## Expander graphs, gonality, and variation of Galois representations

Suppose you have a 1-dimensional family of polarized abelian varieties — or, just to make things concrete, an abelian variety A over Q(t) with no isotrivial factor.

You might have some intuition that abelian varieties over Q don’t usually have rational p-torsion points — to make this precise you might ask that A_t[p](Q) be empty for “most” t.

In fact, we prove (among other results of a similar flavor) the following strong version of this statement.  Let d be an integer, K a number field, and A/K(t) an abelian variety.  Then there is a constant p(A,d) such that, for each prime p > p(A,d), there are only finitely many t such that A_t[p] has a point over a degree-d extension of K.

The idea is to study the geometry of the curve U_p parametrizing pairs (t,S) where S is a p-torsion point of A_t.  This curve is a finite cover of the projective line; if you can show it has genus bigger than 1, then you know U_p has only finitely many K-rational points, by Faltings’ theorem.

But we want more — we want to know that U_p has only finitely many points over degree-d extensions of K.  This can fail even for high-genus curves:  for instance, the curve

C:   y^2 = x^100000 + x + 1

has really massive genus, but choosing any rational value of x yields a point on C defined over a quadratic extension of Q.  The problem is that C is hyperelliptic — it has a degree-2 map to the projective line.  More generally, if U_p has a degree-d map to P^1,  then U_p has lots of points over degree-d extensions of K.  In fact, Faltings’ theorem can be leveraged to show that a kind of converse is true.

So the relevant task is to show that U_p admits no map to P^1 of degree less than d; in other words, its gonality is at least d.

Now how do you show a curve has large gonality?  Unlike genus, gonality isn’t a topological invariant; somehow you really have to use the geometry of the curve.  The technique that works here is one we learned from an paper of Abramovich; via a theorem of Li and Yau, you can show that the gonality of U_p is big if you can show that the Laplacian operator on the Riemann surface U_p(C) has a spectral gap.  (Abramovich uses this technique to prove the g=1 version of our theorem:  the gonality of classical modular curves increases with the level.)

We get a grip on this Laplacian by approximating it with something discrete.  Namely:  if U is the open subvariety of P^1 over which A has good reduction, then U_p(C) is an unramified cover of U(C), and can be identified with a finite-index subgroup H_p of the fundamental group G = pi_1(U(C)), which is just a free group on finitely many generators g_1, … g_n.  From this data you can cook up a Cayley-Schreier graph, whose vertices are cosets of H_p in G, and whose edges connect g H with g_i g H.  Thanks to work of Burger, we know that this graph is a good “combinatorial model” of U_p(C); in particular, the Laplacian of U_p(C) has a spectral gap if and only if the adjacency matrix of this Cayley-Schreier graph does.

At this point, we have reduced to a spectral problem having to do with special subgroups of free groups.  And if it were 2009, we would be completely stuck.  But it’s 2010!  And we have at hand a whole spray of brand-new results thanks to Helfgott, Gill, Pyber, Szabo, Breuillard, Green, Tao, and others, which guarantee precisely that Cayley-Schreier graphs of this kind, (corresponding to finite covers of U(C) whose Galois closure has Galois group a perfect linear group over a finite field) have spectral gap; that is, they are expander graphs. (Actually, a slightly weaker condition than spectral gap, which we call esperantism, is all we need.)

Sometimes you think about a problem at just the right time.  We would never have guessed that the burst of progress in sum-product estimates in linear groups would make this the right time to think about Galois representations in 1-dimensional families of abelian varieties, but so it turned out to be.  Our good luck.

## Iranian election statistics — never mind the digits?

I blogged last year about claims that fraud in the 2009 Iranian election could be detected by studying irregularities in the distribution of terminal digits.  Eric A. Brill just e-mailed me an article of his which argues against this methodology, pointing out that the provincial vote totals (the ones with the fishy last digits) agree with the sums of the county totals, which in turn agree with the sums of the district totals.  In order for the provincial totals to have been made up, you’d have to change a lot of county totals too (changing the total in just one county by a believable amount presumably wouldn’t make a big enough difference in the provincial totals.)  But if you add Ahmadinejad votes to a county here and a county there, the provincial total would be the sum of a bunch of human-chosen numbers, and there’s no reason to expect such a sum to have non-uniformly distributed last digits.  The Beber-Scacco model requires that the culprits start with a target number at the provincial level and then carefully modify county and district level numbers to make the sums match.  But why would they?

