## Non-simple abelian varieties in a family

Here’s a funny question. Let f in C[x] be a squarefree polynomial of degree at least 6. Let S be the set of complex numbers t such that the Jacobian of the hyperelliptic curve

$y^2 = f(x)(x-t)$

is not simple. Is S always finite? Even more, is there a bound on |S| which doesn’t depend on f, or depends only on the degree of f?

This question comes from the introduction to “Non-simple abelian varieties in a family: geometric and analytic approaches” , a new paper by me, Christian Elsholtz, Chris Hall, and Emmanuel Kowalski. In its original form this was a four-author, six-page paper — fortunately we’ve now added enough material to make the ratio a bit more respectable!

The paper isn’t about complex algebraic geometry at all — it explains how to get bounds on S when f has rational coefficients and t ranges over rational numbers, which is quite a different story. The point of the paper is partly to prove some theorems and partly to make a metamathematical point — that problems of this kind can be approached via either arithmetic geometry or analytic number theory, and that the two approaches have complementary strengths and weaknesses. Bounds from arithmetic geometry are stronger but less uniform; bounds from analytic number theory are weaker but have better uniformity.

Here’s my favorite example of this phenomenon. Let X be a smooth plane curve over Q of degree d at least 4. Then by Faltings’ Theorem we know that X has only finitely many rational points.

On the other hand, a beautiful theorem of Heath-Brown tells us that the number of rational points on X with coordinates of height at most B is at most C B^(2/d), for some constant C depending only on d. At first, this seems to give much less than Faltings. After all, as B gets larger and larger, the upper bound given by Heath-Brown gets arbitrarily large — whereas we know by Faltings that there are only finitely many points on the whole curve, no matter how large we allow the coordinates to be.

But note that the constant in Heath-Brown’s result doesn’t depend on the curve X. It is what’s called a uniform bound. Faltings’ theorem, by contrast, gives an upper bound on the number of points which depends very badly on the choice of X. Depending on what you’re trying to accomplish, you might be willing to sacrifice uniformity to get finiteness — or the reverse. But it’s best to have both options at hand.

Is it possible to have uniformity and finiteness simultaneously? Conjecturally, yes. Caporaso, Harris, and Mazur showed that, conditional on Lang’s conjecture, there is a constant B(g) such that every genus-g curve X/Q has at most B(g) rational points. The Caporaso-Harris-Mazur paper came out when I was in graduate school, and the idea of such a uniform bound was considered so wacky that CHM was thought of as evidence against Lang’s conjecture. Joe Harris used to wander around the department, buttonholing graduate students and encouraging us to cook up examples of genus-g curves with arbitrarily many points, thus disproving Lang. We all tried, and we all failed — as did many more experienced people. And nowadays, the idea that there might be a uniform bound for the number of rational points on a genus-g curve is considered fairly reputable, even among people who have their doubts about Lang’s conjecture. As far as I know, the world record for the number of rational points on a genus-2 curve is 588, due to Kulesz. Can you beat it?

## The great circle of life as CJ sees it

The other day we took CJ to a house with a friendly, but barking, golden retriever. CJ was a little intimidated, and at one point really scared when the dog took a run at him.

Later we had this conversation:

Me: CJ, were you a little scared of the dog?

CJ: I was scared but Daddy was not scared.

Me: Maybe when you’re a little older you won’t be scared.

CJ: When I am a little older and Daddy is a baby we will go in the house with the dog and I will not be scared but Daddy will be scared!

Update:  Toby asks in comments whether it’s common for children to have this misperception.  Fortunately, the indispensable I Used To Believe, the online repository of weird childhood beliefs, is here to help.  We quickly find:

• “I used to belive as i got older my parents would get younger and i would have to take care of them as babies until there body couldn’t age as fast then they would die”
• “I used to believe that when I got to a certain age, my older brother would ‘turn small’ and I would be able to be the older sibling. I believed that this cycle would last forever.”
• “i used to believe that my brother and I were going to switch places with my parents – that as we got bigger, they would get smaller. I was under 5, because my other brother had not been born yet – and it seemed to make sense.”
• “When I was about 3 or 4 years old, I believed that, just as little kids were in the process of growing bigger and older, adults were in the process of growing younger and littler. So I thought that my parents would one day be little children again.”

And that’s just from the first 5 pages out of 27 on the topic of “growing older.”  Much more common, apparently, is the belief that you switch genders as you age – CJ has expressed some thoughts along these lines, too, now that I think about it.

## In which I shake my tiny fist at the heavens

It’s snowing here.

That is all.

Tagged

## Assortative mating

The red book is here! By which I mean: the reunion report of the Harvard Class of 1993 arrived in the mail yesterday. Now that most of us are married, I thought it would be interesting to see whether indeed Harvard locks its graduates into a bubble of privilege, accessible only to other alumni of the fanciest schools — in other words, does Harvard marry Harvard?

So I’m just going to go through this book alphabetically, until I get bored, and write down the alma mater of the spouse of each married Harvard grad. I’ll separate my classmates by gender, in case that makes a difference.

Harvard men found spouses from:

• West Virginia
• Florida Atlantic
• Georgetown
• North Carolina State
• Emory
• St. John’s University
• UC-Davis
• Harvard
• U Mass – Boston
• Duke
• UC-Davis
• UC-Berkeley
• Knox College
• Yale
• Harvard
• Durham
• Harvard
• Duke
• Purdue

And the Harvard women:

• George Washington
• Southern Connecticut
• Evergreen State
• Harvard
• Yale
• Texas A&M
• Harvard
• New Mexico
• Colgate
• Harvard
• York
• Harvard
• Harvard
• Columbia
• West Point
• Sydney
• Harvard
• Brandeis
• Harvard
• Princeton
• Tufts

OK, that’s enough. Hard to draw much from such a small sample size, but am I going to flip through the whole book typing in husbands and wives?  I am not.

