Author Archives: JSE

Naser Talebizadeh Sardari, Hecke eigenvalues, and Chabauty in the deformation space

Naser Sardari is finishing a postdoc at Wisconsin this year and just gave a beautiful talk about his new paper.  Now Naser thinks of this as a paper about automorphic forms — and it is — but I want to argue that it is also a paper which develops an unexpected new form of the Chabauty method!  As I will now explain.  Tell me if you buy it.

First of all, what does Naser prove?  As the title might suggest, it’s a statement about the multiplicity of Hecke eigenvalues; in this post, we’re just going to talk about the eigenvalue zero.  The Hecke operator T_p acts on the space of weight-k modular forms on Gamma_0(N); how many zero eigenvectors can it have, as k goes to infinity with N,p fixed?  If you believe conjectures of Maeda type, you might expect that the Hecke algebra acts irreducibly on the space S_k(Gamma_0(N)); of course this doesn’t rule out that one particular Hecke operator might have some zeroes, but it should make it seem pretty unlikely.

And indeed, Naser proves that the number of zero eigenvectors is bounded independently of k, and even gives an explicit upper bound.  If I understand him correctly, such a uniform bound on the multiplicity of an eigenvalue lambda isn’t known for any lambda other than 0.  (Though a bound of the form o(dim(S_k)) is given for nonzero lambda by Frank Calegari on his blog.)

What I find most striking is the method of proof and its similarity to the Chabauty method!  Let me explain.  The basic idea of Naser’s paper is to set this up in the language of deformation theory, with the goal of bounding the number of weight-k p-adic Galois representations rho which could be the representations attached to weight-k forms with T_p = 0.

We can pin down the possible reductions mod p of such a form to a finite number of possibilities, and this number is independent of k, so let’s fix a residual representation rhobar once and for all.

The argument takes place in R_loc, the ring of deformations of rhobar|G_{Q_p}.  And when I say “the ring of deformations” I mean “the ring of deformations subject to whatever conditions are important,” I’m just drawing a cartoon here.  Anyway, R_loc is some big p-adic power series ring; or we can think of the p-adic affine space Spec R_loc, whose Z_p-points we can think of as the space of deformations of rhobar to p-adic local representations.  This turns out to be 5-dimensional in Naser’s case.

Inside Spec R_loc, we have the space of local representations which extend to global ones; let’s call this locus Spec R_glob.  This is still a p-adic manifold but it’s cut out by global arithmetic conditions and its dimension will be given by some computation in Galois cohomology over Q; it turns out to be 3.

But also inside Spec R_loc, we have a submanifold Z cut out by the condition that a_p is not just 0 mod p, it is 0 on the nose, and that the determinant is the kth power of cyclotomic for the particular k-th power you have in mind.  This manifold, which is 2-dimensional, is something you could define without ever knowing there was such a thing as Q; it’s just some closed locus in the deformation space of rhobar|Gal(Q_p).

But the restriction of rho to Gal(Q_p) is a point psi of R_loc which has to lie in both these two spaces, the local one which expresses the condition “psi looks like the representation of Gal(Q_P) attached to a weight-k modular form with a_p = 0” and the global one which expresses the condition “psi is the restriction to Gal(Q_p) of representation of Gal(Q) unramified away from some specified set of primes.”  So psi lies in the intersection of the 3-dimensional locus and the 2-dimensional locus in 5-space, and the miracle is that you can prove this intersection is transverse, which means it consists of a finite set of points, and what’s more, it is a set of points whose cardinality you can explicitly bound!

If this sounds familiar, it’s because it’s just like Chabauty.  There, you have a curve C and its Jacobian J.  The analogue of R_loc is J(Q_p), or rather let’s say a neighborhood of the identity in J(Q_p) which looks like affine space Q_p^g.

The analogue of R_glob is (the p-adic closure of) J(Q), which is a proper subspace of dimension r, where r is the rank of J(Q), something you can compute or at least bound by Galois cohomology over Q.  (Of course it can’t be a proper subspace of dimension r if r >= g, which is why Chabauty doesn’t work in that case!)

The analogue of Z is C(Q_p); this is something defined purely p-adically, a locus you could talk about even if you had no idea your C/Q_p were secretly the local manifestation of a curve over Q.

And any rational point of C(Q), considered as a point in J(Q_p), has to lie in both C(Q_p) and J(Q), whose dimensions 1 and at most g-1, and once again the key technical tool is that this intersection can be shown to be transverse, whence finite, so C(Q) is finite and you have Mordell’s conjecture in the case r < g.  And, as Coleman observed decades after Chabauty, this method even allows you to get an explicit bound on the number of points of C(Q), though not an effective way to compute them.

I think this is a very cool confluence indeed!  In the last ten years we've seen a huge amount of work refining Chabauty; Matt Baker discusses some of it on his blog, and then there’s the whole nonabelian Chabauty direction launched by Minhyong Kim and pushed forward by Jen Balakrishnan and Netan Dogra and many others.  Are there other situations in which we can get meaningful results from “deformation-theoretic Chabauty,” and are the new technical advances in Chabauty methods relevant in this context?

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Did I like Bleak House?

It’s like asking if I like New York.  It’s big!  A lot of different things are in it.  Some things are monumental and wonderful, some things have an offhand arresting beauty, some things smell bad.

Minor thoughts after break — this book just came out 165 years ago and I want to spare you spoilers.

Continue reading

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NLCS game 2: Dodgers 4, Brewers 3

In 35 years of watching baseball I had never been to a postseason game, until this Saturday, when I was able to get two tickets to Game 2 of the National League Championship Series through a wonderful terrific beautiful friend with connections.

First of all, I salute whoever the free spirit was who slammed a Zima right before entering Miller Park.

The game started at 3pm; in late afternoon with the roof shut at Miller Park there’s a slant-line of sunlight across the field which is lovely to look at and probably terrible to hit in.

And indeed there wasn’t a lot of hitting to start with. Wade Miley, once a bad Oriole, now a good Brewer, never looked dominant, giving up lots of hard-hit balls including a shot by Jeremy Freese in the first that Lorenzo Cain hauled back in from over the wall, but somehow pitched 5 2/3 only allowing 2 hits (and collecting a single himself.) Hyun-jin Ryu matched him zero for zero. Every seat in Miller Park full, everyone attentive to the game, a level of attention I’ve never seen there. The guy behind us kept saying “NASTY, throw something NASTY.” CJ believes he sees Marlins Man in the front row — he’s right! Brewers get runners on second and third with one out, Dodgers intentionally walk Yelich to load the bases, (wave of boos), Braun delivers the RBI groundout but can’t score any more. Travis Shaw hits a solo shot to deepest center, the Brewers go up 3-0, and people start to smell win, but the Dodgers lineup has good hitters all the way down to #8 and the usually reliable Milwaukee bullpen starts to crack. Jeremy Jeffress comes in with runners on first and second and nobody out, immediately gives up a single to Joc Pederson, now they’re loaded, still nobody out, Brewers up 3-1. Manny Machado, on third base, keeps jumping off the bag, trying to distract Jeffress. But Jeffress strikes out Yasiel Puig, who’s so angry he smashes his bat over his knee. Crowd exults. Then he walks light-hitting catcher Austin Barnes to force in a run. Nobody’s up in the bullpen. Crowd panics. Yasmini Grandal comes in to hit in the pitcher’s spot and Jeffress somehow gets the double play ball and is out of it. But the next inning, Jeffress stays in a little too long; Chris Taylor leads off with a lucky little dink of an infield single and then Turner muscles a ball out to the short corner in left field; 4-3 Dodgers and it stays that way.

But the Brewers do threaten. 43,000 Brewers fans want to see Yelich get one more chance to be the hero. Hernan Perez draws a walk in the bottom of the ninth, steals while Cain strikes out. So Yelich gets to bat with 2 outs and a runner in scoring position. He grounds out. Crowd deflates. But that’s all you can ask of a baseball game, right? The hitter you want in the situation you want with the game on the line and whatever happens happens. Great baseball. Great team. I hope they win it all.  Maybe I’ll try to be there when they do.

