Hain-Matsumoto, “Galois actions on fundamental groups of curves…”

I recently had occasion to spend some time with Richard Hain and Makoto Matsumoto’s 2005 paper “Galois actions on fundamental groups and the cycle C – C^-,” which I’d always meant to delve into.  It’s really beautiful!  I cannot say I’ve really delved — maybe something more like scratched — but I wanted to share some very interesting things I learned.

Serre proved long ago that the image of the l-adic Galois representation on an elliptic curve E/Q is open in GL_2(Z_l), so long as E doesn’t have CM.  This is a geometric condition on E, which is to say it only depends on the basechange of E to an algebraic closure of Q, or even to C.

What’s the analogue for higher genus curves X?  You might start by asking about the image of the Galois representation G_Q -> GSp_2g(Z_l) attached to the Tate module of the Jacobian of X.  This image lands in GSp_{2g}(Z_l).  Just as with elliptic curves, any extra endomorphisms of Jac(X) may force the image to be much smaller than GSp_{2g}(Z_l).  But the question of whether the image of rho must be open in GSp_2g(Z_l) whenever no “obvious” geometric obstruction forbids it is difficult, and still not completely understood.  (I believe it’s still unknown when g is a multiple of 4…?)  One thing we do know in general, though, is that when X is the generic curve of genus g (that is, the universal curve over the function field Q(M_g) of M_g) the resulting representation

$\rho^{univ}: G_{Q(M_g)} \rightarrow GSp_{2g}(\mathbf{Z}_\ell)$

is surjective.

Hain and Matsumoto generalize in a different direction.  When X is a curve of genus greater than 1 over a field K, the Galois group of K acts on more than just the Tate modules (or l-adic H_1) of X; it acts on the whole pro-l geometric fundamental group of X, which we denote pi.  So we get a morphism

$\rho_{X/K}: G_K \rightarrow Aut(\pi)$

What does it mean to ask this representation to have “big image”?

The Chess Master and the Computer

Garry Kasparov has a thoughtful and educational piece in the New York Review of Books about his transition from best chess player in the world to best human chess player in the world, and what computers mean for the future of chess.  (Spoiler:  Kasparov thinks chess does have a future.)  Mentioned in passing is Jonathan Schaeffer’s unbeatable checkers program, Chinook.  If you enjoy hopeless enterprises you can play against Chinook online.

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Two great tastes that taste great together

Why yes, this is a pastrami sandwich with two potato pancakes in place of the bread. Courtesy of Kaufman’s Bagel and Delicatessen in Skokie, IL.

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George Saunders, “The Bohemians”

I remember being really charmed by his book Pastoralia, which is all about garbled management-speak and commercial items with wacky MultiCapNames and the basic human inability to step off stage ever.  In Persuasion Nation is just like that too, but it starts to feel like a schtick; yeah, yeah, in the future people think the most meaningful thing they can do is view advertisements, it’s comic yet eerily like our present condition, I get it.  But then again there’s “The Bohemians,” the best story here and a completely different thing:

Eddie Sr. rushed to the hospital with his Purple Heart and some photos of Eddie as a grinning wet-chinned kid on a pony.  He found Eddie handcuffed to the bed, with an IV drip and a smashed face.  Apparently, he’d bitten one of the Armenians.  Bail was set at three hundred.  The tailor shop made zilch.  Eddie Sr.’s fabrics were a lexicon of yesteryear.  Dust coated a bright-yellow sign that read “Zippers Repaired in Jiffy.”

“Jail for that kid, I admit, don’t make total sense,” the judge said.  “Three months in the Anston.  Best I can do.”

There’s really no other explanation for this but that George Saunders woke up one day and said “I want to write a Grace Paley story.” Well, why shouldn’t he?  Rock bands should cover the Velvet Underground and short story writers should try to write Grace Paley stories, though inevitably, in both cases, most will fail.

You can read “The Bohemians” online at the New Yorker.  Or watch him read it at Housing Works in NYC.  He plays for yuks more than I think is correct.

Part 1:

Part 2:  (the quoted paragraph is right at the beginning of this part.)

Deforming Galois representations in Rwanda

I just now learned that my friend Ravi Ramakrishna from Cornell spent a sabbatical term last spring at the Kigali Institute of Science and Technology in Rwanda.  And he blogged his semester.  Good reading for anyone interested in math in the developing world, or who likes awesome pictures of gorillas and volcanoes.  Ravi made a side trip to Uganda with Teach and Tour Sojourners; seems like a nice program, though note that you pay your own way to the continent.

