## Probabilistic number theory

During the fall semester, besides the Linear Algebra course for incoming Mathematics and Physics students, I was teaching a small course on Probabilistic Number Theory, or more precisely on a few aspects of probabilistic number theory that I find especially enjoyable.

Roughly speaking, I concentrated on a small number of examples of statements of convergence in law of sequences of arithmetically-defined probability measures: (1) the Erdös-Kac Theorem; (2) the Bohr-Jessen-Bagchi distribution theorems for the Riemann zeta function on the right of the critical line ; (3) Selberg’s Theorem on the normal behavior of the Riemann zeta function on the critical line ; (4) my own work with W. Sawin on Kloosterman paths. My goal in each case was to do with as little arithmetic, and as much probability, as possible. The reason for this is that I wanted to get and give as clear a picture as possible of which arithmetic facts are involved in each case.

This led to some strange and maybe pedantic-looking discussions at the beginning. But overall I think this was a useful point of view. I think it went especially well in the discussion of Bagchi’s Theorem, which concerns the “functional” limit theorem for vertical translates of the Riemann zeta function in the critical strip, but to the right of the critical line. The proof I gave is, I think, simpler and better motivated than those I have seen (all sources I know follow Bagchi very closely). In the section on Selberg’s Theorem, I used the recent proof found by Radziwill and Soundararajan, which is very elegant (although I didn’t have much time to cover full details during the class).

I have started writing notes for the course, which can be found here. Note that, as usual, the notes are potentially full of mistakes! But those parts which are written are fairly complete, both from the arithmetic and probabilistic point of view.

## Ventotene

I just came back from a week on the Italian island of Ventotene, where I participated in a very nice conference on “Manifolds and groups”. This is of course not my usual topic, and besides giving a minicourse on expanders and coverings (with a focus on geometric applications of expanders having to do with coverings…), I learnt a lot of interesting things. I particularly enjoyed the other two minicourses, by T. Gelander on invariant random subgroups and by R. Sauer on Lücks’s Approximation Theorem. (For both of these, as well as for a number of other talks, the slides are available on the conference web page; my own — handwritten — notes on my course should appear there soon, after I scan them).

Being an island, Ventotene is reached by boat. One of the interesting things about the trip to Ventotene was to observe how birds

would follow us until a very definite point, and then suddenly disappear.

Also, the instructions to launch a lifeboat are rather daunting…

The organizers had scheduled a lot of free time during the conference besides the scientific programme. I was therefore able to take a few pictures, such as some of the local lizards,

and some of the wonderful local cats.

The end of the week also coincided with the beginning of the ten-day long celebration of the island’s patron saint, Santa Candida. Among the festivities were the evening launches

(over a week) of huge hot-air balloons (“Mongolfieri”), with some fireworks

(note the amusing effects of my camera’s fireworks scene setting without tripod…). Some Galois-theoretic persiflage was also notable…

Update The scans of my lectures can be found here.

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,

were I took many pictures of Chinese Dragons…

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

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.)