André Weil

Foundations of Algebraic Geometry

Algebraic preliminaries Algebraic theory of specializations Analytic theory of specializations The geometric language Intersection-multiplicities (special case) General intersection-theory The geometry on abstract varieties The calculus of cycles Divisors and linear systems Comments and discussions Appendix I. Normal varieties and normalization Appendix II. Characterization of the

Elliptic functions according to Eisenstein and Kronecker

i

Remarks on the Cohomology of Groups

-symbol by its properties Appendix III. Varieties over topological fields Index of definitions.

Arithmetic on Algebraic Varieties

I EISENSTEIN.- I Introduction.- II Trigonometric functions.- III The basic elliptic functions.- IV Basic relations and infinite products.- V Variation I.- VI Variation II.- II KRONECKER.- VII Prelude to Kronecker.- VIII Kroneckers double series.- IX Finale: Allegro con brio (Pells equation and the Chowla-Selberg formula).- Index of Notations.

Fonction zêta et distributions

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Who Betrayed Euclid

D. G. Northcott has recently contributed some interesting new theorems ([4a], [4b]) to a subject which I introduced in my thesis [1] under the above-given title, and which had been further developed by Siegel [2] and myself [3]. It is my purpose here, by making explicit some concepts which had remained implicit in these papers, to supply what seems to be the proper algebraic foundations for that theory, and to give a comprehensive account of its results, including some new ones, up to date.

Mathematical Teaching in Universities

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Rue d’Ulm

Some time ago your Archive printed a paper on Greek mathematics which, in tone and style as well as in content, fell significantly below the usual standards of that journal. As it has already quite adequately (if perhaps too gently) been refuted there by V.D. Waerden and by Freudenthal, there is no need for referring to it by name. My only purpose in this letter is to point out that we have here almost a textbook illustration of the very thesis which the author (let us call him Z) sought to discredit, viz., that it is well to know mathematics before concerning oneself with its history; just as it is well to know Greek before dealing with Greek mathematics.

Algebraic number-fields

exist exactly 9 pairs q(a) = r(a) = b where 1 < b <9. This is illustrated in the table given below. The following is the outline of a lecture once given by the author at a joint meeting of the Nancago Mathematical Society and of the Poldavian Mathematical Association. It is printed here at the editors request, as the principles stated there seem to be of general application. 1. Improvements in the mathematical teaching in Poldavian Universities depend largely upon general improvements in the educational system in Pol-davia. Mathematicians should devote themselves to the task of making such improvements as lie within their power at present, and thus contributing their share towards general reforms, which in turn will enable them to make further progress. 2. No satisfactory results can be achieved unless reforms are made both in school-teaching and in University teaching. So far as school-teaching is concerned , the efforts of mathematicians in the country should be mainly directed towards necessary changes in the curricula and towards the training of better teachers. 3. University teaching in mathematics should: (a) answer the requirements of all those who need mathematics for practical purposes; (b) train specialists in the subject; (c) give to all students that intellectual and moral training which any University, worthy of the name, has the duty to impart. These objects are not contradictory but complementary to each other. Thus, a training for practical purposes can be made to play the same part in mathematics as experiments play in physics or chemistry. Thus again, personal and independent thinking cannot be encouraged without at the same time fostering the spirit of research. 4. The study of mathematics, as well as of any other science, consists in the acquisition of useful reflexes and in that of independent habits of thought. The acquisition of useful reflexes should never be separated from the perception of their usefulness.

Fermat and His Correspondents

A l’Ecole (comme nous disions) les eleves etaient groupes par «turnes» ou salles d’etude. Des avant la rentree mon premier souci fut de rechercher des «coturnes» sympathiques. Nous fumes cinq, Laberenne, Delsarte, Yves Rocard, Barbotte («cacique», c’est-a-dire premier de la promotion) et moi. Laberenne avait ete mon camarade en taupe chez Grevy; c’etait un grand garcon a l’esprit ouvert, bon camarade, assez peu scolaire. Delsarte arrivait de Rouen, apres une seule annee de taupe comme moi. Rocard venait de Louis-le-Grand et installa aussitot dans son casier de grands registres cartonnes de noir, couverts d’une ecriture menue mais fort lisible, pleins deja d’idees personnelles et de calculs sur la theorie cinetique des gaz. Tous les quatre nous avions, comme on dit, mauvais esprit. On ne pouvait en dire autant de notre cacique, venu de la taupe de Versailles; fils de militaire, respectueux des autorites, peu apte au canular, il fut toujours bien vu de la direction; ce n’etait pas sous ce jour que me l’avaient depeint des camarades versaillais. Il finit par se trouver assez mal a l’aise en notre compagnie; nous sentions un peu trop le fagot.