where (1) is a (normalized) hyper-Kloosterman sum (for , this is a classical Kloosterman sum) modulo a prime ; and (2) the ranges and are such that we have non-trivial bounds even if is a bit smaller than in logarithmic scale. In other words, we obtain non-trivial results below the “Pólya-Vinogradov range”.

The basic strategy to get this result is not new: it was devised by Fouvry and Michel a number of years ago (inspired at least in part by earlier work of Friedlander-Iwaniec and by the Vinogradov-Karatsuba-style “shift” method to estimate certain short exponential sums). What was missing (despite the strong motivation provided by applications that were known to follow from such a result, one of which is described in a recent preprint of Blomer, Fouvry, Milicevic, Michel and myself) was a way to prove certain estimates for (complete) sums over finite fields, of the type

where

unless the parameters are in some “diagonal” positions. *And* we cannot afford too many diagonal cases…

The main contribution of our paper (much of which comes from the ideas of Will!) is to find a relatively robust approach to such estimates.

This relies, as one can expect, from extensive algebraic-geometric arguments to apply the Riemann Hypothesis over finite fields. In fact, from this point of view, this paper is by far the most complicated I’ve ever been involved in. We use, among other things:

- The Riemann Hypothesis over finite fields, in its most general version of Deligne — indeed, we use it multiple times;
- The interpretation of the sum over (in the sum above) as itself a trace function of sbome sheaf on the space of parameters ; this follows from the formalism of étale cohomology, which is also used in many other ways (e.g., to detect irreducibility of sheaves by properties of the top-degree cohomology);
- A very general version of the Euler-Poincaré characteristic formula in étale cohomology – this comes from SGA5;
- The formalism and properties of vanishing and nearby cycles in étale cohomology, and in particular their relations with local monodromy representations of sheaves on curves;
- The global -adic Fourier transform of Deligne as well as the local Fourier transform of Laumon;
- A special case of the homogeneous Fourier transform of Laumon (which we might be able to avoid, although with an argument involving perverse sheaves);
- Katz’s theory of Kloosterman and hypergeometric sheaves, in particular with respect to the computation of their geometric monodromy groups (and its implication through the Goursat-Kolchin-Ribet Criterion), but also (and equally importantly) with respect to their local monodromy properties;
- The diophantine criterion for geometric irreducibility (which is again a case of the Riemann Hypothesis)…

Many of these are results and ideas that I was aware of but had never actually used before, and I learnt a lot by seeing how Will exploited and combined them. I will try to write a few more posts later to (attempt to) explain and motivate them (and how we use them) from an analytic nunber theorist’s viewpoint. The theory of vanishing cycles, in particular, should have many more applications in extending the range of applicability of Deligne’s Riemann Hypothesis to problems in analytic number theory.

The paper is dedicated to Henryk Iwaniec, who has been over the years the most eloquent and powerful advocate for a deeper use of the work of Deligne (and Katz and others) in applications to analytic number theory.

]]>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.

of a whodunit by D.L. Sayers, which at least makes it clear to the dimmest reader who indeed did it?)

From Half-Price Books and Black Oak Books, I was well rewarded. In particular, the latter *boutique* has a rather remarkable selection of mathematical books. I skipped the three copies of Gauss sums, Kloosterman sums and monodromy groups (I got mine for five dollars from the Rutgers University special-deals cart a long time ago), but acquired Freiman’s Foundations of a structural theory of set addition for 7 or 8 dollars (the review by B. Gordon that I link to is quite fun to read), and found a little gem in Littlewood’s rather obscure book *The elements of the theory of real functions*:

Actually, the obscurity of this book is maybe understandable. It’s rather depressing to read

These lectures are intended to introduce third year and the more advanced second year men to the modern theory of functions

in the preface. But more importantly maybe, despite the promising title, the content of the book has very little to do with functions, real or otherwise. It’s really a semi-rigorous treatment of set theory up to very basic facts about subsets of , that does little to excite or attract attention. (Maybe the citation above reflects the fact that women students in Littlewood’s time were simply too clever to find this of any interest, and prefered to spend their time reading Gödel or Bourbaki?)

