Latest adventures

The last two weeks were quite eventful…

First I spent four days in China for the conference in honor of N. Katz’s 71st birthday. I was lucky with jetlag and was able to really enjoy this trip, despite its short length. The talks themselves were quite interesting, even if most of them were rather far from my areas of expertise. I talked about my work with W. Sawin on Kloosterman paths; the slides are now online.

I only had time to participate in one of the excursions, to the Forbidden City,

Forbidden City
Forbidden City

were I took many pictures of Chinese Dragons…

Chinese Dragon
Chinese Dragon

That same evening, with F. Rodriguez Villegas and C. Hall, I explored a small part of the Beijing subway,

Subway map
Subway map

trying to interpret and recognize various Chinese characters, before spending a fair amount of time in a huge bookstore


(where I got some comic books in Chinese for fun).

Upon coming back on Thursday, I first found in my office the two volumes of the letters between Serre and Tate that the SMF has just published, and which I had ordered a few days before taking the plane. Reading the beginning of the first volume was very enjoyable in the train on Friday morning from Zürich to Lausanne, where the traditional Number Theory Days were organized this year. All talks were excellent again — we’re now looking forward to next year’s edition, which will be back in Zürich! And I’ll write later some more comments about the Serre-Tate letters…

And then, from last Monday to Friday, we had in Zürich the conference “Analytic Aspects of Number Theory”, organized by H. Iwaniec, Ph. Michel and myself with the help of FIM. It was great fun, and there were really superb and impressive talks. One interesting experience was the talk by J. Bellaïche : for health reasons, he couldn’t travel to Zürich, but we organized his talk by video (using a software called Scopia), watching it from a teleconference room at ETH. This went rather well.

Kummer extensions, Hilbert’s Theorem 90 and judicious expansion

This semester, I am teaching “Algebra II” for the first time. After “Algebra I” which covers standard “Groups, rings and fields”, this follow-up is largely Galois theory. In particular, I have to classify cyclic extensions.

In the simplest case where L/K is a cyclic extension of degree n\geq 1 and K contains all n-th roots of unity (and n is coprime to the characteristic of K), this essentially means proving that if L/K has cyclic Galois group of order n, then there is some b\in L with L=K(b) and b^n=a belongs to K^{\times}.

Indeed, the converse is relatively simple (in the technical sense that I can do it on paper or on the blackboard without having to think about it in advance, by just following the general principles that I remember).

I had however the memory that the second step is trickier, and didn’t remember exactly how it was done. The texts I use (the notes of M. Reid, Lang’s “Algebra” and Chambert-Loir’s delightful “Algèbre corporelle”, or rather its English translation) all give “the formula” for the element b but they do not really motivate it. This is certainly rather quick, but since I can’t remember it, and yet I would like to motivate as much as possible all steps in this construction, I looked at the question a bit more carefully.

As it turns out, a judicious expansion and lengthening of the argument makes it (to me) more memorable and understandable.

The first step (which is standard and motivated by the converse) is to recognize that it is enough to find some element x in L^{\times} such that \sigma(x)=\xi x, where \sigma is a generator of the Galois group G=\mathrm{Gal}(L/K) and \xi is a primitive n-th root of unity in L. This is a statement about the K-linear action of G on L, or in other words about the representation of G on the K-vector space L. So, as usual, the first question is to see what we know about this representation.

And we know quite a bit! Indeed, the normal basis theorem states that L is isomorphic to the left-regular representation of G on the vector space V of K-valued functions \varphi\,:\, G\longrightarrow K, which is given by
(\sigma\cdot \varphi)(\tau)=\varphi(\sigma^{-1}\tau).
(It is more usual to use the group algebra K[G], but both are isomorphic).

The desired equation implies (because G is generated by \sigma) that Kx is a sub-representation of L. In V, we have an explicit decomposition in direct sum
V=\bigoplus_{\chi} K\chi,
where \chi runs over all characters \chi\,:\, G\longrightarrow K (these really run over all characters of G over an algebraic closure of K, because K contains all n-th roots of unity and G has exponent n). So x (if it is to exist) must correspond to some character. The only thing to check now is whether we can find one with the right \sigma eigenvalue.

So we just see what happens (or we remember that it works). For a character \chi\in V such that \chi(\sigma) = \omega, and x\in L^{\times} the element corresponding to \chi under the G-isomorphism L\simeq V, we obtain \sigma(x)=\omega^{-1}x. But by easy character theory (recall that G is cyclic of order n) we can find \chi with \chi(\sigma)=\xi^{-1}, and we are done.

I noticed that Lang hides the formula in Hilbert’s Theorem 90: an element of norm 1 in a cyclic extension, with \sigma a generator of the Galois group, is of the form \sigma(x)/x for some non-zero x; this is applied to the n-th root of unity in L. The proof of Hilbert’s Theorem 90 uses something with the same flavor as the representation theory argument: Artin’s Lemma to the effect that the elements of G are linearly independent as linear maps on L. I haven’t completely elucidated the parallel however.

(P.S. Chambert-Loir’s blog has some recent very interesting posts on elementary Galois theory, which are highly recommended.)

More conferences

It seems that most of my posts these days are devoted to announcing conferences in which I am involved as organizer… Indeed, there are two coming up this year (actually three, if I count the MSRI summer school):

(1) May 14 and 15, we will have the Number Theory Days 2015 at EPF Lausanne; the speakers are Gaetan Chenevier, Henryk Iwaniec, Alena Pirutka, Chris Skinner and Zhiwei Yun; this is co-organized by Ph. Michel and myself.

