One Semester of Elliptic Curves (EMS Series of Lectures in Mathematics) (EMS Series of Lectures in Mathematics)

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 TM 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. Preface......Page 6 Elliptic integrals......Page 10 Elliptic curves......Page 19 The shape of elliptic curves......Page 21 Elliptic integrals revisited......Page 27 Elliptic functions......Page 33 Poles and zeroes......Page 41 A projective interlude......Page 50 The group structure on an elliptic curve......Page 56 Equivalences of elliptic curves......Page 61 Equivalence of periods......Page 63 The j-function......Page 66 Formulaire......Page 77 Finite fields......Page 79 The curve y2=x3+1......Page 82 Normal forms and twists......Page 84 Division polynomials......Page 87 Algebraic version......Page 91 Torsion points......Page 97 Lattice inclusions......Page 103 Complex multiplication......Page 109 Complex multiplication lattices.......Page 111 Elliptic curve interpretation......Page 113 Modular forms......Page 118 Elliptic curve interpretation......Page 120 Modular forms of higher level......Page 123 Sums of four squares.......Page 129 Hints to exercises......Page 134 Some further reading......Page 136 Index......Page 137 Torsten Ekedahl. Includes Bibliographical References (p. [128]) And Index.