Small Scale Distribution of Heegner Points

George A. Boxer

Mathematics

“For future thesis writers, my primary word of advice is to start early! Rushing to complete your thesis at the last minute is not fun.”

As a freshman, I was quite intimidated by the thesis. I was just beginning my study of mathematics, and I had serious doubts that in only a few years I could contribute anything new. Nonetheless, I was able to do just that, thanks to ample preparation over three years from my professors, peers, and many problem sets, as well as a very generous amount of support from my adviser.

My thesis was in the discipline of number theory, the study of the integers. Why study numbers? For one thing, it never ceases to amaze me how many basic aspects of numbers we still don’t understand, in spite of hundreds of years of research. Another reason is that it leads to a great deal of beautiful mathematics, much of which seems quite far removed from the study of numbers.

Let me briefly describe what my thesis was about. I should first explain what a Heegner point is. For a fixed negative integer d, a Heegner point of discriminant d is nothing more than a triple of integers (a, b, c) satisfying the equation b2–4ac=d. This is not the place to discuss why one might be interested in these points, but it suffices to say that their study goes back at least as far as the great mathematician Carl Friedrich Gauss. Moreover, in a more sophisticated guise they continue to play an important role in modern number theory.

In my thesis I investigated the distribution of these points. My contribution was an upper bound on the “pair correlation” of the Heegner points. For example, my bound implies that the Heegner points aren’t regularly spaced like the atoms in a crystal, and that they don’t come in close pairs, like the oxygen atoms in a gas of oxygen molecules. I also did some numerical calculations to find evidence for the overarching conjecture that the Heegner points are distributed like random points.

Compared to my previous work at Princeton, the thesis was quite different. For my junior independent work I had no original results, nor did I make any substantive expository innovations. Instead the goal had just been to learn about some area of mathematics. It was quite similar in spirit to a course. The nature of thesis research is quite different. The process is not nearly as linear. There were false starts and dead ends. It could be slow and frustrating at times, but this made progress even more exhilarating. When everything came together at the end, it was quite satisfying. Finally, the importance of the thesis adviser cannot be overstated. Without my adviser’s guidance I could not have even come up with a research topic, let alone any results of substance. My adviser’s expertise played a crucial role at every stage.

Let me now describe the process. I didn’t actually have a specific thesis topic until the spring semester. I spent the fall learning things, not necessarily directly related to my eventual topic. At the start of my senior year, my adviser gave me and two other advisees some rough ideas of potential topics, and some advice on what to read. There was also a “special year” on analytic number theory at the Institute for Advanced Study. We attended many of the regular seminars and tried to understand what was going on. This was of course not easy because many of the talks were given for an audience of experts. The other two students and I met somewhat regularly to discuss what we were reading and work on some problems that our adviser gave us.

I settled on the topic of my thesis around the beginning of spring semester. I should note that this is very late. Some departments even require a substantive amount of writing by this point. However it is somewhat more reasonable for a math thesis. Math theses generally aren’t nearly as long as others, and the writing is relatively quick once you have results. There are no activities such as lab work that necessarily consume a lot of time. Nonetheless, I would recommend finding a thesis topic and starting earlier than I did.

Let me describe how things looked as I focused on my topic. The major known result about the distribution of Heegner points was the 1988 theorem of Duke, which states that they are “equidistributed.” This roughly means that they are evenly spread out. However it doesn’t provide much information about how things look at smaller scales. For example, it is perfectly consistent with the Heegner points being well spaced like a crystal lattice, something we suspected wasn’t the case.

For reasons I can’t explain here, there is an analogy between Heegner points and lattice points on two dimensional spheres (that is, for r fixed, solutions to x2+y2+z2=r in x, y, z integers.) Work of my adviser and his collaborators had suggested that these other points are distributed like random points. It thus seemed reasonable to suspect that things would be similar for Heegner points. My goal then was to provide some new information about what was going on at small scales, and make modest progress on the conjecture that Heegner points really are distributed like random points.

