## Can the trace hear the shape of its field?

The most natural arithmetic invariant of a number field K is its discriminant D_K, an integer congruent to either 0 or 1 mod 4 whose prime factors are precisely the primes where K/Q is ramified.  Oftentimes D_K is a squarefree, in which case it’s just the product of the ramified primes; even if not, the multiplicity of a prime factor of D_K can be described quite cleanly in terms of the p-inertia subgroup of Gal(K/Q).

The situation is especially handsome for quadratic field.  The discriminants of quadratic fields are just those integers congruent to 0 or 1 mod 4 which have no square factor larger than 4.  Better still, the discriminant specifies the field uniquely!  So in order to describe a quadratic field it suffices to write down a single integer.

Life gets worse in higher degree.  There can be lots of number fields with the same discriminant D.  For example:  if D is squarefree and K is a cubic field with discriminant D, then the Galois closure L of K is an unramified (Z/3Z)-extension of the quadratic field M with discriminant D.  So if the ideal class group of M has a lot of (Z/3Z) in it, there are going to be a lot of cubic fields with discriminant D!

Just how bad is this multiplicity?  It’s widely believed that, for every n, there are at most D^eps number fields of discriminant D.  But I think nobody has a good idea about how to prove this, even for n=3.

So it’s naturally interesting to ask whether there are other invariants which might uniquely specify a number field.  Here’s one natural candidate.  The ring of integers O_K is a free rank n Z-module, endowed with a natural quadratic form q(x,y) = Tr_{K/Q}(xy), called the trace form.  The discriminant of this trace form is, up to known factors, the discriminant of K.  So you can think of the isomorphism class of the trace form as a refinement of the discriminant.  The question is:  is it such a good refinement that it actually specifies the field?

My former student Guillermo Mantilla-Soler, now at UBC,  just posted a preprint offering the first real insight into this question, which he colorfully phrases “Can the trace hear the shape of its field?”  He shows that the answer to the original question is no:  for instance, he displays two non-isomorphic quintic fields of discriminant 34129 which have isomorphic trace form.  More generally, he gives a necessary condition which I would expect is satisfied by examples in every degree (though it might be hard to prove this.)

But in some sense this is a local issue; the fields in these examples are not totally real, so the trace forms aren’t definite, and so, as Guillermo observes, it suffices to show the forms lie in the same spinor genus.  In the definite case, it’s “harder” for quadratic forms to be isomorphic.  Are there two non-isomorphic totally real number fields with isomorphic trace forms?  Guillermo includes the results of a fairly large computer search which finds no examples in degree < 10 and discriminant < 10^9.

## Angry Birds is less popular than television

Andrew Sullivan, via Hillel Fuld, reports:

Another mind boggling statistic about Angry Birds, and you should sit down for this one, is that there are 200 million minutes played a day on a global scale. As Peter [Verterbacka, Angry Birds creator] put it, that number compares favorably to anything, including prime time TV, which indicates that 2011 will be a big year in the shift of advertisers’ attention from TV to mobile.

Americans alone watch a mean of 5 hours of television per day.  Let’s say half of that is prime time.   300 million Americans times 150 minutes is 45 billion minutes a day, and we haven’t counted any TV usage anywhere else in the world.  The popularity of Angry Birds does not “compare favorably” to the popularity of TV.

Update:   On the other hand, it’s quite reasonable to suppose that there are lots of popular TV shows that occupy fewer global person-minutes than Angry Birds does.  So the claim that advertisers should be taking the idea of advertising on mobile games, just as they do on TV shows, sounds fair to me.

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## Is Zipcar offering too little insurance?

Ron Lieber in the New York Times says Zipcar is underinsuring its customers:

Today, customers who are 21 or older have \$300,000 of liability coverageper accident. That would have to cover mangled limbs, brain damage, pain and suffering and anything else that might befall all the people that a Zipcar vehicle mowed down or plowed into.

