Rational points on solvable curves over Q via non-abelian Chabauty (with Daniel Hast)

New paper up!  With my Ph.D. student Daniel Hast (last seen on the blog here.)

We prove that hyperelliptic curves over Q of genus at least 2 have only finitely many rational points.  Actually, we prove this for a more general class of high-genus curves over Q, including all solvable covers of P^1.

But wait, don’t we already know that, by Faltings?  Of course we do.  So the point of the paper is to show that you can get this finiteness in a different way, via the non-abelian Chabauty method pioneered by Kim.  And I think it seems possible in principle to get Faltings for all curves over Q this way; though I don’t know how to do it.

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Multiple height zeta functions?

Idle speculation ensues.

Let X be a projective variety over a global field K, which is Fano — that is, its anticanonical bundle is ample.  Then we expect, and in lots of cases know, that X has lots of rational points over K.  We can put these points together into a height zeta function

\zeta_X(s) = \sum_{x \in X(K)} H(x)^{-s}

where H(x) is the height of x with respect to the given projective embedding.  The height zeta function organizes information about the distribution of the rational points of X, and which in favorable circumstances (e.g. if X is a homogeneous space) has the handsome analytic properties we have come to expect from something called a zeta function.  (Nice survey by Chambert-Loir.)

What if X is a variety with two (or more) natural ample line bundles, e.g. a variety that sits inside P^m x P^n?  Then there are two natural height functions H_1 and H_2 on X(K), and we can form a “multiple height zeta function”

\zeta_X(s,t) = \sum_{x \in X(K)} H_1(x)^{-s} H_2(x)^{-t}

There is a whole story of “multiple Dirichlet series” which studies functions like

\sum_{m,n} (\frac{m}{n}) m^{-s} n^{-t}

where (\frac{m}{n}) denotes the Legendre symbol.  These often have interesting analytic properties that you wouldn’t see if you fixed one variable and let the other move; for instance, they sometimes have finite groups of functional equations that commingle the s and the t!

So I just wonder:  are there situations where the multiple height zeta function is an “analytically interesting” multiple Dirichlet series?

Here’s a case to consider:  what if X is the subvariety of P^2 x P^2 cut out by the equation

x_0 y_0 + x_1 y_1 + x_2 y_2 = 0?

This has something to do with Eisenstein series on GL_3 but I am a bit confused about what exactly to say.

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Elif Batuman, “The Idiot”

What a novel!  The best I’ve read in quite a while.


One thing I like:  the way this book takes what’s become a standard bundle of complaints against “literary fiction”:

It’s about overprivileged people with boring lives.  Too much writing about writing, and too much writing about college campuses, and worst of all, too much writing about writers on college campuses.   Nothing really happens.  You’re expected to accept minor alterations of feelings in lieu of plot.  

and gleefully makes itself guilty of all of them, while being nevertheless rich in life and incident, hilarious, stirring, and of its time.


Maybe “hilarious” isn’t quite the right word for the way this book is funny, very very funny.  It’s like this:

“Ralph!” I exclaimed, realizing that he was this guy I knew, Ralph.

Whether you find this funny is probably a good test for whether The Idiot is gonna be your thing.


Given this, it’s slightly startling to me that Batuman wrote this essay in n+1, which endorses the standard critique, and in particular the claim that fiction has been pressed into a bloodless sameness by the creative writing workshop.  They bear, as she puts it, “the ghastly imprimatur of the fiction factory.”

What kind of writing bears this stamp?

Guilt leads to the idea that all writing is self-indulgence. Writers, feeling guilty for not doing real work, that mysterious activity—where is it? On Wall Street, at Sloane-Kettering, in Sudan?—turn in shame to the notion of writing as “craft.” (If art is aristocratic, decadent, egotistical, self-indulgent, then craft is useful, humble, ascetic, anorexic—a form of whittling.) “Craft” solicits from them constipated “vignettes”—as if to say: “Well, yes, it’s bad, but at least there isn’t too much of it.” As if writing well consisted of overcoming human weakness and bad habits. As if writers became writers by omitting needless words.

So what’s weird is that Batuman’s writing is exactly the kind that the creative writing workshop leaps to its feet and applauds.  OK, there’s no leaping in creative writing workshop.  It would murmur appreciatively.  Her sentences are pretty damn whittled.  Also clever.  Scenes don’t overspill, they end just before the end.  Batuman’s writing is both crafted and crafty — but not anorexic!  Anorexia isn’t denying yourself what’s needless; it’s a hypertrophy of that impulse, its extension to a more general refusal.

