Course Archives Theoretical Statistics and Mathematics Unit

Course: Elliptic Curves Level: Postgraduate Time: Currently not offered

Syllabus Past Exams Syllabus: [Prerequisites: algebraic number theory (could be simultaneous), algebraic geometry (could be simultaneous)] i) Algebraic curves, divisors, Riemann-Roch theorem. ii) Definition of elliptic curves, Weierstrass form, isogeny, Tate module, Weil pair- ing, Endomorphism ring. iii) Elliptic functions and integrals, Elliptic curves over complex numbers, Uni- formization. iv) Elliptic curves over finite fields, Weil conjectures, Hasse invariant. v) Elliptic curves over local fields, Minimal Weierstrass equation, Torsion, Good and bad reduction and Neron-Ogg-Shafarevich criterion for good reduction. vi) Elliptic curves over global fields, weak Mordell-Weil, Kummer pairing, Mordell- Weil theorem over Q. vii) If time permits: Heights on projective spaces and elliptic curves and Mordell- Weil theorem; Nagell-Lutz theorem. Suggested Texts : (a) J. Silverman, Arithmetic of elliptic curves (chapters 2,3,5,6,7 and sections 8.1 to 8.4), GTM 106, Springer-Verlag (1986). (b) N. Koblitz, Elliptic Curves and Modular Forms, GTM 97, Springer-Verlag 1984. (c) J.W.S. Cassels, Lectures on Elliptic Curves, Cambridge Uni Press 1991. (d) D. Husemoller, Elliptic Curves, Springer Science and Business Media, Vol.111, 1987. Top of the page Past Exams

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