(2) Immediately afterward, from May 18 to 22, comes a conference at FIM, co-organized by H. Iwaniec, Ph. Michel and myself, with the title of “Analytic Aspects of Number Theory”; the current list of speakers is to be found on the web page; here is the poster (which is based on a picture taken by Henryk around Zürich last Fall):

AANT Poster
Analytic Aspects of Number Theory

Most importantly, there is a certain amount of funding available for local expenses of your researchers (doctoral and postdoctoral students). Applications can be made here (before Feburuary 6; the form states January 28, but this is an error that will be corrected).

Bounded gaps between primes: some grittier details

Because I was teaching a course on prime numbers this semester and had just finished a chapter on Vinogradov’s method when his paper appeared, I promptly switched my plans for the last classes in order to present some aspects of Yitang Zhang’s theorem on bounded gaps between primes. In addition, one of the speakers of this week’s conference celebrating 25 years of the “Zahlentheorie Seminar” of ETH had to cancel at short notice, and I replaced her and gave yesterday another survey-style talk. The notes for the latter (such as they are…) can be downloaded in scanned form.

My insights to Zhang’s work remain clearly superficial, but here are some remarks going a bit beyond what I mentioned in the previous post, coming after these lectures, and some discussions with Ph. Michel and P. Nelson.

(1) The most delicate estimates seem to be those for the “Type III” sums. These concern the “good” distribution in invertible residue classes of an arithmetic function f(n) for integers N<n\leq 2N, modulo a "large modulus" q, where f(n) is of the very special type
and the variables m_1, n_1, n_2 and n_3 are, roughly, of size M_1, N_i with M_1N_1N_2N_3=N, and (crucially) q is a bit larger than N^{1/2}: one needs to handle these for q up to N^{1/2+\delta} for some \delta>0 in order to obtain bounded gaps between primes.

The lengths M_1 and N_i are constrained in various ways, and the most critical case seems to be when M_1\approx q^{1/8}, N_i\approx q^{5/8} (the \approx means that one must be able to go a bit beyond such a case, since N is a bit beyond q^2).

(2) Another point is that Zhang manages to bound these sums for each individual residue class a\pmod q (coprime to q). In other words, denoting
\Delta_f(N,q,a)=\sum_{N<n\leq 2N: n\equiv a\pmod{q}}f(n)-\frac{1}{\varphi(q)}\sum_{N<n\leq 2N}f(n),
he proves individual bounds for \Delta_f(N,q,a), instead of average bounds over q (as in the other main part of this argument).

Also, he does not need to use the variable m_1 at all (but since the \alpha(m_1) are mostly unknown coefficients, and the sum is rather short, exploiting it does not seem easy). Hence the result looks enormously like controlling the distribution in residue classes of the ternary divisor function. This is exactly the question that Friedlander and Iwaniec had studied in the famous paper where they proved that the exponent of distribution is at least 1/2+1/230, but their argument is not quite sufficient for Zhang's purpose.

(3) One of the last tricks is Zhang's second use of the structure of the moduli q that are involved in his argument: these were chosen to have only prime factors \leq z=N^{\delta} for some small positive \delta), and Zhang exploits the "granular" (or friable) structure of such moduli in order to obtain flexibility in the possibility of factoring them as q=rs with r of size determined up to a factor at most z. This is particularly important for the “other” sums (it gives bilinear structure and makes it possible to use the dispersion method of Linnik, as already done by Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec in their works on primes in arithmetic progressions). For the "type III" case, it does not seem to be so much of the essence, but Zhang needs to gain a very small amount compared with Friedlander-Iwaniec, and does so by factoring q=rs with r rather small. He then gains from r a factor r^{1/2} which is essential, by exploiting the fact that a Ramanujan sum modulo r is bounded (so he gets more than square-root cancellation from r…)! This is an extremely special situation, and right now, it is what seems the most “miraculous” about this proof (at least to me).

It is for the contribution of the complementary divisor s that Zhang manages to position himself into applying the estimate of Birch and Bombieri for the exponential sums which Friedlander-Iwaniec had also encountered.

(4) This use of Deligne’s work is also very delicate: one can not relax the requirement of square-root cancellation, except by very tiny amounts. For instance, obtaining a bound of size p^2 for the three-variable sum modulo p is useless; in fact, the bound p^2 can be considered here as the trivial estimate, since the sum can be written as an average of one-variable Kloosterman sums. With Zhang’s parameters, one needs an estimate of for the sum which is no larger than p^{3/2+1/2000} (or so) in order to get the desired gain. However, as I explained in the last five minutes of my talk today (and as is explained in this note with Fouvry and Michel) the Birch-Bombieri bound is very well understood from a conceptual point of view.

(5) I was very curious, when first looking at the paper, to see how Zhang would handle the residue classes in the Goldston, Pintz, Yıldırım method, since the most uniform results on primes in arithmetic progressions (those of Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec) are constrained to use (essentially) a single residue class. What happens is that Zhang detects these classes by inserting a factor corresponding to their characteristic function, and by avoiding the Kloostermania approaches that rely on spectral theory of automorphic forms. The important properties of these residue classes are their multiplicative structure (coming from the Chinese Remainder Theorem) and the fact that, on average over moduli, there are not too many of them (the average is bounded by a power of \log N). In particular, his use of the dispersion method is in fact closer in spirit to some of its earliest uses by Fouvry and Iwaniec (for integers without small prime factors instead of primes), which also involved, in the final steps, the classical Weil bound for Kloosterman sums instead of (seemingly stronger) results on sums of Kloosterman sums.