- André Weil, Number theory. An approach through history: From Hammurapi to Legendre (1984).
- André Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88 (Springer-Verlag, Berlin-New York, 1976).
1. Early history of number theory.
According to Jacobi, the theory of elliptic functions was born between the twenty-third of December 1751 and the twenty-seventh of January 1752. On the former date, the Berlin Academy of Sciences handed over to Euler the two volumes of Marchese Fagnano's Produzioni Matematiche, published in Pesaro in 1750 and just received from the author; Euler was requested to examine the book and draft a suitable letter of thanks. On the latter date, Euler, referring explicitly to Fagnano's work on the lemniscate, read to the Academy the first of a series of papers, eventually proving in full generality the addition and multiplication theorems for elliptic integrals.
One might similarly try to record the date of birth of the modern theory of numbers; like the god Bacchus, however, it seems to have been twice-born. Its first birth must have occurred at some point between 1621 and 1636, probably closer to the latter date. In 1621, the Greek text of Diophantus was published by Bachet, along with a useful. Latin translation and an extensive commentary. It is not known when Fermat acquired a copy of this book (the same one, no doubt, into whose margins he was later to jot down some of his best discoveries), nor when he began to read it; but, by 1636, as we learn from his correspondence, he had not only studied it carefully but was already developing ideas of his own about a variety of topics touched upon in that volume.
From then on, "numbers", i.e, number theory, never ceased to be among Fermat's major interests; but his valiant efforts to gain friends for his favorite subject were not, on the whole, crowned with success. "There is no lack of better topics for us to spend our time on" was young Huygen's comment to Wallis. At one time Fermat cherished the thought of devoting a whole book to number theory. On another occasion he tried to persuade Pascal to collaborate with him in writing such a book; of course, he realized the Pascal's gifts of exposition were far superior to his own. To our great loss, Pascal showed no interest and politely declined the suggestion; his views may well have been similar to those of Huygens. After Fermat's death in 1665, there was a great demand for a publication of his writings, hardly any of which had ever appeared in print. In 1670, his son Samuel published a reprint of Bachet's Diophantus, along with Fermat's marginal notes and an essay by the Jesuite father Jacques be Billy on Fermat's methods for solving certain types of so-called Diophantine equations. This was followed in 1679 by the publication, also by Samuel, of a volume of his father's Varia Opera, including few letters of arithmetical content. But it took half a century for this to have any effect, and in the meanwhile number theory seemed to have died off.
As to its rebirth, we can pinpoint it quite accurately. In 1729, young Euler was in St. Petersburg, an adjunct of the newly founded Academy of Sciences; his friend and patron Goldbach was in Moscow. Their correspondence has been carefully preserved and was published by Euler's great-grandson in 1843. Goldbach, in his own amateurish way, was keenly interested in mathematics and particularly in "numbers"; it is in a letter to Euler that he stated the conjecture to which his name has remained attached. On the first of December 1729, Goldbach asked Euler for his views about Fermat's statement that all integers 22n + 1 are primes. In his answer, Euler expresses some doubts; but nothing new occurs until the fourth of June, when Euler reports that he has "just been reading Fermat" and that he has been greatly impressed by Fermat's assertion that every integer is a sum of four squares (and also of 3 triangular numbers, of 5 pentagonal numbers, etc.). After that day, Euler never lost sight of this topic and of number theory in general; eventually Lagrange followed suit, then Legendre, then Gauss with whom number theory reached full maturity. Although never a popular subject, it has been doing well ever since.
Thus an account of number theory since Fermat can do full justice to the inner coherence of the topic as well as to the continuity of its development. In contrast with this, the mere fact that Fermat initially had to draw his main inspiration from a Greek author of the third century, only lately come to light, points to the entirely different character of much of the earlier mathematical work as well as to the frequent disappearance and re-appearance of essential sources of knowledge in former times. About ancient mathematics (whether Greek or Mesopotamian) and medieval mathematics (Western or Oriental), the would-be historian must of necessity confine himself to the description of a comparatively small number of islands accidentally emerging from an ocean of ignorance, and to tenuous conjectural reconstructions of the submerged continents which at one time must have bridged the gaps between them. Lacking the continuity which seems essential to history, his work might better be described by some other name. It is not prehistory, since it depends on written sources; protohistory seems more appropriate.
Of course new texts may turn up; they do from time to time. Our knowledge of Archimedes was greatly enriched in 1906 by the discovery of a palimpsest in Istanbul. What survives of Diophantus consists of six chapters or "books", while thirteen books are announced in the introduction; some new material pertaining to Diophantus, re-worked or perhaps translated from the original text, has been found recently in an Arabic manuscript; more may yet be forthcoming. Important cuneiform texts may still be buried underground in Mesopotamia, or even more probably. (according to Neugebauer) in the dusty basements of our museums. Arabic and Latin medieval manuscripts by the score await identification, even in well-explored libraries. Still, what hope is there of our ever getting, say, a full picture of early Greek geometry? In the third century B.C., Eudemos (not himself a mathematician) wrote in four " books" a history of geometry, some fragments of which have been preserved. But what may have been the contents of his history of arithmetic, comprising at least two books, all but entirely lost? Even if part of it concerned topics which we might regard as algebra, some of it must have been number theory. To try to reconstruct such developments from hints and allusions found in the work of philosophers, even of those who professed a high regard for mathematics, seems as futile as would be an attempt to reconstruct Newton's Principia out of the writings of Locke and Voltaire, or his differential calculus from the criticism of Bishop Berkeley.
