Dear Armin,

  I am told that you are considering the Elliptic Curves
Lecture Course next term, but that you may have taken
fewer of the relevant undergraduate lecture courses than
many of the other students.
  In order to help you to fill in any gaps (and to help
you to decide whether it would be appropriate for you
to take the lecture course), I am attaching the pdf file
of a document I finished writing a few days ago, which
I shall shortly be putting on the web for all students who
think they might take this course. It is intended to refresh
your memory of pre-requisite material, mainly from the first
two undergraduate years. The document starts out at a very
elementary level: groups, quotient groups, rings, fields, 
elementary number theory, etc, and then discusses valuations, 
completions, followed by a few topics in Geometry (singularities 
of curves, projective space). I would expect that, for most
students likely to take Elliptic Curves, most of the topics
in this document will be review of material already covered.
So, if you find that a lot of it looks unfamiliar, then that
is already a warning sign that you would find the course
difficult (although not impossible, if you are able to fill
in gaps beforehand). At the end of the document is some
suggested further reading (which I also repeat below), 
and a problem sheet ("Sheet 0") on the pre-requisite undergraduate material.
  I suggest that you read through the attached document during
the next few days, and then come to see me for a chat, this
coming Tuesday at 5:30 pm, at my College Office, which is
located at: 8 New College Lane (and press the button with the
label "Dr Flynn").
  You should also bear in mind that, as well as the computational
aspects of Elliptic Curves, the lecture course will include
several substantial theorems (on Formal Groups, Isogenies
and the Mordell-Weil Theorem). So, as well as actual prerequisite
topics, there is also the question of whether your total    
amount of previous Pure Mathematics has given you the skill
level to deal with the more theoretical parts of the course.

  Best regards, Victor Flynn

Suggested Background Reading (for prerequisites):

W. Keith Nicholson. Introduction to Abstract Algebra. Wiley, 1999.
Peter J. Cameron. Introduction to Algebra. OUP 1998.
Alan Baker. A Concise Introduction to the Theory of Numbers. CUP, 1985.
I.M. Niven, H.S. Zuckerman and H.L. Montgomery.
  An Introduction to the Theory of Numbers. Wiley, 1991.
W.A. Sutherland. Introduction to Metric and Topological Spaces. OUP, 1975.
Miles Reid. Undergraduate Algebraic Geometry. CUP, 1988.

Reading for the actual Elliptic Curves lecture course itself:

J.W.S. Cassels. Lectures on Elliptic Curves.
LMS-ST 24. Cambridge University Press, Cambridge, 1986.
J.H. Silverman. The Arithmetic of Elliptic Curves.
GTM 106. Springer-Verlag, 1986.