As a followup to the earlier post of the evolution of the paper, here’s another piece of information: the word “conjecture”, in the sense currently used (and sometimes abused), doesn’t appear in the *Annales de l’ENS* before the 1950’s (searching for “conjectur*” in the whole text, and disregarding two early papers of Jacobi and Beltrami which are really discussions of how they were led to some result or other, and some chemistry/mineralogy papers).

## The evolution of the paper

One of the nice consequences of the current development of online archives is that reading the classics has really become much easier than before. Of course, “classics” refers here to those mathematical texts which were published in journals since the middle to late 19th century, and are in easily accessible languages (which, for me, means French or English, and I guess I can look a little bit at Italian papers without feeling too lost too quickly).

As a sometime “flâneur” along those roads, my favorite site is NUMDAM, which contains archives of many journals and seminars, mostly French (there are a few Italian journals and seminars included, as well as Compositio Mathematica). Compared with other sites like JSTOR, Project Euclid, or the Goettingen archive, NUMDAM seems much quicker and easier to browse. It is also freely available (except for the last few years of some journals, but since we speak of classics here, this is not an issue). Moreover, besides the standard PDF format, it has copies of the papers in the djvu format, which is much more compact.

During the last snowy Easter week-end, I’ve looked at NUMDAM to check some points of what might be called the natural evolution of the mathematical paper. I used the Annales Scientifiques de l’École Normale Supérieure as a source, because it has been published continously since 1864.

So one finds, in no particular order (nor guarantee of correctness of the dates mentioned…):

- That proper separate bibliographies did not occur until around 1948, with footnotes then indicating that “Numbers between brackets refer to the bibliography located at the end of the paper”.
- Except that this was mostly in French; in fact, the first papers not in French in this journal appear in 1968 only (series 4, volume 1; it may be, and this would be easy to check, that this new series was the first where languages other than French were permitted). The second such paper is quite famous: it is John Tate’s Residues of differentials on curves.
- What about the first
*joint*paper? The honor goes to Castelnuovo and Enriques, in 1906. But this is an outlier: the next two only occur in 1934 (one is also famous, due to Leray and Schauder), and papers with more than one author can be counted on two hands until 1954. (I am disregarding, here, two earlier papers by Pasteur and Raulin, in 1872, which have to do with the fight against silk-worm diseases, and another one on the construction of the official “mètre étalon”, or yardstick; such non-mathematical papers disappeared around that time, although in 1896, the journal included the discourse given by the renowned Désiré Gernez on the occasion of the inauguration of a statue of Pasteur.) - Then, what about the first joint paper in English? There’s Tate again, with F. Oort in 1970.
- OK, and what about the first paper written by a French person in English? Here, there can be some ambiguity, since someone may well be French without having a name that claims it to the world (…), but Alain Connes may be the first one.
- Another information that could be interesting would be the first article written by a woman, but since first names are typically missing from much of the early tables of contents (and can be ambiguous), this is even harder to decide. The first unambiguous example I saw is due to Jacqueline Ferrand, in 1942.

In an idle moment today, I also looked more quickly at the Bulletin de la Société mathématique de France, which goes back to 1872. The various dates are somewhat similar; there is a single paper not in French before 1952, by Wiener (in 1922). I can envision a vicious fight among the editors to decide if it could be published; at least the first few paragraphs mention that the work on which the paper is based was done in France, and thank profusely Fréchet for his insights… (There are also a few earlier papers *translated in French* from their original language, for instance Heegaard’s thesis appears in 1916, translated, presumably, from the Norwegian). The first joint paper appears in the same year, 1916, and the next one only in 1930.

Reading the titles of article before 1950 or so can be quite amusing; mixed with terminology that still seems very modern and recognizable, there are certain gems like the anallagmatic metric (“métrique anallagmatique”) of R. Lagrange, in 1942. Strangely enough, this is a word recognized by the OED: “Not changed in form by inversion: applied to the surfaces of certain solids, as the sphere”, with quotations from Clifford, in 1869, and Salmon, in 1874. This is too bad, since I was hoping that the date indicated that this paper was an elaborate coded transmission to the Free French… There is also in most titles an element of high and formal seriousness that can be a bit tiring; all those papers starting with “Sur une équation…”, or “Sur une propriété…”, or “Sur *quelque chose*…” (“On *something*…”) do not give a great impression of the fun of doing mathematics.

The authors of the early papers are also divided pretty sharply between names we all know (or have heard about, not only French, but from most countries we think of as having a strong mathematical culture in that period), and completely obscure characters (at least to me). The most magnificent I found is the (probably) redoubtable Gaston Gohierre de Longchamps who published three papers in the *Annales de l’ENS* between 1866 and 1880 (the second of which concerns Bernoulli numbers, and quotes the equally remarkable M. Haton de la Goupillère).