Despite the claim further in the preface that Littlewood *aimed at excluding as far as possible anything that could be called philosophy*, the fluffiness of the statements reminds me strongly of that discipline. Indeed, to see the bland statement

In Prop. 19 take the class to be itself. We obtain a blank contradiction.

shows that we are not in the most fastidious of company from the purely mathematical point of view.

Part of the charm of this book is the really weird terminology and the dizzying array of apparently pointless notational abbreviations (example at random: Prop. 5, p. 56 states *Given any series of terms, and an , , there is a sub-series of similar to “*). Of course, the first edition is apparently from 1925, when some poetic license *might* have been permitted as far as set theory and topology are concerned, but one doesn’t need to be overly formalist to raise one’s eyebrows when understanding (in a “completely revised” third edition of 1954) that a “class” denotes what everyone else calls a set, and that a “set” is what everybody calls a subset of . At least this would go a long way towards explaining the poor track-record of Cambridge Students from that period at having the faintest idea what every mathematician outside England was saying, as far as set theory and topology was concerned. Maybe this was the best way to ensure that they would think about more interesting things?

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.

]]>As another bonus feature, a double tap on the screen will cycle between the types of sums presented: Kloosterman sum to Birch sum to both to Kloosterman…

]]>**Important changes in the new version:**

(1) It was recompiled — hence a new modern look instead of a style reminiscent of the dark ages –, and the resulting binary should work on any Android system with version 4.0.3 or later;

(2) To keep up with recent progress, the program now displays Kloosterman sums and/or Birch sums, instead of Kloosterman and/or Salié sums. On the other hand, the moduli used are now restricted to primes, to simplify things a bit, and only one parameter is used: the sums displayed are

and

(3) In addition to being able to change the modulus by swiping horizontally, a vertical swipe on the screen scrolls among the values of the parameter . As was the case in the previous version (and even more because phones are faster now), the scrolling is usually too fast for a single step, so tapping once close to one of the edges of the window displaying the sum will perform just one step of the corresponding move (e.g., tapping close to the right of the screen goes to the next prime modulus). The value of the sum and its parameters and then displayed quickly.

(4) The plots are presented in orthonormal configurations, to give a more faithful representation of the paths in the plane;

(5) In the “About” dialog, a single click will display (if a PDF viewer is installed) the paper of W. Sawin and myself that explains the limiting distribution of Kloosterman (and Birch) paths…

(6) It is possible to save (or rather to “share” in the usual Android way) a picture of the sums currently displayed, as a PNG file.

(6) And the launcher icon is better.

The installation file is available right now on the updated Kloostermania page!

]]>I had selected expander graphs as the topic of my talk, as being simultaneously a very modern subject, that can be presented with very basic definitions (most students, coming from the French “Classes préparatoires” have had two very solid years of mathematical studies, but of a rather traditional kind), and that has links to many different areas, including more applied ones.

The slides of my presentation can be found here, in French, and I prepared an English translation there (up to the handwritten bits of information on certain pictures, which will remain in French…) Actually, the slides by themselves are not particularly interesting, since there are many remarks and details that I only described during the talk.

The talk begins with a discussion of graphs, continues with basic notions of expansion, and then defines expander graphs, with a bit of the history (especially the paper of Barzdin and Kolmogorov that I like very much), and concludes with two applications (among many…): Apollonian circle packings, and the Gromov-Guth knot-distorsion theorem. If this looks like a lot of ground to cover, this is because the talk was 1h30 long…

While preparing the first part of the talk, I researched some examples of graphs with more care than I had done before. Some of the things I found are extremely interesting, and since they might not be so well-known among my readers, I will present them quickly.

