Winter Quarter 2019, Math 225B, Algebraic Number Theory II
Instructor:
A. Agboola
Lecture:
TuTh 9:30am-10:45am, GIRV 1108
Office:
6724 South Hall
Office hours:
TuTh 11:00am--12:30pm (or by appointment)
Required Text:
J. Silverman, The Arithmetic Of Elliptic Curves, Springer, (1986) (required).
Additional Texts:
The following additional texts may be helpful:
H. Darmon, Rational Points On Modular Elliptic Curves, AMS
(2004). (You can download a copy of this for free from Darmon's
webpage at McGill.)
A. W. Knapp, Elliptic Curves, Princeton University Press (1994).
S. Lang, Elliptic Functions, Springer, (1987).
J. Milne, Elliptic Curves (a version of this, as well as much
else of great value,is available for free from Milne's webpage).
G. Shimura, An Introduction To The Arithmetic Theory Of
Automorphic Functions, Princeton University Press (1984)
J. Silverman, Advanced Topics In The Arithmetic Of Elliptic
Curves, Springer (1994)
J. Silverman, J. Tate, Rational Points On Elliptic Curves, Springer,
(1994).
Examinations:
There will be no examinations given in this course.
Course Outline:
We shall aim to cover the following topics. Additional topics will be
covered if time permits.
A crash course on agebraic varieties and algebraic curves.
Geometry of elliptic curves. Weierstrass equations and the group law. Isogenies.
Elliptic curves over finite fields.
Elliptic curves over local fields: formal groups, minimal models, reduction.
Elliptic curves over global fields: The torsion subgroup and the
Nagel-Lutz theorem.
Heights and the Mordell-Weil theorem.
Galois cohomology. Descent; the Selmer and Tate-Shafarevich groups.