Mathematics – Page 15 – E. Kowalski's blog

The many ways of affineness

Last Saturday, the OED Word of the Day was affineur. Now, I know very well what an affineur is (my favorite is Jean d’Alos, and I especially like his renowned Tome de Bordeaux, the excellence of which can probably be confirmed by Mr. Quomodocumque), but for a few seconds I had in mind the picture of a fearsome algebraic geometer busily transforming all projective varieties into affine ones.

I looked at the adjacent words in the OED; there is quite a list of them involving affine-ness in some way (listed here with dates of first use, as recorded in the OED); actually, affineness is not in the list:

affinage 1656
affinal 1609
affine 1509
affine v 1473
affined 1586, 1907
affineur 1976
affining 1606
affinitative 1855
affinitatively 1825
affinition 1824
affinitive 1579
affinity 1325

It is interesting to think of an algebraico-geometric meaning for each of them (especially the tongue twister affinitatively, and affinition)…

Conferences

Here are two forthcoming conferences that I am co-organizing with Philippe Michel this year:

(1) Quite soon, the traditional Number Theory Days (the eleventh edition of this yearly two-day meeting that alternates between EPF Zürich and ETH Lausanne), will be held in Zürich on March 7 and 8; the web page is available, with the schedule and the titles of the talks; the speakers this year are Raf Cluckers (who is also giving a Nachdiplomvorlesung at FIM on the topic of motivic integration and applications), Lillian Pierce, Trevor Wooley, Tamar Ziegler and (Tamar is probably happily surprised not to come last in alphabetical order!) David Zywina.
Anyone interested in participating should send an email to Mrs Waldburger as soon as possible (see the web page for the address).

(2) In July, intersecting neatly the last stages of the soccer world cup, and beginning in the middle of a week to avoid (thanks to some fancy footwork) starting on the 14th of July, we organize a summer school on analytic number theory at I.H.É.S; people interested in participating should follow the instructions on the web site. The detailed programme will soon be available.

Three little things I learnt recently

In no particular order, and with no relevance whatsoever to the beginning of the year, here are three mathematical facts I learnt in recent months which might belong to the “I should have known this” category:

(1) Which finite fields $k$ have the property that there is a “square root” homomorphism
$s\ :\ (k^{\times})^2\rightarrow k^{\times},$
i.e., a group homomorphism such that $s(x)^2=x$ for all non-zero squares $x$ in $k$ ?

The answer is that such an $s$ exists if and only if either $p=2$ or $-1$ is not a square in $k$ (so, for $k=\mathbf{Z}/p\mathbf{Z}$ , this means that $p\equiv 3\pmod 4$ ).

The proof of this is an elementary exercise. In particular, the necessity of the condition, for $p$ odd, is just the same argument that works for the complex numbers: if $s$ exists and $-1$ is a square, then we have
$1=s(1)=s((-1)\times (-1))=s(-1)^2=-1,$
which is a contradiction (note that $s(-1)$ only exists because of the assumption that $-1$ is a square).

The question, and the similarity with the real and complex cases, immediately suggests the question of determining (if possible) which other fields admit a square-root homomorphism. And, lo and behold, the first Google search reveals a nice 2012 paper by Waterhouse in the American Math. Monthly that shows that the answer is the same: if $K$ is a field of characteristic different from $2$ , then $K$ admits a homomorphism
$s\ :\ (K^{\times})^2\rightarrow K^{\times},$
with $s(x)^2=x$ , if and only if $-1$ is not a square in $K$ .

(The argument for sufficiency is not very hard: one first checks that it is enough to find a subgroup $R$ of $K^{\times}$ such that the homomorphism
$t\, :\, R\times \{\pm 1\}\rightarrow K^{\times}$
given by $t(x,\varepsilon)=\varepsilon x$ is an isomorphism; viewing $K^{\times}/(K^{\times})^2)$ as a vector space over $\mathbf{Z}/2\mathbf{Z}$ , such a subgroup $R$ is obtained as the pre-image in $K^{\times}$ of a complementary subspace to the line generated by $(-1)(K^\times)^2$ , which is a one-dimensional space because $-1$ is assumed to not be a square.)

It seems unlikely that such a basic facts would not have been stated before 2012, but Waterhouse gives no previous reference (and I don’t know any myself!)

(2) While reviewing the Polymath8 paper, I learnt the following identity of Lommel for Bessel functions (see page 135 of Watson’s treatise:
$\int_0^u tJ_{\nu}(t)^2dt=\frac{1}{2}u^2\Bigl(J_{\nu}(u)^2-J_{\nu-1}(u)J_{\nu+1}(u)\Bigr)$
where $J_{\mu}$ is the Bessel function of the first kind. This is used to find the optimal weight in the original Goldston-Pintz-Yıldırım argument (a computation first done by B. Conrey, though it was apparently unpublished until a recent paper of Farkas, Pintz and Révész.)

There are rather few “exact” indefinite integrals of functions obtained from Bessel functions or related functions which are known, and again I should probably have heard of this result before. What could be an analogue for Kloosterman sums?

