Elliptic Functions (Graduate Texts in Mathematics, Vol. 112) 2nd Edition by Serge Lang (PDF)

Description

Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century. The book is divided into four parts. In the first, Lang presents the general analytic theory starting from scratch. Most of this can be read by a student with a basic knowledge of complex analysis. The next part treats complex multiplication, including a discussion of Deuring’s theory of l-adic and p-adic representations, and elliptic curves with singular invariants. Part three covers curves with non-integral invariants, and applies the Tate parametrization to give Serre’s results on division points. The last part covers theta functions and the Kronecker Limit Formula. Also included is an appendix by Tate on algebraic formulas in arbitrary charactistic.

User’s Reviews

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

⭐This book gives an overview of the theory of elliptic curves rom a more number-theoretic viewpoint. As such, it is good reading for anyone interesting in applications of elliptic curves to physics, such as integrable models in statistical mechanics and superstring theory. Readers interested in learning the theory in order to apply it to cryptography may be disappointed as the theory of elliptic curves in characteristic p is put in two appendices. The author is well known for his expertise in algebra and number theory and this is brought out nicely in part two of the book, which deals with complex multipication. The most useful part of the book for those interested in physics applications is part four, dealing with the theory of theta functions, even though they are not discussed in the context of being solutions of the heat equation. String theorists will however have to supplement the reading with other sources in order to gain more familiarity with theta functions. There are no exercises in the book, since it qualifies as being more of a monograph, but there are numerous places where one has to fill in the details of the arguments in the proofs.

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