Batuman is really excellent on the convention of the literary short story cold open, which is required to be:

in-your-face in medias res, a maze of names, subordinate clauses, and minor collisions: “The morning after her granddaughter’s frantic phone call, Lorraine skipped her usual coffee session at the Limestone Diner and drove out to the accident scene instead.”  …. A first line like “Lorraine skipped her usual coffee session at the Limestone Diner” is supposed to create the illusion that the reader already knows Lorraine, knows about her usual coffee, and, thus, cares why Lorraine has violated her routine. It’s like a confidence man who rushes up and claps you on the shoulder, trying to make you think you already know him.

Her paradigmatic offender here is the first line of Michael Chabon’s The Amazing Adventures of Kavalier & Clay:

In later years, holding forth to an interviewer or to an audience of aging fans at a comic book convention, Sam Clay liked to declare, apropos of his and Joe Kavalier’s greatest creation, that back when he was a boy, sealed and hog-tied inside the airtight vessel known as Brooklyn New York, he had been haunted by dreams of Harry Houdini.

about which she says:

All the elements are there: the nicknames, the clauses, the five w’s, the physical imprisonment, the nostalgia. (As if a fictional character could have a “greatest creation” by the first sentence—as if he were already entitled to be “holding forth” to “fans.”)

To me this all starts with One Hundred Years of Solitude, which all of us writers read the hell out of in high school, right?  Surely Batuman too?  No kid can read

Many years later, as he faced the firing squad, Colonel Aureliano Buendia was to remember that distant afternoon when his father took him to discover ice.

and not say, oh, that’s how you do it.

Anyway, I’m mostly with Batuman here; once she shows you how it works, the trick is a little corny.  Maybe I already knew this?  Maybe this is why I always preferred the first line of, and for that matter all of, Chabon’s The Mysteries of Pittsburgh to Kavalier & Clay.  Here’s the opening:

At the beginning of summer I had lunch with my father, the gangster, who was in town for the weekend to transact some of his vague business.

In medias res, yes — but not so overstuffed, just one piece of information (the gangster!) presented to start with.  No names.  The word “transact” — boy, there’s nothing I like more than a perfect placement of a boring word.  I think it’s a lot like the first line of The Idiot:

“I didn’t know what email was until I got to college.”

Except Chabon focuses on rhyme (summer-father-gangster) while Batuman is all scansion — perfect trochees!

 


Of course there are a lot of reasons I’m predisposed to like this.  It’s about bookish, ambitious, romantically confused Harvard undergrads, which Batuman and I both were.  There are a lot of jokes in it.  There are some math scenes.

There’s even a biographical overlap:  Batuman, wrote her college novel right after college, just like I did.  And then she finished her Ph.D. and put the manuscript in a drawer for a long time, just like I did.  (I don’t know if she carried out the intermediate step, as I did, of getting the book rejected by every big commercial house in New York.)  And then at some point in the run-up to middle age she looked at those pages again and said words to the effect of “This is not actually that bad…”

So let me say it straight; The Idiot makes me think about the alternate universe where I stayed a novelist instead of going back to grad school in math, a universe where I spent years working really hard to sharpen and strengthen the work I was doing.  This is the kind of novel I would have been aiming my ambition at writing; and I still wouldn’t have done it this well.  The existence of The Idiot releases me from any regrets.

(I don’t have many.  Math, for me, is fun.  Writing fiction is not.)

 

 

 

 

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What is the median length of homeownership?

Well, it’s longer than it used to be, per Conor Dougherty in the New York Times:

The median length of time people have owned their homes rose to 8.7 years in 2016, more than double what it had been 10 years earlier.

The accompanying chart shows that “median length of homeownership” used to hover at  just under 4 years.  That startled me!  Doesn’t 4 years seem like a pretty short length of time to own a house?

When I thought about this a little more, I realized I had no idea what this meant.  What is the “median length of homeownership” in 2017?  Does it mean you go around asking each owner-occupant how long they’ve lived in their house, and take the median of those numbers?  Probably not:  when people were asked that in 2008, the median answer was 10 years, and whatever the Times was measuring was about 3.7 years in 2008.

Does it mean you look at all house sales in 2017, subtract the time since last sale, and take the median of those numbers?

Suppose half of all houses changed hands every year, and the other half changed hands every thirty years.  Are the lengths of ownership we’re medianning half “one year” and half “30 years”, or “30/31 1 year” and 1/31 “30 years”?

There are about 75 million owner-occupied housing units in the US and 4-6 million homes sold per year, so the mean number of sales per unit per year is certainly way less than 1/4; of course, there’s no reason this mean should be close to the median of, well, whatever we’re taking the median of.

Basically I have no idea what’s being measured.  The Times doesn’t link to the Moody’s Analytics study it’s citing, and Dougherty says that study’s not public.  I did some Googling for “median length of homeownership” and as far as I can tell this isn’t a standard term of art with a consensus definition.

As papers run more data-heavy pieces I’d love to see a norm develop that there should be some way for the interested reader to figure out exactly what the numbers in the piece refer to.  Doesn’t even have to be in the main text.  Could be a linked sidebar.  I know not everybody cares about this stuff.  But I do!

 

 

 

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Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields

I’ve gotten behind on blogging about preprints!  Let me tell you about a new one I’m really happy with, joint with TriThang Tran and Craig Westerland, which we posted a few months ago.

Malle’s conjecture concerns the number of number fields with fixed Galois group and bounded discriminant, a question I’ve been interested in for many years now.  We recall how it goes.

Let K be a global field — that is, a number field or the function field of a curve over a finite field.  Any degree-n extension L/K (here L could be a field or just an etale algebra over K — hold that thought) gives you a homomorphism from Gal(K) to S_n, whose image we call, in a slight abuse of notation, the Galois group of L/K.

Let G be a transitive subgroup of S_n, and let N(G,K,X) be the number of degree-n extensions of K whose Galois group is G and whose discriminant has norm at most X.  Every permutation g in G has an index, which is just n – the number of orbits of g.  So the permutations of index 1 are the transpositions, those of index 2 are the three-cycles and the double-flips, etc.  We denote by a(G) the reciprocal of the minimal index of any element of G.  In particular, a(G) is at most 1, and is equal to 1 if and only if G contains a transposition.

(Wait, doesn’t a transitive subgroup of S_n with a transposition have to be the whole group?  No, that’s only for primitive permutation groups.  D_4 is a thing!)

Malle’s conjecture says that, for every \epsilon > 0, there are constants c,c_\epsilon such that

c X^{a(G)} < N(G,K,X) < c_\epsilon X^{a(G)+\epsilon}

We don’t know much about this.  It’s easy for G = S_2.  A theorem of Davenport-Heilbronn (K=Q) and Datskovsky-Wright (general case) proves it for G = S_3.  Results of Bhargava handle S_4 and S_5, Wright proved it for abelian G.  I kind of think this new theorem of Alex Smith implies for K=Q and every dihedral G of 2-power order?  Anyway:  we don’t know much.  S_6?  No idea.  The best upper bounds for general n are still the ones I proved with Venkatesh a long time ago, and are very much weaker than what Malle predicts.

Malle’s conjecture fans will point out that this is only the weak form of Malle’s conjecture; the strong form doesn’t settle for an unspecified X^\epsilon, but specifies an asymptotic X^a (log X)^b.   This conjecture has the slight defect that it’s wrong sometimes; my student Seyfi Turkelli wrote a nice paper which I think resolves this problem, but the revised version of the conjecture is a bit messy to state.

Anyway, here’s the new theorem:

Theorem (E-Tran-Westerland):  Let G be a transitive subgroup of S_n.  Then for all q sufficiently large relative to G, there is a constant c_\epsilon such that

N(G,\mathbf{F}_q(t),X) < c_\epsilon X^{a(G)+\epsilon}

for all X>0.

In other words:

The upper bound in the weak Malle conjecture is true for rational function fields.

A few comments.

  1.  We are still trying to fix the mistake in our 2012 paper about stable cohomology of Hurwitz spaces.  Craig and I discussed what seemed like a promising strategy for this in the summer of 2015.  It didn’t work.  That result is still unproved.  But the strategy developed into this paper, which proves a different and in some respects stronger theorem!  So … keep trying to fix your mistakes, I guess?  There might be payoffs you don’t expect.
  2. We can actually bound that X^\epsilon is actually a power of log, but not the one predicted by Malle.
  3. Is there any chance of getting the strong Malle conjecture?  No, and I’ll explain why.  Consider the case G=S_4.  Then a(G) = 1, and in this case the strong Malle’s conjecture predicts N(S_4,K,X) is on order X, not just X^{1+eps}.   But our method doesn’t really distinguish between quartic fields and other kinds of quartic etale algebras.  So it’s going to count all algebras L_1 x L_2, where L_1 and L_2 are quadratic fields with discriminants X_1 and X_2 respectively, with X_1 X_2 < X.  We already know there’s approximately one quadratic field per discriminant, on average, so the number of such algebras is about the number of pairs (X_1, X_2) with X_1 X_2 < X, which is about X log X.  So there’s no way around it:  our method is only going to touch weak Malle.  Note, by the way, that for quartic extensions, the strong Malle conjecture was proved by Bhargava, and he observes the same phenomenon:

    …inherent in the zeta function is a sum over all etale extensions” of Q, including the “reducible” extensions that correspond to direct sums of quadratic extensions. These reducible quartic extensions far outnumber the irreducible ones; indeed, the number of reducible quartic extensions of absolute discriminant at most X is asymptotic to X log X, while we show that the number of quartic field extensions of absolute discriminant at most X is only O(X).