Note:  I don’t know if Ravi actually deformed any Galois representations while in Rwanda.  But come on, if you know the guy, you know he probably did.  He can’t leave those things alone.

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2009 book list

Here’s the list of books I read in 2009.  Last year I only read three books published in 2008:  this year I’ve got seven 2009 pubs on the list, plus two late-2008 books I read early in the year.  So I’m getting up to date.

Best of the year:  the obvious choice, 2666, is the right one.

Book I should have blogged:  No One Belongs Here More Than You. I hate twee little stories with quirky yet meaningful dialogue.  Or I think I do!  But I was crazy for this book, especially the unassumingly heartbreaking “This Person.”

Also, Rads: not actually a great piece of writing, but extremely informative about the utter alienness of the world of 40 years ago.

What is the What:  possesses the simple novelistic virtues and is about virtue.  Why didn’t people like it?  Maybe because it was “based on a true story.”  But it never occurred to me while reading this that there was a real Valentino Achak Deng, and in my interior universe there still isn’t.

Worst of the year:  I keep trying to read contemporary SF people tell me is good, and it keeps on being a few interesting conceits wrapped around dudes hatching plans to save the world by shooting other dudes in a daring fashion, as in Glasshouse.

31 Dec 2009:Chronic City, by Jonathan Lethem.
22 Dec 2009:Funny Peculiar: Gershon Legman and the Psychopathology of Humor, by Mikita Brottman.
28 Nov 2009:The Will to Whatevs, by Eugene Mirman.
24 Nov 2009:Rads, by Tom Bates.
18 Oct 2009:Brothers, by Yu Hua.
22 Sep 2009:The Stardust Lounge, by Deborah Digges.
15 Sep 2009:The Duplicate, by William Sleator.
12 Sep 2009:Granta 71.
28 Aug 2009:War Trash, by Ha Jin.
24 Aug 2009:The White Mountains, by John Christopher.
17 Aug 2009:Don’t Cry, by Mary Gaitskill.
31 Jul 2009:Lives of Girls and Women, by Alice Munro.
16 Jul 2009:The Dwarf, by Pär Lagerkvist (tr. Alexandra Dick).
11 Jul 2009:Transition 101.
10 Jul 2009:Mathematicians, by Mariana Cook.
5 Jul 2009:In the Land of Invented Languages, by Arika Okrent.
2 Jul 2009:End of I., by Stephen Dixon.
25 Jun 2009:The Magicians, by Lev Grossman.
20 Jun 2009:Oscar and Lucinda, by Peter Carey.
25 May 2009:Seventy Times Seven, by John Sanford.
23 May 2009:Drop City, by T. Coraghessan Boyle.
30 Apr 2009:Consider Phlebas, by Iain M. Banks.
26 Apr 2009:No One Belongs Here More Than You, by Miranda July.
22 Apr 2009:The Size of Thoughts, by Nicholson Baker.
18 Apr 2009:Green-Eyed Thieves, by Imraan Coovadia.
10 Apr 2009:Ms. Hempel Chronicles, by Sarah Shun-Lien Bynum.
25 Feb 2009:Glasshouse, by Charles Stross.
20 Feb 2009:The Believer 60.
11 Feb 2009:What is the What? by Dave Eggers.
16 Jan 2009:2666, by Roberto Bolano.

What do roots of random polynomials look like?

I found this picture, made by Sam Derbyshire, on John Baez’s This Week in Mathematical Physics, progenitor of all math blogs and still teaching me things 16 years into its run.  It depicts the roots of polynomials of degree at most 24 with all coefficients $\pm 1$.  The image above is a closeup of a neighborhood of $(1/2)e^{i/5}$.  (Or so Baez says; judging by the picture I wonder if he meant $(4/5)e^{i/5}$.)  The full picture looks like this:

You can see why there’s a big hole around zero:  if |z| is small, $\pm 1 \pm z \pm z^2 \pm \ldots$ will have a hard time being very far from 1.

