## Random pro-p groups, braid groups, and random tame Galois groups

I’ve posted a new paper with Nigel Boston, “Random pro-p groups, braid groups, and random tame Galois groups.”

The paper proposes a kind of “non-abelian Cohen-Lenstra heuristic.”   A typical prediction:  if S is a randomly chosen pair of primes, each of which is congruent to 5 mod 8, and G_S(p) is the Galois group of the maximal pro-2 extension of Q unramified away from S, then G_S(p) is infinite 1/16 of the time.

The usual Cohen-Lenstra conjectures — well, there are a lot of them, but the simplest one asks:  given an odd prime p and a finite abelian p-group A, what is the probability P(A) that a randomly chosen quadratic imaginary field K has a class group whose p-primary part is isomorphic to A?  (Note that the existence of P(A) — which we take to be a limit in X of the corresponding probability as K ranges over quadratic imaginary fields of discriminant at most X — is not at all obvious, and in fact is not known for any p!)

Cohen and Lenstra offered a beautiful conjectural answer to that question:  they suggested that the p-parts of class groups were uniformly distributed among finite abelian p-groups.  And remember — that means that P(A) should be proportional to 1/|Aut(A)|.  (See the end of this post for more on uniform distribution in this categorical setting.)

Later, Friedman and Washington observed that the Cohen-Lenstra conjectures could be arrived at by another means:  if you take K to be the function field of a random hyperelliptic curve X over a finite field instead of a random quadratic imaginary field, then the finite abelian p-group you’re after is just the cokernel of F-1, where F is the matrix corresponding to the action of Frobenius on T_p Jac(X).  If you take the view that F should be a “random” matrix, then you are led to the following question:

Let F be a random element of GL_N(Z_p) in Haar measure:  what is the probability that coker(F-1) is isomorphic to A?

And this probability, it turns out, is precisely the P(A) conjectured by Cohen-Lenstra.

(But now you cry out:  but Frobenius isn’t just any old matrix!  It’s in the generalized symplectic group!  Yes — and Jeff Achter has shown that, at least as far as the probability distribution on A/pA goes, the “right” random matrix model gives you the same answer as the Friedman-Washington quick and dirty model.  Phew.)

Now, in place of a random quadratic imaginary field, pick a prime p and a random set S of g primes, each of which is 1 mod p.  As above, let G_S(p) be the Galois group of the maximal pro-p extension of Q unramified away from S; this is a pro-p group of rank g. What can we say about the probability distribution on G_S(p)?  That is, if G is some pro-p group, can we compute the probability that G_S(p) is isomorphic to G?

Again, there are two approaches.  We could ask that G_S(p) be a “random pro-p group of rank g.”  But this isn’t quite right; G_S(p) has extra structure, imparted to it by the images in G_S(p) of tame inertia at the primes of S.  We define a notion of “pro-p group with inertia data,” and for each pro-p GWID G we guess that the probability that G_S(p) = G is proportional to 1/Aut(G); where Aut(G) refers to the automorphisms of G as GWID, of course.

On the other hand, you could ask what would happen in the function field case if the action of Frobenius on — well, not the Tate module of the Jacobian anymore, but the full pro-p geometric fundamental group of the curve — is “as random as possible.”  (In this case, the group from which Frobenius is drawn isn’t a p-adic symplectic group but Ihara’s “pro-p braid group.”)

And the happy conclusion is that, just as in the Cohen-Lenstra setting, these two heuristic arguments yield the same prediction.  For the relatively few pro-p groups G such that we can compute Pr(G_S(p) = G), our heuristic gives the right answer.  For several more, it gives an answer that seems consistent with numerical experiments.

Maybe it’s correct!

## Math And: Arielle Saiber on Italian poetry and Italian algebra, Friday, Oct 23 at 4pm

Something to do tomorrow (besides eating the Beef n Brew slice): the Math And… seminar is very pleased to welcome Arielle Saiber from Bowdoin for our Fall 2009 lecture.  Arielle is an Italianist of very broad interests, with academic papers on Italian literature, the early history of algebra and geometry, Dali’s illustrations for Dante, and the polyvalent discourse of electronic music.  Tomorrow there will only be time to unite the first two.

23 Oct 2009, 4pm, Van Vleck B239: Arielle Saiber (Bowdoin, Italian)

Title “Nicollo Tartaglia’s Poetic Solution to the Cubic Equation.”

Niccolo Tartaglia’s (1449-1557) solution to solving cubic equations, which renowned mathematician and physician Girolamo Cardano wanted but Tartaglia resisted, led to one of the first intellectual property cases in Western history. Eventually, Tartaglia agreed to give Cardano what he so desired, but only if the latter promised he would not publish it. Cardano promised, and Tartaglia sent him the solution. Wasting little time, however, Cardano published the solution (along with a ‘general’ solution that he himself developed). Tartaglia was, not surprisingly, furious and began a vicious battle with Cardano’s assistant, Ludovico Ferrari (Cardano refused to engage Tartaglia directly). But vitriolic polemics aside, there is something else rather curious about this ordeal: the solution Tartaglia gave Cardano was encrypted in a poem. This talk looks at the motives behind his “poetic solution” and what it says about the close relationship between ‘poeisis’ and ‘mathesis’ in this period of mathematics’ history.