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dBs, Dentists

Two songs that seem very much of a common kind, though I find it hard to articulate exactly why. Maybe the high, incredibly clear vocals. Maybe that they’re songs about love that aren’t about pleasure.

The dBs, “Black and White”

The Dentists, “Charms and the Girl”

The Dentists song was one of those lost songs for me for years, something I’d heard on a mixtape some WHRB friend had made — all I remembered was that opening, the note repeated again and again, then the four-note circle, then the notes repeated, then the four-note circle, then the tenor vocal coming in: “I have heard / a hundred reasons why…” When I finally found it again, thanks to tech magnate and indie-pop culture hero Kardyhm Kelly, it was just as great as I’d remembered.  It’s not on Spotify.  Is that what we define as “obscure” these days?

If I have caught a foul ball it is only by standing on the shoulders of giants

“Combined cap and baseball mitt,” A patent by Richard Villalobos, 1983.

 

This is meant for fans, not players. The idea is that if a foul ball comes towards you, you may not have time to grab a glove you’ve stashed at your feet. Rather, you quickly slip your hand into the cap-mounted glove and snag the foul with your hat still attached to the back of your hand.

25 dishes that define Madison, and my takes on same

Wisconsin State Journal listicle:  25 Dishes That Define Madison

Pretty good list!  I have some comments.

Graze Burger:  It’s a very good burger, but if you go to Graze and either the patty melt or the green chili burger is on the menu, definitely order those, which are truly special.  Madison is not a patty melt town — I get mine at Mickie’s Dairy Bar, and they’re good, but you have to like grease, lots and lots of grease.  Grazes gets the bread shatteringly crispy without making it greasy, a neat trick.

Ian’s mac and cheese pizza:  As a former Ian’s Customer Of the Month (January 2010) it pains me to admit this, but Glass Nickel’s plagiaristic “mac daddy” pizza is better than Ian’s.

Lao Laan-Xang squash curry:  The very dish that 2-year-old CJ dumped on the ground, where it was mistaken for vomit!

Mickie’s dairy bar scrambler:  I never order this, it’s too big, but this is the dish that CJ always admired the football players for eating when he was 3.

The Old Fashioned’s cheese curds:  They are pretty good.  But nowhere near the top fried curds in Madison.  I have opinions.  The best curds in Madison are from the Curd Girl cart.  They are little miracles.  Light, in a way you do not think a deep-fried and battered sphere of cheese could be.  But they are.  Second best curds are at Graze.  Third best are at Steenbock’s on Orchard, where they’re served with grainy mustard and fried sage.  Purists will be annoyed by this, but look, I love mac and cheese pizza, I’m not going to have a problem with sage in my curds.

Parthenon’s gyro sandwich:  No.  Disgusting.  There isn’t a great gyro in Madison.  Plaka Taverna is fine.  Med Cafe is fine but it’s schwarma.

Plaza’s plazaburger:  Just OK.  Like Paisan’s, Plaza is strictly for people who went to college here and want nostalgia food.

Stella’s hot and spicy cheese bread:  It’s not that spicy and every time you get a bite with no cheese you realize the bread is just not that good.  I’d love to see Madison Sourdough or Batch rip off the idea and make a good version.

No matter what new problem it sets for parents

The nursery school, therefore, appears as a counter influence against the almost hidden processes by which society through the parents undertakes the premature exploitation of children’s interests in behalf of its own conventionalized and not very natural program of life.  It thus happens that one of the first considerations in a nursery school program is that after satisfying the expectations of the family with regard to the physical care of children it should keep its further thinking in firm alignment with biological rather than social understandings with regard to the present and future welfare of the child, and this no matter what new problem it sets for parents, and no matter what amount of diversity of opinion may arise between them and the school.

(Frederick W. Ellis, introduction to Harriet M. Johnson’s Children in the Nursery School, 1928.)

Why are the 2018 Orioles so bad, so very, very, very bad?

