping class groups

This is an experimental post:  I’m going to put up something I’ve thought about only vaguely and partially, with the idea that it might be an interesting thing to discuss with people at the upcoming workshop on anabelian geometry at the Newton Institute.  Please forgive (or, better, correct) any and all mistakes.

Here’s an old puzzle of Oort, which I’ve mentioned here previously:

Does there exist, for every g > 3, an abelian variety over Qbar not isogenous to the Jacobian of any smooth genus-g curve?

Here’s one way you could imagine trying to construct an example.  Given a g-dimensional abelian variety A over a number field K, the action of the absolute Galois group G_K on the l-adic Tate module of A provides a Galois representation \rho: G_K \rightarrow Sp_{2g}(Z_\ell).

Now suppose A is in fact the Jacobian of a smooth genus g curve X/K.  Then \rho lifts to a representation \tilde{\rho}: G_K \rightarrow G, where G is the automorphism group of the pro-l geometric fundamental group of X.  (In fact, this is the case even if A is only isogenous to a Jacobian, as long as the degree of the isogeny is prime to l.)

So you can ask: are there abelian varieties A such that \rho doesn’t lift from the symplectic group to G?  More:  such that the restriction \rho | G_L doesn’t lift to G, for any finite extension L/K?  This seems quite difficult; it means you have to find an obstruction which isn’t torsion.

My further thoughts on this are even more disorganized and I’ll keep them to myself.  Oh, except I should say:  what makes this puzzle “anabelian” is that it has something to do with the notion that a section of the map

\pi_1^{et}(M_g/K) \rightarrow G_K

ought to come from a point of M_g(K).

Yes, there is an anabelian puzzle 2, which I’ll try to post in the next couple of days.

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Smarts, luck, wealth

Reuters blogger Felix Salmon wrote Tuesday:

A tweet from Joe Weisenthal yesterday, on the subject of Annie Leibovitz, is I think revealing of a particularly American mindset: call it the Wealth Corollary of the Efficient Market Hypothesis. In a nutshell, it says that if you’ve made lots of money, you must be pretty smart.

Isn’t that totally wrong?  Under the efficient market hypothesis, if you’ve made lots of money on the market, you must be pretty lucky. (Whether it takes smarts to make lots of money as an art photographer, or a baseball player, or a breakfast-cereal magnate, is surely outside the scope of the EMH.)

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Paean

Macaroni and cheese can be found all over Wisconsin.  Deep-fried macaroni and cheese is on the menu of at least one restaurant here in town.

But for deep-fried macaroni and cheese on a stick, I think you have to go to the state fair.

I love the state fair.

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La wei si ji dou, or: eat at Fugu

At last there’s an acceptable, even pretty good, Chinese restaurant in downtown Madison:  Fugu, in the space formerly occupied by the misleadingly named Yummy Buffet.  (OK, to be fair, it was actually a buffet.)  It’s billed as pan-Asian but the management is from Hong Kong, and I’ve done well by sticking to the Chinese portion of the menu.  I particularly liked a dish called “cured meat with string beans,” which consisted mostly of very tender, very flavorful, very salty dry-cooked green beans, lightly sauced and studded with little ovals of something like a cross between Hebrew National salami and beef jerky.  The waiter told me the meat was pork but wasn’t able to give any further description.

Here’s how the dish was identified on the menu:

腊味四季豆

I decided to figure out what this actually meant — partly because I liked the dish so much, partly because I was interested to see if I could still use a Chinese dictionary, something I learned to do when I attempted to learn Chinese in high school.  I spent every Sunday morning of senior year going to Potomac Chinese School, where I was placed in a group consisting of non-Chinese adults and Chinese-American kids who had gotten kicked out of their regular class.  Suboptimal pedagogical environment.  And Chinese is really hard.  So I didn’t learn more than the rudiments, and I could never manage to say anything without waving my head in sync with the inflections.