  4.  I think there is, on the other hand, a chance of getting rid of the “q sufficiently large relative to G” condition and proving something for a fixed F_q(t) and all finite groups G.

 

OK, so how did we prove this?

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I guess Caffe 608 was in trouble

Eight years after I wondered whether the arthouse cinema / cafe in Hilldale could really make a go of it, Sundance 608 is getting bought out by AMC.  I have really come to like this weird little sort-of-arthouse and hope it doesn’t change too much under new management.  It’s a sign of my age, I guess, that I still think of “movie at the mall” as an entertainment option I want to exist.  It’s my Lindy Hop, my vaudeville, my Show of Shows.

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Is academia wrong for you?

Good article by Daniel McCormack in Chronicle of Higher Education on underpublicized aspects of academic life.

For instance:

These iterative failures are, at a very deep level, the essence of creating new knowledge, and are therefore inseparable from the job. If you can’t imagine going to bed at the end of nearly every day with a nagging feeling that you could have done better, academe is not for you.

The academic workplace is a really unusual one.  For instance, it’s one of the few places to work where you’re nobody’s boss and nobody’s your boss.  It really suits some people — I’m one.  But lots of other people feel otherwise: it’s too slow, too lacking in immediate feedback, too content with the way “it’s always been done.”  And a lot of those people have great things to contribute to mathematics and don’t fit in the system we’ve set up to pay people to do math.

Also, this:

So while the ideal career path leads from graduate school to a tenure-track position, the one you will more likely find yourself on leads from graduate school to a series of short-term positions that will require you to move — often.

is less true in math than in many other areas, but still kind of true.  And it works badly not just for people who temperamentally hate moving, but for people who want to have kids and have a limited childbearing window.

McCormack is right:  without catastrophizing, we should always be trying to give our Ph.D. students as real a picture as possible of what academic life is like, and not just the advisor’s life with tenure at an R1 university.  Lots of people will still happily sign up.  But other people will think more seriously about other great ways to do mathematics.

 

 

 

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Intersectionality as nonlinearity

I wonder if the idea of intersectionality would be better-understood in STEMmy circles if we called it “nonlinearity” instead.  Put that way, e.g.

“the condition of being queer and disabled isn’t the sum of the condition of being queer and the condition of being disabled, or even some linear combination of those, it’s just its own thing, which draws input from each of those conditions in some more complicated way and which has features of its own particular to the intersection”

it’s something I think most mathematicians would find extremely natural and intuitive.

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The past was bad

It’s looking tonight like the GOP will manage to pass some version of the AHCA, a bill repealing the Affordable Care Act and creating some kind of return to the pre-ACA status quo; hard to know exactly what, since the vote will be taken without the bill being publicly released, and the House has decided not to wait for the Congressional Budget Office to estimate just how much this bill will cost Americans.

GOP fans will say: “How can this be such a big disaster, crying liberals?  Ten years ago there was no Obamacare, and people did fine.”

Some people did fine!  Some people didn’t do fine.

You’ll hear people say, in the same sad snappish tone of voice, “Parents today are obsessed with safety, in my day kids rode in the way back of the station wagon, they didn’t wear seatbelts, they crossed the street by themselves, and they were fine.”

Some kids were fine!  But just so you know:  in 1975, about 1600 kids 13 and under were killed by cars as pedestrians, and another 1400 were killed in crashes while riding in cars.  In 2015, those numbers were 186 and 663.  Throw in teenagers and that’s another 8700 dead passengers in 1975; down to 2715 in 2015.

People did fine, except for the thousands of kids who got killed back then who wouldn’t get killed now.

A while ago I was reading the reunion book for the Harvard class of 1893, the people who graduated exactly 100 years before me.  You know what you notice in their bios?  A lot of people’s children died.  In 1920, about 8% of American babies died before the age of 1.  It’s now 0.6%.

People were fine!  They had a baby, the baby died, they got on with their life.

But I like it better when babies hardly ever die, when thousands of children don’t get killed in car crashes, and when Americans have access to affordable health insurance even if they’ve been sick before.  The past was fine.  But it was also bad.

 

 

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Fitchburg facts

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