Kowalski – Page 78 – E. Kowalski's blog

Quadratic reciprocity, exercised

Here is an amusing elementary property:

Let n and m be odd integers, and d such that n+m=d², and d is congruent to 2 modulo 4; then
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(\frac{d}{n}\right)=-\left(\frac{d}{m}\right)$
where the left and right-hand sides are Jacobi symbols.

If n and m happen to be odd primes, then we have Legendre symbols, and then this property means that one, and only one, of the equations

$y^2=d\ \mbox{mod}\ n$

and

$y^2=d\ \mbox{mod}\ m$

has a solution. (For general n and m, we can only deduce that one of those equations has no solution).

Here is a hint for the proof: compute

$\left(\frac{d}{d^2-n}\right)$

using the reciprocity law for Jacobi symbols to check that it is –(d/n). This requires to use all cases of the reciprocity law (namely, the “generic” law for odd moduli, and the supplementary laws for (2/d) and (-1/d) if d is odd). There might be another proof than this, of course, but it doesn’t seem easy to use, for instance, the concrete interpretation above, in the case where n and m are primes.

I don’t know if this is a well-known factlet about quadratic reciprocity. I only noticed it using computer experiments (then, of course, it wasn’t very long before I found the easy proof using the reciprocity law), as I wanted to check that, roughly, among the expressions d²=n+m, with d even and n and m odd primes, roughly half of the values of n were such that d is not a quadratic residue modulo n. When d was congruent to 2 modulo 4, the computer indicated that this was true of exactly half of the values, which only seemed plausible if the property above was true…

Incidentally, it is known by work of Perelli that, for any monic integral polynomial P of positive degree, which does not always take odd values, then for “most” positive integers n, P(n) is a Goldbach number if it is even, i.e., it is a sum of two primes, so for most d congruent to 2 modulo 4, we can indeed write d²=n+m with n and m both primes. This result refines, in a sense, the well-known fact – which goes back to van der Corput, Tchudakov and Estermann at least – that most even integers are sums of two primes; one proof of this, which is more or less a consequence of a suitably strong version of Vinogradov’s theorem on representations of odd integers as sums of three primes, is given in Chapter 19 of my book with Henryk Iwaniec.

(As to why I looked at this bizarre question, it is because of a question of Nguyen Ngoc Dong Quan, involving slightly more complicated conditions).

New category

I’ve not been particularly efficient in using tags, categories, and the rest, to organize this blog, but I’d like to point out that I just added one category Exercises which can be used to retrieve all posts which contain rather elementary mathematical facts which may be suitable as exercises for rainy days or when one has to write an exam.

Version control in action

A while back I mentioned the usefulness of version control software in my work (see also the current discussion at the Secret Blogging Seminar). Now here is a True Life example of what it can do: Vijay Patankar wrote to me today, pointing out that in my paper Weil numbers generated by other Weil numbers and torsion fields of abelian varieties, I claim (misquoting slightly for typographical reasons):

Remark 3.9. In [K1], the question of the “splitting behaviour” of a simple abelian variety A/Q at all primes is also raised: is it true that the reduction modulo p of A remains simple for almost all p? In fact, the “horizontal” statements of Chavdarov can already deal with this. For instance, this property holds if A/Q has the property that the Galois group of the field Q(A[n]) generated by the points of n-torsion of A is equal to Sp(2g,Z/nZ) for any sufficiently large prime n.

Here, the citation [K1] refers to my earlier paper Some Local-Global Applications of Kummer Theory, but as he pointed out, the question is not mentioned anywhere there… So where did it go?

I have no memory at all of what happened, but thanks to SVN’s history facility, I have been able to reconstitute the outline of this nanoscopic academic drama: first, in late 2001, I added a remark about this problem in the final section, among other questions suggested by the paper:

r416 | emmanuel | 2001-09-13 08:52:44 +0200 (Thu, 13 Sep 2001) | 3 lines 416 emmanuel \item Given an abelian variety $A/k$ over un number field, how does 416 emmanuel its decomposition in simple factors relate to that of its reductions 416 emmanuel in general? In particular, assuming $A$ to be $k$-simple, is the set 416 emmanuel of primes $\ideal{p}$ with $A_{\ideal{p}}$ reducible finite, or of 416 emmanuel density $0$?
Here, the first line is an excerpt from the log file which records the history of every file under version control (it is produced by the svn log command); the number 416 is the “revision number” which identifies at what point in time the changes corresponding to this “commit” were made.

The next lines (which, in fact, come chronologically first in the unraveling of the mystery, as they tell which revision number to look at to find the exact date) are obtained by the svn blame command: for each line of a file, this indicates (1) at which revision the line was added (or last changed); (2) who did the change.

Then, in late 2002, just before sending the corrected version to the publisher, I commented out this question:
r1831 | emmanuel | 2002-11-14 21:16:41 +0100 (Thu, 14 Nov 2002) | 2 lines 1831 emmanuel %\item Given an abelian variety $A/k$ over un number field, how does 1831 emmanuel % its decomposition in simple factors relate to that of its reductions 1831 emmanuel % in general? In particular, assuming $A$ to be $k$-simple, is the set 1831 emmanuel % of primes $\ideal{p}$ with $A_{\ideal{p}}$ reducible finite, or of 1831 emmanuel % density $0$?

I still don’t know why I ended up doing this; indeed, two years later, I was convinced I had not done so, when I wrote the excerpt above from my other paper (the date can again be determined using SVN). In passing, this confirms the principle (which I try to adhere to usually) that one should always give a complete detailed reference to any outside work – even if it’s your own. If I had taken the trouble of trying to locate the page or section number for this question, I would have realized it was missing…

Finally, if the question seems of interest, this paper of Murty and Patankar develops it further.