I did learn how to look things up in the dictionary, though.  Here’s the trick:  each character has a kind of “fundamental piece,” usually the simplest element of the character.  In the second character above, it’s the little box on the left-hand side.  The fundamental pieces are listed in the dictionary in order of strokes; the little box has just three, so you find it on the list of three-stroke fundamental pieces, then you look at the sublist of “characters which are a little box + a five-stroke secondary piece,” and that’s a short enough list to search by eye, finding that 味 is “wei,” which means flavor.

The second character, 四, is one I remembered — it means “four.”  But I looked it up anyway, and was rewarded with the compound 四 季, “si ji”, which means “four seasons.”

Now here’s the part where it gets easier than it was when I was in high school — you can Google “wei si ji,” and you quickly find a menu offering “chuan wei si ji dou.”  And the character for “dou” is exactly the 豆 you’re looking for.  So you’ve got four out of five.

I tried to use Google magic to figure out the first character, but no use — I had to figure out the fundamental piece and look it up by hand.  This was the hardest part, but I eventually found out it was “la,” which means “sausage.”

So now we’ve got the whole thing:  “la wei si ji dou,” or something like “sausage with four-season flavor and beans.”

But of course this isn’t right — Googling various contiguous chunks of characters, you find that “si ji dou” is just the name of a particular kind of string bean.  According to this page,

The reference in the name “si ji dou”, (lit: four season bean) is likely due to the beans’ heartiness, and farmers’ ability to grow it in almost any season.

And “la wei” is the name of the meat:  according to a Chinese friend of a friend, “smoked or preserved pork sausage, similar to salami.”

In other words, the Chinese name of the dish is “cured meat with string beans.”

I still say the hour I spent doing this was worth it.  You never know when you might need to look something up in the Chinese dictionary, and now my skills are fresh.

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Should MathSciNet be a social network?

Jim Borger makes the following interesting suggestion in comments to the “Do you follow the arXiv?” thread:

What I think would be way better is if the MathSciNet sent out emails with abstracts of newly published papers. With some very basic filtering, based on which authors, subjects, key words, etc you like, they could probably keep the emails reasonably small. That would be much more useful than the archive digests. Then you could truly keep an eye on whatever fields you want without much effort.

This would presumably be easy to implement, given some kind of personal login to MathSciNet — but do you want MathSciNet keeping track of what abstracts you looked at, and which ones induced you to click through to the article?

The right way to do this, I guess, would involve allowing us to tag MathSciNet abstracts, so that overlaid on the AMS subject classification would be an emergent user-generated classification scheme which I’d expect to be much richer and more useful.  And it wouldn’t stop there — I imagine MathSciNet would keep track of everybody’s browsing in order to identify users with similar tastes, and make recommendations accordingly.  “People who looked up Deligne’s “Le Groupe Fondamental de la Droite Projective Moins Trois Points” also liked…”

Three questions:

  1. Would you be into this?
  2. Does Google Scholar already do this for people who use it while logged in to their Google accounts?  Is Google keeping track of my scholarly interests and ordering its Scholar search results accordingly, as it does for web search?
  3. Why doesn’t the arXiv allow tagging?  Or does it, and I can’t find it?  There are already links on each article page for bookmarking at CiteULike, del.icio.us, and digg, all places where you can tag; so arXived articles are tagged, but the tags are scattered across different services used by different populations.  Why not tag where most people read?

By the way, I really like Jim’s new paper “Lambda-rings and the field with one element.” Yes, another definition of the category of schemes over F_1; but this approach smells particularly good to me.  Here’s one thing I like.  As always, you have some notion of which Spec Z – schemes descend to Spec F_1, and what should be meant by “descent data.”  In Jim’s story, you can do the same thing starting with S-schemes, where S is an algebraic curve over some finite field F_q.  But the target of this construction is not, as you might initially think, F_q-schemes — rather, there’s another category, “F_1^S-schemes”, which lies between S-schemes and F_q-schemes.  In this category you have, e.g., some rank 1 Drinfel’d modules.  I take this to be saying that you can’t take the slogan “Spec Z is like an algebraic curve over Spec F_1″ too seriously; maybe there just isn’t anything which is to Spec Z as Spec F_q is to S.

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