(3) In my recent paper with G. Ricotta (extending to automorphic forms on all $GL(n)$ the type of central limit theorem found previously in a joint paper with É. Fouvry, S. Ganguly and Ph. Michel for Hecke eigenvalues of classical modular forms in arithmetic progressions), we use the identity
$\sum_{k\geq 0}\binom{N-1+k}{k}^2 T^k=\frac{P_N(T)}{(1-T)^{2N-1}}$
where $N\geq 1$ is a fixed integer and
$P_N(T)=\sum_{k=0}^{N-1}\binom{N-1}{k}^2T^k.$

This is probably well-known, but we didn’t know it before. Our process in finding and checking this formula is certainly rather typical: small values of $N$ were computed by hand (or using a computer algebra system), leading quickly to a general conjecture, namely the identity above. At least Mathematica can in fact check that this is correct (in the sense of evaluating the left-hand side to a form obviously equivalent to the right-hand side), but as usual it gives no clue as to why this is true (and in particular, how difficult or deep the result is!) However, a bit of looking around and guessing that this had to do with hypergeometric functions (because $P_N$ is close to a Legendre polynomial, which is a special case of a hypergeometric function) reveal that, in fact, we have to deal with about the simplest identity for hypergeometric functions, going back to Euler: precisely, the formula is identical with the transformation
${}_2F_1(-(N-1),-(N-1);1;T)=(1-T)^{2N-1}{}_2F_1(N,N;1;T),$
where
${}_2F_1(\alpha,\beta;1;z)=\sum_{k\geq 0}\frac{\alpha (\alpha+1)\cdots (\alpha+k-1)\beta(\beta+1)\cdots \beta+k-1)} {(k!)^2}z^k$
is (a special case of) the Gauss hypergeometric function.

Reflections on reading the Polymath8(a) paper

Earlier today, I finished reviewing the Polymath8 paper (except for the last section, which is a discussion of possible improvements to some of the results in the paper; this is now a bit obsolete, and will probably be deleted).

Since the discussion on the project (and on the paper in particular) is almost entirely available online on Terry Tao’s blog, one can review the reviewing process, which is somewhat unusual in current mathematics, where reading/reviewing/refereeing a paper is mostly done in the dark.

Looking back, it seems that I began on September 30 (I thought it had been earlier…) At that time, the paper was about 164 pages long, and the current version is 176 pages long, so it is not so surprising that it took a bit more than two months to read through it, when combined with the semester teaching and other duties. (In fact, I changed some line-spacing settings, and the final version would have been probably at least five pages longer otherwise.)

At first, I actually intended to read the paper as if refereeing it, in some sense, with just corrections and minor changes, and things would certainly have gone faster in that case. But it seems that I have some genetic disorder and that I can’t read a paper through without wanting to change all the notation and quite a bit more (for good or ill…) Thus, where quite a few sections are concerned (especially towards the end), I basically re-typed much of the text with many many changes of notation. This was of course slower, but there is one big advantage in proceeding this way: I was much more likely to catch and correct typos and minor slips, and moreover I couldn’t just decide to go over a section with glassy eyes, since all the variables, functions, names of auxiliary quantities, and so on needed to be changed…

As far as mathematical issues in the first draft are concerned, I found only one which required any work to correct. I’d be very surprised if there were any still lurking, since I basically re-did every single computation, including checking numerical constants. And even this issue was really minor in comparison with the complexity of the whole argument, and didn’t affect the “generic” cases of the bounds where it was present.

So I think one can say that this paper has been reviewed in full basically at the thoroughest possible technical level (i.e., excluding philosophical or high-level comments). I actually wonder, in a hypothetical way, if I would ever have accepted to referee this paper, and if I had, whether I would have worked as carefully as I did…

And I am not quite done with bounded gaps between primes, since the Association des collaborateurs de Nicolas Bourbaki has asked me to give a Bourbaki seminar on the work of Zhang (and Maynard); this will be on March 29, 2014, in Paris, and I will now start preparing the text that will accompany the lecture…
But before I begin, I will offer myself a nice whisky this evening.

Trace functions, a survey

At last count, my series of works with Étienne Fouvry and Philippe Michel on trace functions and their applications consists of seven research papers or preprints, amounting to a bit more than 200 pages. To these are added a number of works-in-progress or partial notes (some with results we did not need or use and so took out of earlier drafts of our papers, some with worked-out examples or remarks, etc). We have a relatively firm project of writing a monograph account of the whole theory and applications, which we view in part as a way of making accessible some of the deep consequences of the Riemann Hypothesis over finite fields of Deligne, but this is clearly a long-term project. In the meantime, we have written a first short survey, the first draft of which can be found on my web page. This is in fact the written (and slightly expanded) account of a Colloquio de Giorgi that I gave at the

Scuola Normale Superiore in Pisa earlier this year, and we have included an unusual representation of the fundamental domain of the modular group acting on the upper half-plane as an homage and acknowledgement of this occasion.