Drivers under 21 get much less coverage. Zipcar would have to pay a lot of money to provide \$300,000 in coverage to less-experienced college-age drivers, and it figures that most of its users in this age group are covered by their parents’ auto policies anyway. So Zipcar does as little as possible here, offering each state’s minimum requirements and no more….

Zipcar members who do not read the disclosures on the company’s Web site would never know about any of this. And many of them don’t, since the company has persisted with the claim elsewhere on its site that its insurance is “comprehensive.”

Wouldn’t a lawyer for an injured person or the family of an accident victim go after Zipcar first, since that’s where the money is? They could try, but a federal law shields rental car companies in many instances, and Zipcar has already cited it in at least one legal skirmish over someone injured in an accident involving a Zipcar.

Just in case, however, Zipcar still insures itself. In a filing accompanying its initial public offering, the company noted that in the event that it was responsible for an accident, say because it failed to maintain its cars, it had coverage up to \$5 million in the United States. That is more than 16 times the maximum protection that it offers its members.

I have no position on how much liability insurance a driver ought to have.  But every advice on this I’ve ever read agrees that the amount of insurance you want goes up with your assets.  It would be truly weird if Zipcar didn’t insure itself against much, much greater losses than its customers do.

According to a Zipcar spokeswoman, Colleen McCormick, the \$300,000 in coverage has been adequate for every accident since it began operations. She added that more than half of accidents involve only the Zipcar vehicle itself. When another car is involved, 93 percent of the accidents have resulted in claims of less than \$10,000, and 99.3 percent result in claims of less than \$50,000.

That makes the company pretty lucky. Sure, accidents with injuries are rare, but what happens when they do occur? According to ISO, a data provider to insurance companies, about 2 percent of bodily injury liability insurance claims in the United States are for more than \$300,000; in the State of New York, it’s 3 percent.

For brain damage in a vehicular accident, the median jury award in 2008, the most recent year for which data was available, was \$289,793, according to Jury Verdict Research, which compiles the data and publishes it. For leg injuries, the median was \$192,775.

Not one but two devious coordinate changes here.  Zipcar says that claims of more than \$50,000 make up .7% of half of all claims, so about .4%.  Lieber argues that this makes them very lucky:  3% of claims in New York State are for more than \$300,000.  But look closely — the second number only applies to claims involving bodily injury — obviously these will be the most expensive class.  How frequent are bodily injury claims?  Lieber gives no hint.  But he does go on, rather slyly in my view, to narrow his focus to an even smaller class of claims, those involving brain injuries — now the median size of claim is already close to \$300K.  But how many bodily injuries involve brain injuries?  Again, we’re in the dark.

I heartily endorse the genre of business piece that uses numbers as part of an argument.  But you need to include all the relevant numbers!

One number that’s plainly important:  how often do Zipcar customers end up facing liabilities that exceed what they’re insured for?  Credit to Lieber for answering this one — it’s “never.”

“Never in 10 million drives has a single person had to come out of pocket” for a liability claim, said Rob Weisberg, Zipcar’s chief marketing officer. “Our coverage is two times our next-largest competitor, and our coverage is greater than most Americans have who insure their personally owned vehicles.”

That doesn’t make those Americans adequately covered. And the logic here strikes me as backward. Insurance is supposed to be for things that would be financially catastrophic. To sell protection against a three-figure fee while leaving members exposed to a seven-figure judgment doesn’t make much sense.

So if you’re a Zipcar member, as I am, now you know what the worst case looks like. Still feeling comfortable with the company’s coverage?

To sum up:  Zipcar is mistreating its customers by failing to offer a service that most of them don’t want, and which, if offered, would not have been used in the company’s entire history.

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## Living in the past

CJ and I played Galaga last night.  Galaga is great.  When those spaceships go off the bottom of the screen, except they haven’t quite gone all the way off the bottom of the screen, and then they curl back up and destroy you — that is classic.

Another thing I cannot deny still liking is “Living in the Past,” by Jethro Tull.

And here is Galaga.  I was surprised to find I still knew the background music by heart, after all these years.

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## Living in the future

I just learned that my father orders his breakfast cereal from Amazon.