It was clear at the beginning that having some sort of control on the pair correlation would be crucial for making any substantial progress. My result naturally splits up into two steps. The first is to obtain a formula for the number of pairs of Heegner points a given distance apart. By combining a nice trick with a deep theorem, Siegel’s mass formula, I can reduce this problem to a different one: a calculation of some so-called “local densities.” These calculations, while not overly conceptually demanding, were fairly lengthy and involved. This was probably the hardest part of my thesis, at least in terms of how much work was required.

Once this was done, I had my formula, but it fluctuated too wildly and unpredictably to be of direct use. Hence the next step was to average it over an interval to smooth it out and try to obtain a useful upper bound. This involved the mathematical technique known as “sieving.” I won’t say much more about this, except that the most exciting moment is when you put everything together. The complicated formula coming from Siegel’s theorem and local density calculations goes through a complicated sieving process and a single factor pops out. This factor winds up being quite simple and is exactly what I wanted! Once this was all in place, writing everything up was fairly straightforward.

For future thesis writers, my primary word of advice is to start early! Rushing to complete your thesis at the last minute is not fun. Your last months at Princeton will be substantially more enjoyable if you finish well before the deadline. Moreover, I would urge you to approach your thesis from the right mind set. Do not view it as one last hurdle before graduation. Take full advantage of the opportunities that are available to you at Princeton, especially the expertise of the faculty. Make sure to choose a topic that you can remain passionate about for a year, and, most importantly, try to have fun!

Small Scale Distribution of Heegner Points

George A. Boxer

Peter C. Sarnak

Eugene Higgins Professor of Mathematics

“One reason that I enjoy advising senior theses is that it is very satisfying to watch a young person develop some mathematics to a point where they discover something new, and just as importantly, to see them realize that they have the potential to make a long-lasting mark on something to which they devote themselves.”

The techniques at the heart of most research in modern mathematics are quite specialized and sophisticated. This makes it a challenge for a senior thesis in mathematics to contain original research. Ideally, this should be a component of the work because it is what makes the thesis very different from classwork, and it also is a means for the student to get a taste of making a new insight or a discovery. Having advised a number of senior theses over the years, I have found a recipe that seems to work quite well, especially with our strongest students (and we are very lucky to have a good number of them!). The trick is to identify unsolved problems and phenomena that are of current interest, that are not well understood, whose formulation does not require advanced notions, and for which there is room for computer experimentation.

It turns out that in certain fields, such as number theory and probability theory, there is a good supply of such problems. These can make for excellent senior thesis projects. Firstly, by reading a background paper or two, the student learns some basic theoretical techniques that might be relevant for the problem. In rare cases, this might lead them to apply these techniques successfully and to establish a new result, but, in any case, by probing the problem experimentally they will either confirm or refute some expectation or, more often than not, discover some novel features. It is not appreciated widely enough that many of the most striking discoveries in pure mathematics were found by hand by looking at examples. Mathematics is very concrete, even if the tools for proving theorems are often very abstract and sophisticated.

George Boxer was one of the seniors whose theses I advised last spring. His thesis was one of the best that I have seen in some years. Developing and combining quite different techniques, he established some very interesting results about the local distribution of solutions to certain diophantine equations. His results and related experimentation give compelling evidence that these solutions behave much like points chosen at random, a feature that was not expected in this context. His findings are suitable for publication in a very good mathematics journal.

One reason that I enjoy advising senior theses is that it is very satisfying to watch a young person develop some mathematics to a point where they discover something new, and just as importantly, to see them realize that they have the potential to make a long-lasting mark on something to which they devote themselves. It also is a means for them to see that typically their teacher is at a loss for good ideas on a difficult unsolved problem, and as with most things, progress is slow and hard earned. My one piece of advice to seniors writing theses is not to leave it to the last minute. Almost all students do so and amazingly produce something very good in a rush, but things could be much easier if they weren’t done under such pressure.