Finally, one observes with interest that the poor reputation of rigor of these older mathematicians is an unwarranted slander; no erratum is needed for the entire corpus of the *Annales de l’ENS* until 1953, with the single exceptions of one in 1907 (a paper by Émile Merlin), and of a remonstrance by Brouwer pointing out a few mistakes in a paper of Zoretti in 1910, which the latter rather grudgingly accepts (by claiming that another mathematician had priority in finding those)…

## Gossip

Here is a minor post about one of the minor pleasures of life: etymology. I realized recently that “a gossip” (as a person) and “une commère”, which are more-or-less translations of each other in French and English, have the same higher-level etymology. In other words, they evolved in the same way from words having the same meaning, that of “godmather”, or “marraine” respectively, although they do not share a common older Latin, or Greek, or other, root (the English word comes from “God” and “sib”, of teutonic origins related to “kin”; the French comes from Latin “co” and “mater”). In English, the forms “Godsibbas”, “godzybbe”, go as far back as the 11th century, but the modern meaning is attested in the OED only from the late 16th century on. In French, the original meaning is seen from the late 12th century, and the modern one from the 14th century.

(I will leave aside any discussion about the historical or psychological insights provided by this parallel evolution of godmather).

There’s not any mathematics in there, of course, except for having decided to look for the relevant information (in the OED and the *Grand Robert de la Langue Française*, or the Trésor de la langue française, respectively) after discussions with colleagues at ETH where the question arose of the right translation of “gossips” in French. Note that although “commérages” is in a sense obvious, the undercurrents seem to not be quite the same…

## Deligne’s proof of the Weil conjectures

Today, I will jump on the coat-tails of Terry Tao’s link to Brian Osserman’s article on the Weil Conjectures for the (upcoming) Princeton Companion to Mathematics (edited by Tim Gowers and June Barrow-Green), and use this as an occasion to mention an old text of mine which was meant to be an introduction to Deligne’s (first) proof of the Riemann Hypothesis over finite fields. [The date of 2008 on the PDF file simply reflects that I recompiled it recently to incorporate a PDF version of the single figure in it, which I may as well include here to break the textual monotony…

this is supposed to illustrate the famous “cycle évanescent”, a terminology which I find somewhat more poetic in French, where “évanescent” is rather old-fashioned and formal, than in the English “vanishing cycle”, although the etymology is in fact the same].

I say “text” because it’s a bit hard to define in what category this could/should be put. I wrote it as a Graduate Student at Rutgers in 1997, as the text for two lectures I gave at the end of a graduate course by my advisor, Henryk Iwaniec, on the general topic of solutions of equations over finite fields. His lectures had ended with a detailed proof of the Riemann Hypothesis for curves (and their all-important consequence, for analytic number theorists, the Weil bound for Kloosterman sums), following Bombieri’s adaptation of Stepanov’s method.

I took this background as starting point to give as complete a sketch as I could (given my understanding of the algebraic geometry at the time…). This means the text is not really self-contained: the basic theory of algebraic curves is assumed to be known, at least informally. However, the somewhat original feature (which is why I am mentioning this text here) was that I tried to explain the origin, the formal definition and some of the properties of étale cohomology “from scratch”. This culminates, at the end of the first part, with an almost complete and rigorous computation of the étale cohomology of an elliptic curve over an algebraically closed field (based on the basic theory of isogenies of elliptic curves). This is in fact not difficult at all, but I had not seen it done elsewhere in a self-contained way, and it certainly seemed enlightening to me.

As implied in the previous paragraph, there are two parts to the paper. The first is the preparation (i.e., supposedly leading to where things stood before Deligne’s breakthrough), and it still seems to have some value for people interested in starting to learn this important theory. The second tries to follow Deligne’s proof very closely, and is in fact probably hard to read, and therefore of less interest. (Moreover, Deligne’s first proof was largely subsumed in his *second* proof, which leads to much more general statements, and in particular to his Equidistribution Theorem; this second proof is really quite a bit more involved than the first one, in terms of algebraic geometry).

## Averages of singular series, or: when Poisson is everywhere

Here is another post where the mediocre mathematical abilities of HTML will require inserting images with some TeX-produced text…

I have recently posted on my web page a preprint concerning some averages of “singular series” (another example of pretty bad mathematical terminology…) arising in the prime *k*-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by Goldston-Pintz-Yildirim that there are infinitely many primes *p* for which the gap *q-p* between *p* and the next prime *q* is smaller than *ε* times the average gap, for arbitrary small *ε>0*.

This result refers to the behavior, on average over * h=(h_1,…,h_k)*, of the constant

*S(*which is supposed to be the leading coefficient in the conjecture

**h**) |{*n*<*X* | *n*+*h_i* is prime for *i*=1,…,*k*}|~*S( h)*

*X*(log

*X*)

^{-k}

Gallagher showed that the average value of *S( h)* is equal to

*1*, and I’ve extended this in two ways…