The Orioles held on to win one tonight, 5-3 over the A’s, getting out of a bases-loaded one-out jam in the top of the 8th, so maybe for once I’m emotionally able to take a look at this loss of a season.

Baltimore was not supposed to be great this year.  But they weren’t supposed to be terrible, either.  The Orioles thought they had an outside chance at a wild card in Machado’s walk year and signed free-agent pitchers Alex Cobb and Andrew Cashner. Before the season, Fangraphs projected them to win 75 games or so and battle with the Rays for fourth place in the AL East.

They’re now 42-104 and en route to the worst record in the team history.

How was everyone so wrong?

Here’s my take.  Nobody was wrong.  The projections were the right projections to make.  Sometimes you get unlucky and everything goes to shit at the same time.

First: the Orioles are not as bad as that record; they’ve been unlucky.  Their Pythagorean record is 50-96.  That’s not good.  But it’s not historically bad.

Second:  let’s look at the players who contributed at least 2 WAR to the 2017 Orioles.

  • Jonathan Schoop
  • Manny Machado
  • Adam Jones
  • Trey Mancini
  • Welington Castillo
  • Tim Beckham (in just 50 games!)
  • Dylan Bundy
  • Mychal Givens
  • Kevin Gausman

Let’s throw in Cobb and Cashner too, since they delivered that much WAR to their 2017 teams.  This is a pretty long list of players from whom the Orioles were counting on some production (except Castillo, who was cut loose.)

Of these, Cobb, Schoop, Beckham, Bundy, and Givens each had the worst season of their career.  So did Mancini, though his career’s only two years long.  Cashner was back to his 2016 level of bad after a good 2017.  Gausman and Machado played about as well as you might expect.  (Machado’s hitting improved a lot, but the move to shortstop made him less defensively valuable.)  Jones hit as usual but baseball-reference rates his defense as having degraded enough to essentially eliminate his value.  And Chris Davis, of course, who was just sort of OK in 2017 but delivered a lot of value in 2015 and 2016, is turning in one of the worst seasons in major league history; his average currently sits at .174.  Or Chris Tillman, a very good pitcher as recently as 2016, who stunk in 2017 and unfathomably stunk even more this year until finally being taken out back and released.  (I saw what may end up being his last major league win.)

So you’re basically taking this entire list of players, who together might have been expected to constitute the core of an respectably mediocre ballclub, and saying that not one of them will play better than you expect, and more than half of them will play worse than you could have reasonably imagined.

I think the Orioles just got snakebit.

But what happens now?  The Orioles haven’t been bad for very long, so they don’t have recent high draft picks.  Machado, Schoop, and Gausman are gone, along with half the bullpen.  Jones might go.  I think in 2019 we are just going to watch Jonathan Villar and Cedric Mullins cavort in front of almost nobody.

I’ll watch that.  And I’ll enjoy it, because I always do.  The losses don’t mean much to me.  Every win makes me proud.

 

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Sinking Ships — The Cinema Clock

This post is to remind myself of this great 1980 release from a band that apparently nobody knows anything about.  I like the way it starts out like a Joy Division rip and then gets much bigger and more interesting (not more interesting than Joy Division, more interesting than a Joy Division rip.)

 

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Hmoob

I’ve lived in Madison, a city with a really big Hmong community, for more than a decade, and I only just now learned something kinda basic about the most common transliteration system for the Hmong language.

Hmong, like Chinese, is tonal.  When you write Chinese in pinyin, you draw a tone mark over each syllable to indicate tone; like mā (‘mother’) or mǎ (‘horse.’)

In Hmong, the tone is indicated by an extra character placed at the end of the syllable.  The character looks like a Roman consonant, but it’s not — it’s a tone mark.  So “Hmoob,” which is the Hmong word for the Hmong language isn’t pronounced to rhyme with “tube” — the syllable ends with a nasalized vowel, and the character “b” is just there to tell you to pronounce the word in a high tone.  “Hmoov” (“flour”) differs from “Hmoob” only in tone.

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