(1) The worm is *Caenorhabditis elegans*, more chummily known as *C. elegans*, one of the stars of neurosciences, as being the only animal whose entire nervous system has been mapped at the level of all individual neurons and connections between them. This was done in 1986 by White, Southgate, Thomson and Brenner, with minor corrections since then, and an important update and representation in 2011 by Varshney, Chen, Paniagua and Chklovskii. The resulting graph has 302 neurons (this number is apparently constant over all individuals), and about 8000 edges. (Interestingly, it is naturally a mix of directed and un-directed edges, depending on the type of biological connection). The paper of Varshney, Chen, Paniagua and Chklovskii is quite fascinating, as it investigates many mathematical invariants of the graph, and for instance compares the number of small subgraphs of various types with the expected number for random graphs with comparable parameters (certain subgraphs arise with higher frequency…)

Much more about *C. elegans* is found on dedicated websites, including Openworm, which seems to have as a goal to recreate the animal virtually…

(2) The brain is the humain brain; it can’t be mapped at the level of *C. elegans* (there are about neurons and synapses, from what I’ve seen), but I read a very interesting survey by Valiant about attempts to understand the computational model underpinning the reasoning capacities of the brain. He presents four basic tasks that must be among those that the brain performs, and explains how he succeeded in earlier work (from 1994 to 2005) in finding realistic algorithms and models for these problems. He comments:

In [5,14] it is shown that algorithms for the four random

access tasks described above can be performed on the

neuroidal model with realistic values of the numerical

parameters. The algorithms used are all of the vicinal

style. Their basic steps are all local in that they only

change synaptic strengths between pairs of neurons that

are directly connected. Yet they need to achieve the more

global objectives of random access. In order that they be

able to do this certain graph theoretic connectivity prop-

erties are required of the network. The property of

expansion [15], that any set of a certain number of

neurons have between them substantially more neighbors

than their own number, is an archetypal such property.

(This property, widely studied in computer science, was

apparently first discussed in a neuroscience setting [16].)

The vicinal algorithms for the four tasks considered here

need some such connectivity properties. In each case

random graphs with appropriate realistic parameters have

it, but pure randomness is not necessarily essential.

Here reference [16] is to the Barzdin-Kolmogorov paper.

(3) Lastly, I was wondering what is the size (number of vertices) of the largest graphs for which the first non-zero Laplace eigenvalue has been computed, approximately. The best I found is a paper of Kang, Breed, Papalexakis, and Faloutsos, which describes an algorithm that they have applied successfully to real-world graphs with about a billion vertices. This is rather impressive…

]]>In the simplest case where is a cyclic extension of degree and contains all -th roots of unity (and is coprime to the characteristic of ), this essentially means proving that if has cyclic Galois group of order , then there is some with and belongs to .

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 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 in such that , where is a generator of the Galois group and is a primitive -th root of unity in . This is a statement about the -linear *action* of on , or in other words about the representation of on the -vector space . 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 is isomorphic to the left-regular representation of on the vector space of -valued functions , which is given by

.

(It is more usual to use the group algebra , but both are isomorphic).

The desired equation implies (because is generated by ) that is a sub-representation of . In , we have an explicit decomposition in direct sum

where runs over all characters (these really run over all characters of over an algebraic closure of , because contains all -th roots of unity and has exponent ). So (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 eigenvalue.

So we just see what happens (or we remember that it works). For a character such that , and the element corresponding to under the -isomorphism , we obtain . But by easy character theory (recall that is cyclic of order ) we can find with , and we are done.

I noticed that Lang hides the formula in Hilbert’s Theorem 90: an element of norm in a cyclic extension, with a generator of the Galois group, is of the form for some non-zero ; this is applied to the -th root of unity in . 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 are linearly independent as linear maps on . 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.)

]]>Algebra would look very different without her (“successive sets of symbols with the same second suffix“).

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