Tagged ,

## How many points does the average curve have?

Felipe Voloch complained that I didn’t list Ruby’s BBQ in my last post as one of the charms of visiting UT. I’ll make it up to him by observing that one of the charms of visiting UT is talking math with Felipe! He asked me an interesting question, about which we had different intuitions — I’ll present the question here and those readers who have an opinion are encouraged to voice it. (Math below the fold to avoid shocking the modesty of non-mathy readers.)

## Mickie’s corned beef hash, herring with lingonberries, Elgin hot sausage

(No, not all together, though I wouldn’t swear that such a combo never crossed Steve‘s lips in bachelor days.)

The in-laws from Israel, last seen here, were back last week for the run-up to Passover. I always try to hook them up with a little Americana, so we had an excessive breakfast at Mickie’s Dairy Bar, whose excessive breakfasts are even better — and more excessive — than the ones at The OP. A momentary worry that my breakfast would not be excessive enough prompted me to order a side of corned beef hash, the true test of any diner. This was superb — maybe the best I’ve had. (The red flannel hash at Henrietta’s Table is the other contender, but it’s so socioeconomically elevated as to not really be the same dish.) Mickie’s corned beef hash is hardly hashed at all — the beef comes in big oblong chunks, and the potatoes in fat, crispy discs. All are coated in a smoky, peppery rub. It is the kind of corned beef hash I imagine cowboys might eat, outdoors, at the beginning of a day they know will be long and exhausting, especially if they are worried that their breakfast might otherwise not be excessive enough.

Then Passover started. We spent it in Hempstead, Texas, ancestral seat of Mrs. Q’s maternal side. It is the Watermelon Capital of Texas (more precisely, one of at least five municipal claimants to the title) and the only way to get wireless there is to pull up in the parking lot of the Super 8 and check e-mail until a chambermaid notices you — thus limited blogging. The good thing about Passover in Texas is that it’s a great place to eat big pieces of meat. More specifically: the good thing about giving a talk at UT on Passover is that you can stop for Elgin hot sausage on the way back. Now I like a Wisconsin brat as much as the next man, but there’s no comparison to the satisfying snap of the casing on an Elgin hot, or the tender, peppery beef filling, with a texture closer to meatball than kielbasa. And you can have it shipped to your house!

We’re back home since Tuesday night. On CJ’s request, we stopped at Brennan’s on the way back from daycare yesterday; he likes the cheese samples, I like the herring with lingonberries from Hughes Seafood.

I have, just once, been to a smorgasbord in Sweden. There must have been a wide variety of food on offer, but in my memory, that’s all been crowded out by the awe-inspiring combinatorial explosion of herring. Herring in mustard, herring in wine, herring and onions, herring and curry — and, most exotic of all, herring with fruit. People of America, follow the Swedes — you don’t have to adopt cross-country skiing or six weeks of paid vacation, but please, listen to me, and start sweetening your herring. If you need further instructions, buy a pound of herring with lingonberries from Hughes Seafood. And you too will sing my herring-sweetening tune.

## Scocca blogs!

His public demanded it, and now we have it:  the Tom Scocca Blog, featuring a cute baby and vignettes about expat life in Beijing as the Olympics get ready to roll.  Excellent use of the generally underutilized “I was a Chinese Housewife” tag.  If you just want to cut straight to the child abuse, here’s where he feeds his baby durian.

## Terrace season, cart season

Spring, finally — so I can sit outside and do math on the Terrace, one of the great perks of working at UW. (Photo snapped by my laptop cam in mid-blog!) Warm weather also means the return of the food carts to Library Mall. There are two newcomers this year — FIB (Italian beef) and Santa Fe Trailer (New Mex-Mex.) I tried the Italian beef today: tasty, though the “dipped” version, submerged in the au jus before serving, was a little too soggy to eat as a sandwich per se. I’m looking forward to trying Santa Fe Trailer; Mexican food from the carts is otherwise limited to Senor Pepper’s, which you can tell by the name — right? — is a bit disappointing. If you’re a newcomer to the carts, the best one is Kakilima, the Indonesian cart. Buraka (East African) and Taste of Jamaica are also very good. Many people like Zen Sushi, which serves some interesting non-standard Japanese lunch stuff (like curry noodles) but isn’t my bag. The Chinese and Vietnamese carts are, well, OK. The Vietnamese cart is called “I’m Here,” which is on the one hand charming but on the other hand honestly articulates the only reason to eat there. The Sukho Thai cart, alas, is gone. Well, not exactly — I just learned from the linked article that Santa Fe Trailer is in fact the same physical cart as Sukho Thai, sold, renovated, and festooned with green chilis. Now if I can just convince them to keep selling crazy noodles!

Did you ever hear of a McDonald’s directly adjoining a college campus that couldn’t stay in business? It happened to ours. The wide variety of appealing, superfast lunch choices on the Mall must be one of the reasons.

## Great moments in college photojournalism

From the front page of today’s Badger Herald, “First ‘State of the ASM’ not well-attended.” Subtitle: “Student government chair unsure whether event was publicized.” Caption: “Members of the Associated Students of Madison prepare to discuss the “State of the ASM” with audience members Tuesday at Chamberlin Hall.”