# Elliptic Curves

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*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
groups.

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.

#### Readership

Graduate students and research mathematicians interested in elliptic curves.