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Canada’s chief statistician resigned last month in protest of the government’s decision to replace the long-form census questionnaire, previously mandatory for 20 percent of the population, with a voluntary version.   I imagine the point is that a voluntary questionnaire can’t possibly be delivering anything like a random sample of the population, though the linked article doesn’t make the statistical issues very clear.  The new head census-taker says the voluntary survey will be “usable and useful” but not comparable to the previous census.   Do I have any Canadian readers who can explain to me why the government thought this was a good idea?

And for Americans:  did you know that the response rate for the Canadian census is 96%?  Whoa.

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## In which I make oblique reference to a change in my personal circumstances

When you fill out a birth certificate in Wisconsin, there’s a “Mother’s Information” section and a “Husband’s Information” section.  If you’re unmarried, you’re not allowed to put the father’s name on the birth certificate.  You have to leave it blank, and petition the State Vital Records Office after the fact to get the father included.  And if you are married, you have to put the husband on the birth certificate, whether or not he’s the father of the child.   In fact, if you’re married to Mr. X, conceive a child by Mr. Y, and subsequently get divorced from Mr. X, the ex-Mrs.X still has to put Mr. X on the birth certificate.  He can only be removed by court order.

What can the rationale for this be?  I guess it must arise from acrimonious cases where the paternity of baby X is really unknown, but Mr. X, angry at having been cheated on and dumped, insists, rightly or not, that the baby is not his, and refuses to pay child support.

It is not at all clear how you’re supposed to fill out the form if you’ve been married to more than one man over the course of the pregnancy.

(Mrs. Q would like me to clarify that the abovementioned change in my personal circumstances is the one which entails filling out a birth certificate request with the State of Wisconsin, and does not involve any alterations in marital status, unknown biological parentage, or outstanding claims of child support.)

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## Math busking

Now I know what to say next time the dean asks us for some innovative fundraising ideas for the math department.

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## Moleeds

A clip of  Charles Fleischer,  a stand-up comic, wearing an endigitted blazer and performing a routine with a lot of numerology in it:

I think the very first joke in this is funny and concise, but it quickly degenerates into a kind of sub-Robin-Williams “I talk loudly and quickly and change accents a lot and am kind of manic, is it funny yet?  No?  LOUDER, QUICKER, MORE ACCENTS!” schtick.

But the joke is on us, because Fleischer’s not kidding about his theory of “moleeds.”  In 2005 he gave a TED talk about it.  This is a weird and in some ways uncomfortable thing to watch — the audience still thinks they’re watching a comedy routine, and just keeps chuckling while Fleischer argues, with ever-increasing fervor,  that the equation 27 x 37 = 999 somehow explains mirror symmetry and the theory of Calabi-Yau manifolds.

The talk doesn’t cast TED in the best light, to be honest.  Don’t they have someone on staff who can do some minimal vetting of talks that claim to be about math?

(Note:  there is always the possibility that Fleischer’s whole act is an extravagantly thorough Kaufmannesque send-up of people’s tendency to attach themselves to meaningless patterns and theories.  But it doesn’t read that way to me.)

## California prayer agency, Earth-2 Lawrence v. Texas

Continuing from the last post, here’s a test case for the view that judges applying the “rational basis” must defer completely to referenda.

Suppose the voters of California pass a referendum instituting a state agency which employed people to pray to Jesus for the health of sick Californians.  Can a judge declare this a 1st amendment violation, or not?  Surely prolonging the lives of thousands of citizens constitutes a legitimate state interest, and, per Kennedy’s opinion in the last post, it is not the government’s responsibility to provide evidence that the referendum would aid that interest, nor the judge’s responsibility to consider such evidence.  On what basis could the referendum be unconstitutional?

Actually, looking this up, it seems that a law violating the Establishment Clause triggers (at least sometimes) “strict scrutiny,” a more stringent requirement than “rational basis.”  I expect the referendum above would be axed on that basis.   Racially discriminatory laws have to pass strict scrutiny as well.   But discrimination against gays triggers only the weaker rational basis test.

Justice O’Connor wrote in her concurrence in Lawrence v. Texas that Texas’s law forbidding same-sex sodomy failed the rational basis test, because it was motivated solely by moral disapproval, rather than by a legitimate state interest.”  (By the way, O’Connor writes in the same opinion that “preserving the traditional institution of marriage” is a legitimate state interest.)

Question: Suppose the lawyers for Texas had argued, without providing any evidence, that the state felt same-sex sodomy was more likely than opposite-sex sodomy to promote unspecified disease.  Would the law have still been held unconstitutional?  Or would it have met the rational basis standard?

## Gay marriage and the null hypothesis

Two controversial topics in one post!

