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3rd year level algebraic curves course (B3b) advisable but not essential. 3rd year algebraic number theory course (B9b) advisable but not essential.
Elliptic curves give the simplest examples of many of the most interesting phenomena which can occur in algebraic curves, they have an incredibly rich structure and have been the testing ground for many developments in algebraic geometry whilst the theory is still full of deep unsolved conjectures, some of which are amongst the oldest unsolved problems in Mathematics. The course will concentrate on arithmetic aspects of elliptic curves defined over the rationals, with the study of the group of rational points, and explicit determination of the rank, being the primary focus. Using elliptic curves over the rationals as an example, we will be able to introduce many of the basic tools for studying arithmetic properties of algebraic varieties.
Non-singular cubics and the group law; Weierstrass equations.
Elliptic curves over finite fields; Hasse estimate (stated without proof).
p-adic fields (basic definitions and properties).
1-dimensional formal groups (basic definitions and properties).
Curves over p-adic fields and reduction mod p.
Computation of torsion groups over Q; the Nagell-Lutz theorem.
2-isogenies on elliptic curves defined over Q, with a Q-rational point of order 2.
Weak Mordell-Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.
Height functions on abelian groups and basic properties.
Heights of points on elliptic curves defined over Q; statement (without proof) that this gives a height function on the Mordell-Weil group.
Mordell-Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.
Explicit computation of rank using descent via 2-isogeny.
Public keys in cryptography; Pollard's p-1 method and the elliptic curve method of factorisation.
4 examination questions.
J.W.S. Cassels, Lectures on Elliptic Curves, LMS Student Texts 24, Cambridge University Press, 1991.
N. Koblitz, A Course in Number Theory and Cryptography, Graduate Texts in Mathematics 114, Springer, 1987.
J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer, 1992.
J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer, 1986.
A. Knapp, Elliptic Curves. Mathematical Notes 40, Princeton University Press, 1992.
G. Cornell, J.H. Silverman and G. Stevans (editors), Modular Forms and Fermat's Last Theorem, Springer, 1997.
J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer, 199
Last changed 2006-05-10