## Two good things I ate this week

Yesterday, a bowl of gumbo from New Orleans Take Out on Monroe.  So richly spiced as to be almost black, so thick with roux and file that it was almost not a soup.  This is non-traditional but I crumbled up and added my sweet cornbread to make of it a kind of granular black spicy undefined entity that was the best thing I ate this week.

Today, the Beef n Brew special slice at Ian’s Pizza.  Available only through tomorrow.  Thin-sliced coffee-rubbed steak from Fountain Prairie, roasted wild mushrooms, caramelized onions, and gravy.  Autumnal, superb.

## Modern game

We went to Cambridge, WI yesterday for a very worthwhile tour of Hinchley’s Dairy Farm.  To give you a bit of a break from cow-cow-cow they have a little menagerie of goats, rabbits, and poultry you can feed, including a modern game bantam.  The still picture doesn’t really capture how weird these things are to look at.  Mrs. Hinchley told us that little kids often say they look like dinosaurs — and that’s true, but much more accurate is Mrs. Q.’s observation that, of all chickens, the modern game bantam is the one which most resembles a guy in a chicken suit.

## Brothers

Just finished and very much enjoyed Yu Hua’s Brothers, China’s all-time best-selling novel.  If you’re going to read one long translated work of fiction in the social-magical-realism mode this year, make Hua’s book the one.  Especially if you like lots of bathroom jokes swirled into your multigenerational sagas of love and capital.  The translation, by Eileen Chow and Carlos Rojas, feels very natural without reading like colloquial English; I can’t speak to its faithfulness, of course.

The book is in some ways a standard melodrama; people get rich, people get poor, people get politically oppressed and beaten, two people want to marry the same person, people disappear on long journeys only to reappear at just the right time, people get artificial breasts and hymens surgically attached (OK, that last part is somewhat less standard, but by the end of the 600+ pages of Brothers it’s started to seem standard.)  I think my social prejudices would work against my buying an American novel that functioned like Brothers. Or America’s all-time best-selling novel, whatever it is.   But when a book is in translation all snobbery falls away.  Maybe because it is “improving” to read foreign books.  (Those scare quotes are meant to distract you from the fact that I actually kind of believe this to be the case.)

One sentence from a direction I decided not to pursue in this post:  “The brothers in Brothers are the driven and insatiable Baldy Li and his meeker, gentler brother Song Gang; the inexorable rise to wealth, prominence, and sexual irresistibility of the former, and the corresponding decline into government dependence, ill-health, and gynecomastia of the latter, is the kind of story Ayn Rand would have told if a) she were funny, and b) she thought that the successful brother was horrible and doomed, but recognized that the alternative to this kind of success was even worse.”

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## Why Math Overflow works, and why it might not

I spent a bunch of time yesterday playing with Math Overflow, the new math Q&A website launched last week by Berkeley grad students David Brown and Anton Gerashchenko. The site is built on the popular Stack Exchange platform, giving users the power not only to ask and answer questions but to vote on other people’s answers, giving those users “reputation points” which they can use to unlock more features of the site.

I was chatting with Tim Gowers last month, in the context of PolyMath, about what made a website “sticky,” or, to put it more pungently, “addictive” — what makes users willing to go back to the same site multiple times a day, and keep it up for weeks or months?  Math Overflow seems to have this quality in a particularly pure form.  Unlike PolyMath — where showing up half a day late might well give you no chance of catching up and making a contribution — Math Overflow offers a constantly changing array of new questions.  Questions to which you might know the answer right off the top of your head — or at least if you take ten minutes to think about it, or just a quick half-hour to scan through some references or…

Now at this point you might say “I could answer this, but I don’t really have the time right now.”  But then somebody else would answer it first! And then you wouldn’t get that warm feeling of helping somebody out!

I think this quality of rightnowness is what’s kind of great about Math Overflow, the thing that will get a lot of people to look at it consistently and thus make it a useful place to ask questions.  But there’s also something worrisome about it.  It shouldn’t be important to be the first one to answer.  A much more rational response to that “right now” feeling would be:  “I don’t need the warm feeling.  An earnest, hard-working grad student will come along and give the same answer I would have given; except the E,HWGS will spend more time and give a more thorough answer with more details included.”  And maybe giving a terse, dashed-off answer as soon as you see the question will actually prevent that E,HWGS from ever writing the ideal answer!

But then, a terse, dashed-off answer is a lot better than no answer.  At the moment I’m very high on this site.  I hope a lot of people — even earnest, hard-working senior faculty — will put a shoulder to it, and see what happens.

