Yet another property of quadratic fields with extra units

Here is an amusing exercise that is suitable for a course on basic algebraic number theory: let p be a prime number. Consider integral solutions (a,f) to

$4p=a^2+3f^2$

with a and f positive. The claim is that, if p is congruent to one mod 3, there are three distinct solutions (a,f), (b,g), (c,h), and if they are ordered

$1\leq a

then we have

$c=a+b$

For example, in the case p=541,  we find a=17, b=29, c=46:

$4p=2164=17^2+3\times 5^5=29^2+3^3\times 7^2=46^2+3\times 2^4$

The pedagogic value of the exercise is that while it looks like something that one could prove by a simple brute force computation, this is not so easy to do, while it becomes elementary knowing the basic facts about factorizations of primes in quadratic fields (and units of imaginary quadratic fields, in this case Q(√-3).

Indeed, the equation means that p is the norm of the integral ideal generated by

$\frac{a}{2}+ \frac{f\sqrt{-3}}{2}$

It is known that only primes congruent to 1 modulo 3 are norms in this field, hence the first condition on p. Then, it is known that the ideal above is unique up to conjugation. So the only possible extra solutions, given one of them, are obtained by multiplying by a unit of the field, and isolating the “coefficient of 1”, or in other words taking the trace to Q. Since the units are

$\pm 1,\, \pm j=\frac{\mp 1\pm \sqrt{-3}}{2},\, \pm j^2$

simply multiplying shows there are three positive solutions:

$a,\,\frac{a+3f}{2},\,\frac{|a-3f|}{2}$

Depending on the sign of a-3f, the conclusion follows by considering two cases.

[This minor property of quadratic fields was motivated by the question of finding interesting examples of relations between the zeros of zeta functions of algebraic curves over finite fields; for quite a bit more about this – both results of independence and examples of relations -, see my preprint on the subject, in particular Section 6 for the examples.]

Équidistribution, or équirépartition ?

For years, I have been convinced that the proper French translation of “equidistribution” was not the faux ami (false friend) “équidistribution”, but rather the word “équirépartition”. The latter is for instance used by Serre (and Bourbaki).

But then  I realized recently that Deligne uses “équidistribution” in his great paper containing his second proof of the Riemann Hypothesis over finite fields, which contains in particular his famous equidistribution theorem (see Section 3.5, entitled “Application: théorèmes d’équidistribution”).

Since, in fact, neither word appears in the French dictionaries I have available (unsurprisingly: “equidistribution” is not in the OED), and since moreover “distribution” and “répartition” do appear and are identified as synonyms, it seems now that in fact both words should be acceptable…

There is no hyperbolic Minkowski theorem

Minkowski’s classic theorem of “geometry of numbers” states that any convex subset of Rn which is symmetric (with respect to the origin) and of volume (with respect to Lebesgue measure) larger than 2n contains a non-zero integral point.

This theorem is used, in particular, in the classical treatment of Dirichlet’s Unit Theorem in algebraic number theory. While teaching this topic last year, I wondered whether there was an hyperbolic analogue, in the following sense, where H is the hyperbolic plane in the Poincaré model:

does there exist a constant C such that any geodesically convex subset B of the hyperbolic plane H with hyperbolic area at least C which is geodesically symmetric with respect to the point i contains at least one point z of the form g.i, where g is an element of SL(2,Z) and g.i refers to the usual action by fractional linear transformations, with z not equal to i.

Here, the subset B is geodesically convex if it contains the geodesic segment between any two points, and symmetric if, whenever x is in B, the point on the geodesic from i to x which is at distance d(i,x) from i, but in the opposite direction, is also in B.

It turns out that the answer is “No”. Indeed, C. Bavard gave the following example:

let B be a euclidean half-cone with base vertex at 0, axis the vertical axis, and angle at the origin small enough, then B does not contain any “integral” point except i, but has infinite hyperbolic area. Moreover, it is easily seen that B is convex and symmetric in the hyperbolic sense, since hyperbolic geodesics are vertical half-lines and half-circles meeting the real line at right angles.

To check the claim, it is enough to show that for any integral point z=g.i distinct from  i, the ratio |x|/y has a positive lower bound, where z=x+iy (this will show that the angle from the vertical axis is bounded from below, so the point is not in a cone like the one above with sufficiently small angle). But z is given by (ai+b)/(ci+d) with a, b, c, d integers and ad-bc=1, and this ratio is simply |ac+bd|. Being an integer, either it is 0, or it is at least 1. Manipulating things, one checks that the first case only occurs for matrices in SL(2,Z) which are orthogonal matrices, and those fix i, so the point is then z=i. Hence, except for this case, the ratio is at least 1 and this concludes the argument.

It is interesting to see what breaks down in the (very simple) proofs of Minkowski’s theorem in the plane. In the first proof found on page 33 of the 5th edition of Hardy and Wright’s “An introduction to the theory of numbers” (visible here), the problem is that there is no way to dilate the convex region B in a homogeneous way compatible with the SL(2,Z) action. In other words, SL(2,Z) is essentially a maximal discrete subgroup of SL(2,R) (maybe it is maximal? I can’t find a reference).

Peano paragraphing

Every mathematician has heard of the Peano axioms of arithmetic. Here is a lesser known contribution of Giuseppe Peano: the “Peano paragraphing method”. This is a numbering system for sections/subsections/etc in books where the different items are identified by a decimal number (e.g., 9.132), where the integral part is the chapter number, and the decimal part is arranged in increasing order within each chapter. So for instance 9.301 is a subsubsection lying between 9.3 and 9.31.

I had noticed this system in Titchmarsh’s book “The theory of functions”, from 1932, without understanding it (it is not explained, nor attributed to Peano). Then I saw it again just recently as I was looking up a reference in Whittaker and Watson’s “A course of modern analysis” from 1927, where the explanation and attribution are given in a remark at the beginning. This greatly clarified my previous perplexity in navigating the book of Titchmarsh, which I had found extremely confusing; for instance in Chapter 9, we have

9.1, 9.11 up to 9.15, 9.2, 9.3, 9.31, 9.32, 9.4, 9.41 to 9.45, 9.5, 9.51 to 9.55, 9.6, 9.61, 9.62, 9.621 to 9.623, 9.7…

Looking into other classical books, I can see this system in Watson’s treatise on Bessel functions, but it is not used in either Hardy and Wright’s “Introduction to the theory of numbers”, nor in Titchmarsh’s “The theory of the Riemann zeta function”. It is also absent from Zygmund’s “Trigonometric series” (which, on the other hand, uses a continuous numbering scheme X.Y (Chapter.Item) both for equations, theorems, etc), and from Hardy and Rogosinsky’s “Fourier series”.

Note finally that it seems rather euphemistic to say that this is “lesser known”: neither Google nor Wikipedia seem to be able to give a reference or explanation!

One-upmanship

A quip of S. Lang states that “analysis is number theory at the place infinity”.  (Which implies, correctly, that analytic number theory is some particularly exalted form of number theory).

The equally quipful E. Witten goes rather further in reducing mathematics to its essentials: during the conference organized by the Mathematics Department of Princeton University in honor of the 250th anniversary of Princeton University, he said something like: “Most of 20th century mathematics is the study of the harmonic oscillator”. (This can be seen, in a slightly different and weakened form, on page 120 of the Google Book preview linked above; my memory is that he did state, during his lecture, something closer to what I wrote; but that was a while ago, so I may be over-reacting in hindsight…)

P.S. For the obligatory etymological epilogue: the word “one-upmanship” is quite recent (1952), but “quip” goes back to the early 16th century. I didn’t know about the charming derivative “quipful” before looking in the OED.