Pete L.
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I am quite amenable to booking ``extra'' office hours. The ground rules are: (i) please give me at least 24 hours notice. (ii) Please send me an email the night before a morning appointment or the morning of a later appointment to remind me that we are meeting. (iii) If I do not show for an appointment (empirically, the chance that I will fail to show seems to be about 5-10%), feel free to call me on my cell phone. Probably I'm not too far away. (iv) If we do book an appointment, please do show up or call or email to let me know you're not coming!

(i) Undergraduate level number theory, especially congruences (especially, quadratic reciprocity). Take a look at my 4400/6400 course page for this, especially Handouts 1-4, 9.5 and 14-18.

(ii) Basic graduate algebra (Math 8000), especially Galois theory.

(iii) Some familiarity with intermediate level algebraic number theory topics, such as rings of integers of number fields, primes splitting/remaining inert/ramifying in extensions, ideals in Dedekind rings, p-adic numbers and/or adeles would be helpful.

(iv) Similarly, some passing acquaintance with elliptic curves would be nice: anyone who took Prof. Lorenzini's course or my VIGRE research group last semester knows (or should know) more than enough.

The study of this question naturally leads to the consideration of several important areas of modern number theory: quadratic forms, class field theory, modular functions, and elliptic curves. Each one of these is itself a substantial and multifaceted branch of number theory, such that one could not only spend an entire course on that topic, but that there is the dangerous possibility that one could come away from that entire course not really having seen the light of day. (Class field theory in particular is a notorious black hole, one of those unfortunate subjects where learning the proofs of the theorems has little to do with one's ability to appreciate and apply them. First courses on quadratic forms or modular functions could be very pleasant, but one needs a very stern dose of class field theory -- in the representation-theoretic style of Tate and Langlands -- in order to make contact with most modern work. Admittedly elliptic curves are nice.)

As I take it, the point of Cox's book is that a better way to gain an appreciation of these subjects than to dutifully study each one separately is to learn just enough about each one to see it applied to a particular nontrivial problem, and thus to use this problem as a bridge to ferry information and insight between these various subfields of number theory. I am a firm believer in the method of problem solving by transportation: what is true but intricate and obscure in one branch of mathematics often becomes natural and obvious when translated into the framework of another.

In this course, the two distinct terrains are that of on the one hand binary quadratic forms and on the other hand ideals in quadratic rings. More precisely, to the positive integer n we associate two sets: the first is the set of primitive binary quadratic forms of discriminant n, and the second is the set of invertible ideals in the ring Z[\sqrt{-n}]. Each of these sets is infinite, but under a natural equivalence relation they become finite. These two finite sets, call them H

The essential questions are then what can we say about the structure of the class group, and how explicitly can we make the condition that p maps to the identity? For instance, if the class group is trivial, then we are done, and this is the misleadingly simple situation that obtains for (only) a few very small values of n, like n = 1,2,3. The understanding of the class group H

Here are some papers on related topics, which might be suitable for the major presentation: