# One Semester of Elliptic Curves

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*Torsten Ekedahl*

A publication of the European Mathematical Society

These lecture notes grew out of a one semester introductory course on
elliptic curves given to an audience of computer science and
mathematics students, and assume only minimal background
knowledge. After having covered basic analytic and algebraic aspects,
putting special emphasis on explaining the interplay between algebraic
and analytic formulas, they go on to some more specialized
topics. These include the \(j\)-function from an algebraic and analytic
perspective, a discussion of elliptic curves over finite fields,
derivation of recursion formulas for the division polynomials, the
algebraic structure of the torsion points of an elliptic curve,
complex multiplication, and modular forms.

In an effort to motivate basic problems the book starts very slowly
but considers some aspects such as modular forms of higher level which
are not usually treated. It presents more than 100 exercises and a
Mathematica ™ notebook that treats a number of calculations
involving elliptic curves.

The book is aimed at students of mathematics with a general interest in
elliptic curves but also at students of computer science
interested in their cryptographic aspects.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Undergraduate and graduate students interested in elliptic curves.