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## Anabelian puzzle 5: isogenies between Jacobians and metabelian fundamental groups

Another question that came up at the Newton Insitute:  can two different curves X,Y over F_q have the same geometrically metabelian pro-l fundamental group?

I would think not, and here’s why.  First of all, the actions of Frob_q on H_1(X,Z_ell) and on H_1(Y,Z_ell) agree.  This already implies that Jac(X) and Jac(Y) are isogenous.  Can this actually happen in large genus?  Yes:  a recent arXiv preprint by Ben Smith gives lots of explicit examples of pairs of hyperelliptic curves with isogenous Jacobians.  From Smith’s paper I learned about the recent construction by J. F. Mestre of pairs of hyperelliptic curves in every genus with isogenous Jacobians.

In other words, the geometrically abelian fundamental group need not distinguish X from Y.

But the fact that the geometrically metabelian pro-l fundamental groups agree implies the following much stronger fact.  Let X_n be the maximal abelian cover of X/F_qbar whose Galois group has exponent l^n, and define Y_n similarly.  Then X_n and Y_n have isogenous Jacobians for all n.  I would think this would be impossible if X and Y were not isomorphic; but I don’t have the slightest idea for a proof.

Baby version of this question:  do there exist non-isomorphic curves X and Y of large genus (say, for the moment, over C) whose Jacobians are isogenous, and such that each Prym of X is isogenous to a Prym of Y?

Steve was talking about the future of poetry at the Twin Cities Book Fest this weekend, so CJ and I hopped up for the weekend to see him and his family.  A few notes:

• Priceline works!  I’ve never used them before, I suppose because it’s rare I’m traveling not for work and not staying with relatives.  I worried there’d be no free rooms Saturday night with a Twins-Yankees playoff game the next day; but in fact Priceline found us a \$60 room at the Holiday Inn Metrodome.  Why were there still rooms available next door to the stadium?  Because, as Steve explained, the Twins reserve most playoff tickets for locals, with only 3,000 seats available to New York fans.  I both approve of this practice (on grounds that it sticks it to New York fans) and disapprove (on grounds that stadium owners extract all kinds of concessions from cities and states with the promises of massive hotel, bar, and restaurant sales to visiting fans, and surely the city of Minneapolis forwent a pile of revenue from Yankee fans who would have been staying in CJ’s and my hotel room, had they been able to get tickets for the game.)
• The crowd in the lobby Saturday night was about equally mixed between belogoed Gopher fans, the afterparty from a hotel wedding, and ravenous zombies.  Lots of aggression between the beeriest groomsmen and the most in-character zombies, which looked like it might get physical; rather than witness this CJ and I tucked ourselves into our big comfy bed and watched the Discovery Channel until we fell asleep.  We learned a lot about walnuts.
• You probably already know this, but if you’re driving from Madison to Minneapolis you should stop at Norske Nook in Osseo and get pie.  They sell other food but it’s little more than an unneccesary delay of pie.
• I never found out what the future of poetry was, but if it has one it will surely involve Minneapolis-based Coffee House Press, which, per the chatter at the book people party Saturday night, is one of the few literary entries everybody in po-biz endorses and admires.  Buy some books!
• We made it back to Madison about 15 minutes before the start of yesterday’s all-ages They Might Be Giants show at the Barrymore.  It’s twenty years, to the month I think, since I first saw them play.  I thought there would be a lot of eight-year-olds there but the crowd actually skewed younger than CJ.  Maybe the eight-year-olds were up in the mosh pit.  Spirited short set, almost all drawn from the kids’ records — very nice, though, to hear a bit of “The Famous Polka.” Assertion:  the songs from the standard TMBG catalogue that read as kids’ songs (“Istanbul not Constantinople,” “Particle Man,” “Why Does the Sun Shine?” “Dr. Worm,” “Older”) are better kids’ songs than the official kids’ songs.  Discuss in comments.

Nerve.com finally consults the real experts.

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## How it was with tobacco

I was looking through old issues of Fact for a book review by Gershon Legman, and came across this, from the March-April 1964 issue:

Writing about lung cancer in Cosmopolitan a few years ago, Gordon and Kenneth Boggs reported:  “Now that the furor has died down and experts have had time to examine the supposedly damning statistics, the cigarette seems to be all but exonerated.”  Besides, “filters have removed much of the sting from the general condemnation.”

What do Gordon and Kenneth Boggs say today?  Not a word.  They can’t.

The article was originally submitted to Cosmopolitan by two writers who said nothing whatsoever about cigarettes being “all but exonerated” and about any protection afforded by filters.  The editors asked the writers to insert a few sentences to that effect; they refused.  The editors themselves added the sentences.  Onto the article they put the by-line “By Gordon and Kenneth Boggs,” who do not exist.

Is this actually the way things used to work?

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