Elliptic Functions according to Eisenstein and Kronecker by Andre Weil (PDF)

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Drawn from the Foreword: (…) On the other hand, since much of the material in this volume seems suitable for inclusion in elementary courses, it may not be superfluous to point out that it is almost entirely self-contained. Even the basic facts about trigonometric functions are treated ab initio in Ch. II, according to Eisenstein’s method. It would have been both logical and convenient to treat the gamma -function similarly in Ch. VII; for the sake of brevity, this has not been done, and a knowledge of some elementary properties of T(s) has been assumed. One further prerequisite in Part II is Dirichlet’s theorem on Fourier series, together with the method of Poisson summation which is only a special case of that theorem; in the case under consideration (essentially no more than the transformation formula for the theta-function) this presupposes the calculation of some classical integrals. (…) As to the final chapter, it concerns applications to number theory (…).

User’s Reviews

Editorial Reviews: Review E. Hlwka, Monatshefte für Mathematik, Bd. 83, 1977, Heft 3: “This book does not fit into any normal framework and is not, as one might think, a historical work. It is one that contains too personal thoughts of A. Weil and that will most certainly have a fundamental role to play in the history of mathematics. It is warmly recommended reading for anyone interested to learn what is happening in mathematics now.” About the Author Biography of André WeilAndré Weil was born on May 6, 1906 in Paris. After studying mathematics at the École Normale Supérieure and receiving a doctoral degree from the University of Paris in 1928, he held professorial positions in India, France, the United States and Brazil before being appointed to the Institute for Advanced Study, Princeton in 1958, where he remained until he died on August 6, 1998.André Weil’s work laid the foundation for abstract algebraic geometry and the modern theory of abelian varieties. A great deal of his work was directed towards establishing the links between number theory and algebraic geometry and devising modern methods in analytic number theory. Weil was one of the founders, around 1934, of the group that published, under the collective name of N. Bourbaki, the highly influential multi-volume treatise Eléments de mathématique.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Nice quality ,price

⭐Beautiful book.

⭐Weil has written a beautiful preface to this work: “‘When kings are building’, says the German poet, ‘Carters have work to do.’ Kronecker quoted this, in his letter to Cantor of September 1891, only to add, thinking of himself no doubt, that each mathematician has to be king and carter at the same time.”But carters need roads. Not seldom, in the history of our science, has it happened that a king opened up a new road into the promised land and that his successors, intent upon their own paths, allowed it to be overrun by brambles and become unfit for transit.”The mathematical literature would be well served by more books like this that give organized modern presentations of the methods and results of past mathematical authors. Books along these lines are Edwards’ “Galois Theory” and “Riemann’s zeta function”.Even for a general mathematical reader (who is willing to spend time working out calculations), Chapter II would be worth reading, to see Eisenstein’s extraordinary construction of the trigonometric functions. In Chapter III, Weil shows Eisenstein’s construction of the elliptic functions, following the approach of Chapter II. Chapter IV proves relations between different Eisenstein series. The only other chapter that I have spent serious time with is the final one, which gives the only presentation of the Chowla-Selberg formula (giving special values of the Dedekind eta function) in a monograph I know of.

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