Science and mathematics

Quite by chance, I’ve stumbled in the archive of Nature (alas, not freely available) on a paper by J. Sylvester (dated December 30, 1869) concerning, roughly, the status of mathematics among sciences. He says his text was a reaction to earlier talks and articles by Huxley (the biologist, not the limericks writer…). His esteemed opponent having stated

Mathematics “is that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation”, but knows only deduction,

Sylvester argues strongly in the opposite direction:

I think no statement could have been made more opposite to the fact,

and goes on to give examples from his own work, in particular, where conclusions were reached, and entire theories were constructed, based on simple apparently accidental remarks, by processes of observation, induction and imagination.

Besides this discussion, reading this paper is quite fascinating. Mostly, it must be said, because it is rather incredibly hard to read. Not only physically (the font size is small, and the footnotes even smaller, and printed 2-up, it really exercises your eyesight), but also because of the language, which I believe should cause many a lover of the English language to either faint or burst out laughing; the Gothic Victorian style (cleverly ridiculed by Jane Austen in “Northanger abbey”) is here put into overdrive for the purpose of scientific discussion. There are rather frightening mathematical terms which, presumably, a few living readers can still interpret,

canonisant, octodecadic skew invariant, invariantive criteria, amphigenous surface, a catena of morphological processes

and there are Latin, Greek and French quotations, untranslated, and a German one (which, strangely, Sylvester feels to be in need of translation). The following passage is quite typical:

Now this gigantic outcome of modern analytical thought, itself, only the precursor and progenitor of a future still more heaven-reaching theory, which will comprise a complete study of the interoperation, the actions and reactions, of algebraic forms (Analytical Morphology in its absolute sense), how did this originate? In the accidental observation by Eisenstein, some twenty or more years ago, of a single invariant (the Quadrinvariant of a Binary Quartic) which he met with in the course of certain researches just as accidentally and unexpectedly as M. Du Chaillu might meet a Gorilla in the country of the Fantees, or any one of us in London a White Polar Bear escaped from the Zoological Gardens. Fortunately, he pounced upon his prey and preserved it for the contemplation and study of future mathematicians…

But there are also interesting things, like a discussion of the status of higher-dimensional geometry, and indeed a forecast of Flatland (the book of that title was only published 15 years later):

for as we can conceive beings (like infinitely attenuated book-worms in an infinitely thin sheet of paper) which possess only the notion of space of two dimensions…

The follow-up paper is much in the same style (with beauties such as “the Eikosi-heptagram“, and flights of fancy like “my own latest researches in a field where Geometry, Algebra and the Theory of Numbers melt in a surprising manner into one another, like sunset tints or the colours of the dying dolphin” – this theory is that of “the Reducible Cyclodes”), and also quite insightful sometimes. For instance, there is en passant, the following very convincing footnote:

Is it not the same disregard of principles, the indifference to truth for its own sake, which prompts the question “Where’s the good of it?” in reference to speculative science, and “Where’s the harm of it?” in reference to white lies and pious frauds? In my own experience I have found that the very same people who delight to put the first question are in the habit of acting upon the denial implied in the second. Abit in mores incuria.

(Sylvester writes the “i” in the word “in” in the last quotation as a dotless i; I doubt it’s a typographical error, but I can’t find an indication that this is proper Latin grammar; does any reader here have an insight on this?)

An introductory course in integration and probability

A while ago (in 2002 to be precise), I taught an introductory course of integration, Fourier analysis, and probability in Bordeaux (for third-year university students). While giving the lectures during the first year, I typed them more or less in parallel (in fact, sufficiently close to the course that I could probably have done it also on a weblog, as T. Tao has done this year with his course on Ergodic Theory, and his course on Perelman’s proof of the Poincaré Conjecture…).

Except for using those notes two more years in the same course (or a very similar one the third year), I did not work on them anymore. But since I had spent a fair amount of time to bring the text to a reasonable state of polish, and since I’ve found myself using it a few times as a convenient reference for basic facts that I wanted to show to other students, it seems more than reasonable to put this course on the web somewhere. And so, here it is: Un cours d’intégration.

The text is in French, which diminishes its current interest, but as it is likely that I will teach this topic again in English at ETH, I’ll probably use it as a basis for a translation and/or adaptation.

As the introduction indicates, the only noticeable feature of this integration course (though I don’t think it is at all unique) is that I have treated probability theory in parallel with measure theory, instead of treating probabilistic language and its basic results after the main development, as is often done. I find this is better for a first course, because even students who never go on to another probability course can get, if they are attentive, a good immersion in the special langage and frame of mind of probabilists. (And this was a practical concern in Bordeaux because, typically, there wasn’t another probability course following this one, at that time at least).

The course itself is roughly 140 pages long, and is fairly standard. In measure theory, most of the basic results are treated, though not the Radon-Nykodim theorem. In probability, things go as far as the Central Limit Theorem, but there are no martingales.

The last 40 pages contain the exams given the first two years I gave the course, with corrections. Some of the problems are dedicated to proofs of quite nice results, including the Radon-Nykodim theorem on [0,1] and Lebesgue’s differentiation theorem, and some are more standard. (Those who have never seen a French-style three/four hours long exam can have a look to get an idea of how this type of things are done in this strange country; with hindsight, it’s clear the exams were quite a bit too difficult for the students I had…)