I wonder what this picture looks like as the maximum degree of the polynomial gets higher and higher?  I suppose this would have something to do with the amount of time spent near zero by the random walk which starts at 0, then moves by either z^k or -z^k at time k.  When |z| < 1/2, the limiting measure of this random walk is supported on a kind of Cantor set:  in any event, it never returns to 0.  When |z| > 1, the random walk is going to get farther and farther away; again, we should not expect to return to 0.

When 1/2 < |z| < 1, I would have thought this random walk was better understood.  But not so!  Boris Solonyak proved in 1995 (JSTOR link)  that the random walk converges to an absolutely continuous measure m_z for almost all real z in (1/2,1).  It’s conjectured that this is the case for all but countably many z, but this remains out of reach.  There are certainly exceptions:  Erdos showed, for instance, that when $z = (1/2)(1+\sqrt{5})$, the limiting measure is purely singular.  (Let’s hope Dan Brown doesn’t find out about this.)

All this reminds me of a beautiful lecture Balint Virag gave here at UW a few months ago, about the roots of random complex power series

$\sum_{i=0}^\infty a_i z^i$

where the coefficients a_i are independent complex Gaussians with mean 0 and variance 1.  In this much smoother case, the density of zeros in the open unit disc is very simple:  it’s $(1/\pi) (1-|z|^2){-2}$.  In a 2005 paper, Virag and Yuval Peres do much more:  they show that the joint intensity function for an n-tuple z_1, …. z_n in the unit disc is

$\pi^{-n} det [(1-z_i \bar{z}_j))^{-2}]_{i,j}$

In fact, the distribution of zeroes turns out to be invariant under the whole group of Mobius transformations on the disc!

Actually, I lied — what Virag talked about in Madison was more recent work about random power series of the form

$\sum_{i=0}^\infty a_i (1/\sqrt{i!}) z^i$.

These guys converge on the whole complex plane, not just on the open unit disc:  and it turns out that the point process measuring the locations of their zeroes is now invariant under complex translations!  Virag and his collaborators have a very precise understanding of this process:  the repulsion between zeroes, the distribution of the deviation from the expected number of zeroes in larger and larger dilates of a nice region, etc.

Of course I had to ask myself:  what is the p-adic version of all this?  And here I got a bit stuck.  One might of course study power series of the form

$P = \sum_{i=0}^\infty a_i z^i$

where now a_i is distributed on Haar measure on Z_p.  But of course this is more like a power series with real Gaussian coefficient, which, Balint tells me, is not nearly as nice as the complex case.  Unfortunately, Q_p doesn’t have an algebraic closure that’s locally compact!  So I’m not sure what, if anything, the p-adic analogue of the complex Gaussian could possibly be.

Still, it seems fun to understand the zeroes of a polynomial P as a above, with random coefficients in Z_p.  The distances of the zeroes from 0, for instance, is governed by the Newton polygon of P.  Is there a nice description of the resulting distribution on the set of Newton polygons?  It’s easy to see that the total number of zeroes in the unit disc — that is, the total length of the Newton polygon — is geometrically distributed with parameter 1/p.

Is there a p-adic analogue of $1/\sqrt{i!}$?  That is, can you build a decay into the coefficients of P to get a random p-adic power series defined on all of C_p, and which, let’s say, is invariant under translation by Q_p?

A true-false quiz for the world

Google’s autocomplete suggestions for searches starting with “Is”:

• Is Lady Gaga a man
• Is Lady Gaga a hermaphrodite
• Is the world going to end in 2012
• Is bronchitis contagious
• Is Santa real
• Is pneumonia contagious
• Is Khloe Kardashian pregnant
• Is Walmart open on Christmas 2009
• Is Wendy Williams a man
• Is Limewire illegal

My answers:  No. No.  No.  Yes.  No.  Yes.  Don’t know.  I’d think so.  Judging from the name, no.  Judging from the name, yes.

Reader survey: do you double-space after a period?

I do, just like they taught me in 10th grade typing class.  But a quick sample of my incoming e-mail suggests I’m the only one, apart from Mrs. Q and my mom.  Mrs. Q informs me the APA styleguide demands one space after a period: she uses the double space in e-mail but not in papers.  LaTeX splits the difference, using an intersentence space about half again as large as the space between words.  The iPhone automatically drops in a period when you double-space after a word, which seems designed to suit habitual double-spacers like me; on the other hand, the text it actually produces has just one post-period space.

So I’m confused — has this tradition really fallen out of use and no one told me?  Or was it never a tradition at all, outside my high-school typing class?