## “Le Groupe Fondamental de la Droite Projective Moins Trois Points” is now online

The three papers that influenced me the most at the beginning of my mathematical career were “Rational Isogenies of Prime Degree,” by my advisor, Barry Mazur; Serre’s “Sur les représentations modulaires de degré 2 de $\text {Gal}({\overline {\Bbb Q}}/{\Bbb Q})$;” and Deligne’s 200-page monograph on the fundamental group of the projective line minus three points.  The year after I got my Ph.D. I used to carry around a battered Xerox of this paper wherever I went, together with a notebook in which I recorded my confusions, questions, and insights about what I was reading.  This was the paper where I learned what a motive was, or at least some of the things a motive should be; where I first encountered the idea of a Tannakian category; where I first learned the definition of a Hodge structure, and what was meant by “periods.” Most importantly, I learned Deligne’s philosophy about the fundamental group:  that the grand questions proposed by Grothendieck in the “Esquisse d’un Programme” regarding the action of Gal(Q) on the etale fundamental group $\pi := \pi_1^{et}(\mathbf{P}^1/\overline{\mathbf{Q}} - 0,1,\infty)$ were simply beyond our current reach, but that the nilpotent completion of $\pi$ — which seems like only a tiny, tentative step into the non-abelian world! — nonetheless contains a huge amount of arithmetic information.  My favorite contemporary manifestation of this philosophy is Minhyong Kim’s remarkable work on non-abelian Chabauty.

Anyway:  Deligne’s article appears in the MSRI volume Galois Groups over Q, which is long out of print; I bought a copy at MSRI in 1999 and I don’t know anyone who’s gotten their hands on one since.  Kirsten Wickelgren, a young master of the nilpotent fundamental group, asked me the obvious-in-retrospect question of whether it was possible to get Deligne’s article back in print.  I talked to MSRI about this and it turns out that, since Springer owns the copyright, the book can’t be reprinted; but Deligne himself is allowed to make a scan of the article available on his personal web page.  Deligne graciously agreed:  and now, here it is, a publicly available .pdf scan of “Le Groupe Fondamental de la Droite Projective Moins Trois Points.”

Enjoy!

## Southern California Number Theory Day, the airport Chili’s, Evan Longoria counterfactuals

I came back this morning from a very brief trip to California to speak at Southern California Number Theory Day, hosted this year at UC Irvine. The other speakers were terrific, well worth undergoing the pain of a red-eye flight back Midwest. (Non-math material follows below the SCNTD sum-up, for those readers who don’t cotton to the number theory.)

• Brian Conrad talked about his work (some of it with Gabber and G. Prasad) on finite class numbers for algebraic groups, and an alternative to the notion of “reductive group” over global function fields of characteristic p, where the usual notion doesn’t behave quite as well as you expect. Very clear, and very much in Brian’s style in its admirable refusal to concede any “simplifying assumptions.” Well, except the occasional avoidance of characteristic 2.
• Jeff Achter talked about a circle of results, many joint with Pries, about the geography of the moduli space of curves in characteristic p. Here you have lots of interesting subvarieties that don’t have any characteristic 0 analogue, such as the “p-rank r stratum” of curves whose Jacobians have exactly p^r physical p-torsion points. Typical interesting theorem: the monodromy representation of the non-ordinary locus (a divisor in M_g) surjects onto Sp_2g, just as the monodromy representation of M_g itself does. I asked Jeff whether we know what the fundamental group of the non-ordinary locus is — he didn’t know, which means probably nobody does.
• Christian Popescu closed it out with a beautiful talk arguing that we should replace the slogan “Iwasawa theory over function fields is about the action of Frobenius on the Tate module of a Jacobian” with “Iwasawa theory over function fields is about the action of Frobenius on the l-adic realization of a 1-motive related to the Jacobian.” This point of view — joint work of Popescu and Greither — cleans up a lot of things that are customarily messy, and shows that different-looking popular conjectures at the bottom of the Iwasawa tower are in fact all consequences of a suitably formulated Main Conjecture at the top.

On the way over I’d eaten a dispiriting lunch at the St. Louis airport Chili’s, where I waited twenty minutes for a hamburger I can only describe as withered. Last night, I got to LAX with an hour and a half to spare, and the Rays and Phillies in the 7th inning of a close game 3. And the only place to watch it was Chili’s. This time I was smart enough just to order a Diet Coke and grab a seat with a view of the plasma screen.

The airport Chili’s, late on a Word Series night, turns out to be a pretty pleasant place. People talk to you, and they talk about baseball. On one side of me was a pair of fifty-something women on their way to Australia to hang out with tigers in a nature preserve. One was a lapsed Orioles fan from Prince George’s County, the other had no team. On the other was a guy from Chicago in a tweed jacket who writes for the Daily Racing Form. He liked the Mets. We all cheered for Philadelphia, and pounded the table and cussed when Jayson Werth got picked off second in the 8th in what seemed at the time the Phils’ best chance to score. (Werth, you might remember, used to be the Orioles’ “catcher of the future”; in the end, he never played a major-league game for the Orioles, or behind the plate.)

The game went into the bottom of the 9th tied 4-4, about a half hour before I was supposed to board. I figured I’d miss the end. But a hit batsman, a wild pitch, and an off-line throw to second put Eric Bruntlett on third with nobody out. Tampa Bay intentionally walked the next two hitters to get to Carlos Ruiz.

Question 1: Was this wise? I understand you set up the force, and I understand you want to put the worst Phillies hitters in the critical spot. But even a pretty bad hitter suddenly turns pretty good if you can’t walk him. And the extra two baserunners mean that Tampa Bay is still in big trouble even if Bruntlett is out at the plate after a tag-up. Mitchel Lichtman of The Hardball Times says Joe Maddon blew this decision.

And then: well, you probably saw this on TV, but Ruiz hits a slow, goofy chopper up the third-base line. Evan Longoria charges it, but by the time he gets there Bruntlett is almost home; Longoria heaves a desperate moonball in the general direction of home plate, only much, much higher; Phillies win.

Question 2a: Would Longoria have had a play if he’d stopped, set, and thrown, instead of trying to fling the ball to the catcher mid-dive?

Question 2b: Should Longoria have tried to make the play at all? Suppose he’d just stood at third, recognizing he had no play. Maybe Bruntlett scores and the Phillies win; but maybe the ball rolls foul, sending everyone back to their base with the game still tied. My Racing Form neighbor was convinced the ball was headed foul, and that Longoria had blown the game by picking it up. Subquestion: Would any human being alive have the self-control not to charge the ball in this situation?

Question 2c: A commenter on Baseball Think Factory proposed a counterfactual ending for this game even more outlandish than what actually occurred. Say Longoria runs towards the ball, sees he has no play, decides not to pick it up and hope it rolls foul. The ball rolls past Longoria, headed towards third base, as Bruntlett crosses the plate. If the ball stays fair, Phillies win; if not, Ruiz bats again. So the ball’s rolling along the line, and meanwhile, Shane Victorino, who started on second, is rounding third — and as he passes the ball he kicks it fair. Now Victorino is clearly out for interfering with the ball in play. But in this scenario, has Philadelphia won the game?