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## Stephen Burt interviewed in Publishers Weekly

Steve Burt interviewed in the PW series, “The Art of the Review:”

Classes can reveal the properties of their members more fully (to understand the differences between calcium and magnesium, for example, you should know why they are both alkaline earths) but classes can also obscure them (the Pagans and the Germs were both American punk rock bands, but to me their songs sound nothing alike). Classes should be used with care everywhere; there’s probably no way to fully avoid them.

But you aren’t asking about classes in general; you are asking why poetry critics and reviewers seem to classify and classify, whereas fiction reviews try to avoid it. Perhaps it’s because few books of poetry can count on a buzz produced by their authors, or by a publicity campaign, or by grassroots, independent-bookstore-sales-driven chatter, all of which can justify (to assigning editors, to casual readers) space and time for extensive reviews of single volumes. Poetry reviewers, poetry critics, even very academic ones, need other pegs on which to hang their claims.

Novelists, necessarily, work in sustained solitude, when they are working (however gregarious they become otherwise), whereas poets can work in solitude in short bursts and then come together to discuss—and make programs and slogans about—what they made.

Poets also seem to attach themselves and their work more often either to their peer group, or to their teachers; some poets can tell you where and with whom they studied almost in the way that classical musicians can tell you about their teachers, and their teachers’ teachers.  If novelists do that, I haven’t seen it.

For more, buy Steve’s book, Close Calls With Nonsense.

## Close election? Flip for it

I have an op-ed in today’s Washington Post advocating the use of randomization devices to determine the winner of close elections.

Some will balk at the idea of choosing our leaders by chance. But that’s actually the coin flip’s most important benefit! Close elections are already determined by chance. Bad weather in the big city, a busted voting machine in an outlying town, a poorly designed ballot leading elderly Jews to vote for Pat Buchanan — any of these chance events can make the difference when the electorate is stuck at 50-50.

A note for the many people who either e-mailed me or posted comments to say that I was a nutty leftist who would never have written this if the more liberal candidate in the Wisconsin supreme court election were ahead: in fact, I pitched this when Kloppenburg appeared to be leading by 200 votes.  The correction of the Waukesha numbers, which made the election much less close, was thus inconvenient for both Kloppenburg and me.  But I just rejiggered the piece to place a greater emphasis on very close votes from the past (Franken-Coleman, Bush-Gore.)  Not sure why I neglected to include the equally tight Gregoire-Rossi WA-GOV race.  Or how I failed to notice that Charles Seife, whose book I quote in the piece, also wrote essentially the same editorial in the New York Times two years ago.

## It matters little whether we progress with understanding the diophantine approximation

As to the relative importance of soft and hard science for humanity’s future, there can be no comparison.  It matters little whether we progress with understanding the diophantine approximation.  Our survival depends on whether we progress with understanding how people behave, why some societies become frustrated, whether their governments tend to become unstable, and how political leaders make decisions like whether to press a red button.

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## Under the bridge

CJ:  “Trolls are not nice.”

Me:  “That’s true, but they aren’t real.”

CJ:  “Some trolls are real, they are on the computer and they start flame wars.”

## Two more maps of Wisconsin

Looking now like David Prosser will hold on to a narrow victory in the final count with about 50.2% of the vote.

From the Huffington Post, a map showing the county-by-county change from Scott Walker’s share of the vote to David Prosser’s. The image on the original post has a nice mouse-over where you can see the actual numbers.  At first glance, the picture doesn’t bode well for Wisconsin politics — the most Democratic parts of the state got more Democratic, and the most Republican parts got more Republican.  (Or so it looks — I didn’t, y’know, actually make a spreadsheet.)  Combined with the inevitable bitterness that follows a close election, I think we’re looking at another couple of years of state politics carried out in Manichean deathmatch mode.  People with direct knowledge tell me the State Supreme Court has been operating that way for years already.

And here’s a map showing the geographic distribution of ethnicities in Wisconsin in 1900.  The Democratic/Republican line from northwest to southeast is also the Norwegian/German line.