Course Archives Theoretical Statistics and Mathematics Unit
Course: Elliptic Curves
Level: Postgraduate
Time: Currently not offered
Past Exams


[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.

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