2. Weil on 'Kronecker on Eisenstein'.
In 1891, Kronecker agreed to give the inaugural lecture at the first meeting of the newly founded German Mathematical Society. He cancelled the plan after losing his wife, but in a letter to Cantor, president of the Society, expressed the hope that he would be able to supply a written text for the lecture which he described as follows: 'The talk ... should have the short title "About Eisenstein" .... Here, then, besides the pure-arithmetic and the analytic-arithmetic more particularly its pure analytical investigations on elliptic functions must be identified, which have been completely lost to the consciousness of the present time ... .' Soon after that Kronecker died; he never wrote up that lecture. However he had already discussed Eisenstein's work at some length (pointing out how Eisenstein had anticipated some of Weierstrass' best-known innovations and gone well beyond them) in his last major paper on elliptic functions in 1891 by the Berlin Academy. This is how he comments upon it: 'Essentially new points of view ... particularly concerning the transformation theory of theta-functions ... were introduced by Eisenstein in the fundamental but seldom quoted "Beiträge zur Theorie der elliptischen Funktionen" published in Crelle's Journal in 1847, which are based upon entirely original ideas ... ' ... . Well could Kronecker say of that paper that it was 'seldom quoted'; ... Eisenstein's ideas could indeed seem 'as good as lost'. It is not merely out of an antiquarian interest that an attempt will be made here to resurrect them. Not only do they provide the best introduction to much of the work of Hecke but we hope they can be applied quite profitably to some current problems, particularly if they can be used in conjunction with Kronecker's late work, which is their natural continuation ... . As any reader of Eisenstein must realize, he felt hard pressed for time during the whole of his short mathematical career. As a young man he complains of nervous ailments which often compel him to interrupt his work; later he developed tuberculosis, and died of it in 1852 at the age of 29. His papers, although brilliantly conceived, must have been written by fits and starts, with the details worked out only as the occasion arose; sometimes a development is cut short, only to be taken up again at a later stage. Occasionally Crelle let him send part of a paper to the press before the whole was finished. One is frequently reminded of Galois' tragic remark 'Je n'ai pas le temps'.
3. Weil on Kronecker.
Kronecker was born in 1823, the same year as Eisenstein; they were students in Berlin at the same time. In 1847 Kronecker had to leave Berlin to take care of the business interests of his family; by the time he came back to settle there permanently, Eisenstein was dead. The first signs of an awakening interest in elliptic functions on the part of Kronecker appear in 1853; there he merely mentions the leminiscatic case as providing the generalization to the Gaussian field Q(i) of his theorem on the abelian extensions of Q. Undoubtedly he must then have studied, besides Abel, the work of Eisenstein on the division of the leminiscate; but this (even Eisenstein's great paper of 1850) was based on Abel's formulas and notations and bore no close relation to the 'Genaue Untersuchung' of 1847 .... In 1856 we find Kronecker extending his investigations to the general case of elliptic functions with complex multiplication.... In 1863, under the influence of Dirichlet's work, he introduces new functions..., states for the first time a partial result on his 'limit formula' and deduces from it a solution of Pell's equation (i.e. the determination of a unit of a real quadratic field) by elliptic functions.
4. Weil on the beginnings of elliptic functions.
Eisenstein, having laid the foundations for a theory of elliptic functions, was able to carry out much of his design for the building itself, and to indicate how he wished it completed. Kronecker, having conceived ambitious plans for a vastly enlarged edifice, started, rather late in life, to dig deeper foundations but found time for little else. It is idle to speculate about the kind of continuation he had in mind; perhaps he did not know it himself. On the other hand, the number-theoretic motivation for the huge efforts of his later years can be discerned clearly and deserves a short digression here. Kummer, then Dirichlet, had been his teachers and remained his lifelong friends. Kummer's arithmetical work centered around cyclotomic fields, their ideal classes and their class-numbers. As Dirichlet had first discovered in a different setting, those class-numbers depend upon the values of Dirichlet's L-functions at s = 1, i.e. ultimately upon the values, for suitable values of the arguments, of the simple series discussed above in our Chapter 7. Kummer, having reestablished these results in the language of his ideal-theory, proceeded to investigate the p-adic properties of those values, beginning with his celebrated congruences for Bernoulli numbers. This had in fact been the deepest part of his work, more valuable perhaps than the more spectacular applications to Fermat's problem and even to the reciprocity laws. Early in life (in 1853) Kronecker found the conceptual justification for Kummer's singling out the cyclotomic fields; not only are they (as Gauss had discovered) abelian extensions of the rational number-field, but they are the only ones. Already in the same paper  where he announced this momentous discovery, Kronecker stated the corresponding result for the Gaussian field Q(i) and the lemniscatic functions. Out of this grew the idea that the division of elliptic functions with complex multiplication must play the same role, for the corresponding imaginary quadratic fields, as the division of the circle plays for Q, and that of the lemniscate for Q(i). Allusions to this occur in the letter to Dirichlet of May 1857. This, as he wrote later to Dedekind, had been the most cherished dream of his youth ('Mein liebster Jugendtraum').