Tata Institute of Fundamental Research Publications
Volume: 9; 2006; 400 pp; Hardcover
MSC: Primary 60; 43;
Print ISBN: 978-81-7319-502-0
Product Code: TIFR/9
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Elliptic CurvesShare this page
R. V. Gurjar; Kirti Joshi; N. Mohan Kumar; Kapil H. Paranjape; A. Ramanathan; T. N. Shorey; R. R. Simha; V. Srinivas
A publication of the Tata Institute of Fundamental Research
These notes constitute a lucid introduction to
“Elliptic Curves”, one of the central and vigorous areas
of current mathematical research. The subject has been studied from
diverse viewpoints—analytic, algebraic, and arithmetical. These
notes offer the reader glimpses of all three aspects and present some
of the basic important theorems in all of them. The first part
introduces a little of the theory of Riemann surfaces and goes on to
the study of tori and their projective embeddings as cubics. This part
ends with a discussion of the identification of the moduli space of
complex tori with the quotient of the upper half plane by the modular
The second part handles the algebraic geometry of elliptic curves. It begins with a rapid introduction to some basic algebraic geometry and then focuses on elliptic curves. The Rieman–Roch theorem and the Riemann hypothesis for elliptic curves are proved, and the structure of the endomorphism ring of an elliptic curve is described.
The third and last part is on the arithmetic of elliptic curves over \(Q\). The Mordell–Weil theorem, Mazur's theorem on torsion in rational points of an elliptic curve over \(Q\), and theorems of Thue and Siegel are among the results which are presented. There is a brief discussion of theta functions, Eisenstein series and cusp forms with an application to representation of natural numbers as sums of squares.
The notes end with the formulation of the Birch and Swinnerton–Dyer conjectures. There is an additional brief chapter (Appendix C), written in July 2004 by Kirti Joshi, describing some developments since the original notes were written up in the present form in 1992.
A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.
Table of Contents
Table of Contents
Graduate students and research mathematicians interested in elliptic curves.