Several of the key factual findings in Judge Walker’s opinion are in the form of predictions, not facts. For example, Judge Walker finds that “permitting same-sex couples to marry will not . . . otherwise affect the stability of opposite-sex marriages.” But real predictions have confidence levels. You might think you’re going to get an “A” on an exam next week, but that’s not a fact. It’s just a prediction, and there’s a hidden confidence level: Maybe there’s an 80% chance you’ll get that grade, or a 60% chance. Judge Walker’s prediction-facts have no confidence levels, however. He doesn’t say that there is an 87% chance that permitting same-sex marriage will not affect the stability of opposite-sex marriages. He says that it is now a fact — with 100% certainty — that that will happen.

I think Kerr is incorrect about Walker’s meaning.  When we say, for instance, that a clinical trial shows that a treatment “has no effect” on a disease, we are certainly not saying that, with 100% certainty, the treatment will not change a patient’s condition in any way.  How could we be?  We’re saying, instead, that the evidence before us gives us no compelling reason to rule out “the null hypothesis” that the drug has no effect.  Elliott Sober writes well about this in Evidence and Evolution.  It’s unsettling at first — the meat and potatoes of statistical analysis is deciding whether or not to rule out the null hypothesis, which as a literal assertion is certainly false!  It’s not the case that not a single opposite-sex marriage, potential or actual, will be affected by the legality of same-sex marriage; Walker is making the more modest claim that the evidence we have doesn’t provide us any ability to meaningfully predict the size of that effect, or whether it will on the whole be positive or negative.

This doesn’t speak to Kerr’s larger point, which is that Walker’s finding of fact might not be relevant to the case — California can outlaw whatever it wants without any evidence that the outlawed thing causes any harm, as long as it has a “rational basis” for doing so.  The key ruling here seems to be Justice Kennedy’s in Heller v. Doe, which says:

A State, moreover, has no obligation to produce evidence to sustain the rationality of a statutory classification. “[A] legislative choice is not subject to courtroom factfinding and may be based on rational speculation unsupported by evidence or empirical data.”

and later:

True, even the standard of rationality as we so often have defined it must find some footing in the realities of the subject addressed by the legislation.

I’m in the dark about what Kennedy can mean here.  If speculation is unsupported by evidence, in what sense is it rational?  And what “footing in the realities of the subject” can it be said to have?

More confusing still:   in the present case, the legislation at issue comes from a referendum, not the legislature.  So we have no record to tell us what kind of speculation, rational or not, lies behind it — or, for that matter, whether the law is intended to serve a legitimate government interest at all.  Maybe there is no choice under the circumstances but for the “rational basis” test to be no test at all, and for the courts to defer completely to referenda, however irrational they may seem to the judge?

(Good, long discussion of related points, esp. “to what extend should judges try to read voters’ minds,” at Crooked Timber.)

## Turkey burgers, gazpacho, Paul Robeson

It’s hard to make a turkey burger taste good. You kind of need to season the hell out of it. We mixed a pound of ground turkey with a minced half-onion, a couple of cloves of garlic, an egg, and — CJ’s idea — 1/2 tsp each cinnamon and cumin.  Kind of a turkofta.  Onions keep it from getting dry, spices keep it from getting bland.  I blog it in order to remember it.

In other news, this New York Times gazpacho smoothie is ace and we’ve been making it three times a week.  The suggested pecorino crackers are too salty and unnecessary.

We’re in the heart of tomato season now and I’m buying about 10 lb a week.  Did you know there was a Paul Robeson tomato?  Once you sang on Broadway and battled for civil rights, Paul Robeson.  Now you are in my smoothie.

## Show report: New Pornographers at the Orpheum

New Pornographers played the Orpheum last night.

• Boston’s “Foreplay” on the sound system before the band comes on.  Comes off as witty.
• On the records Dan Bejar’s singing doesn’t stand out as much as it does live.  Something about the way he approaches the microphone makes him look like he’s always about to rap rather than sing.  Bejar leaves the stage during the songs he’s not singing.  This seems churlish to me.  He couldn’t just stand there and bang a tambourine on his hip?
• “My Slow Descent into Alcoholism,” the best song they ever wrote — probably my favorite song anybody released last decade — appears in the encore.  It is great, but all live versions lack the precision which is part of the glory of the studio version — precision married to absolutely unmoderated rocking-out-ness.  See:
• Kathryn Calder, once an occasional vocal stand-in for Neko Case, is now a full member of the band, playing keyboards and singing backup.  Both facially and in manner she reminds me very powerfully of Doris Finsecker.

Calder:

Finsecker:

• Show ends with “Testament to Youth in Verse.”  Openers the Dodos come on stage, everybody’s singing the big “no no no…” at the end of the song, swaying, waving goodbye, drinking beers.  The cellist in the back picks up a saxophone.  It’s an almost exact replica of the credit sequence of Saturday Night Live. On purpose?