Pete  L. Clark
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 Math 8430 -- Topics in Arithmetic Geometry II: TuTh 12:30-1:45, 326 Boyd

Instructor: Assistant Professor Pete L. Clark, pete (at) math (at) uga (dot) edu

Course webpage: (i.e., right here)

Office Hours: During an hour long discussion section each week in which all students are strongly encouraged to attend. Otherwise by appointment.

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!

Course text: Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication, by David A. Cox. It is expensive -- I mean required. (Seriously, this is one math book which is clearly worth its price.)

Discussion Section: I would like to have an extra discussion section, one hour a week, for discussion of problems and presentation of projects.

How Your Grade is Computed: There will be homework, some assigned from Cox's book and some posted here. Moreover, every student should expect to give at least two presentations in the class, a MINOR presentation discussing some aspect of the course closely related to what has been discussed in the lectures (e.g. solution of one or more important homework problems, coverage of "omitted" material from Cox's book) and a MAJOR presentation, which will report on the student's attempted assimilation of some more substantial outside reading concerning (presumably) quadratic forms, class field theory, elliptic curves and/or elliptic modular functions.

Course Prerequisites: (listed here to give you a heads up on what you may wish to book up on, not to discourage you from taking the course.)
(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.

Course Content: Our primary goal in the course, as in David Cox's book, is to address various aspects of the following charmingly simple question: fix a positive integer n. Which prime numbers p are of the form x2 + ny2?

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 Hq(n) and Hi(n) are naturally in bijection. What is more, each one of them can be endowed with the structure of an abelian group, the class group and the canonical bijection puts these two finite abelian groups into isomorphism. Now we consider any odd prime p, prime to n, which satisfies the easy necessary condition that -n is a quadratic residue modulo p. Then it turns out that on both sides there is a natural way to assign to p an element of the class group, such that the success of our problem -- namely the representability of p by x2+ny2 -- is equivalent to p mapping to the identity element of the group.

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 Hi(n) given by class field theory is quite deep but rather difficult to make specific: for instance, it follows readily that there exists a polynomial f(x) with integral coefficients, of degree equal to h(n), the size of the class group, such that an odd prime p with -n a residue mod p is represented by our quadratic form iff f(x) has a root modulo p. Moreover, class field theory tells us that the density of the set of primes so represented is 1/2h(n). It does not make it very easy to construct the polynomial though, and this is where one brings in the theory of modular forms. As regards the size of the class group, it turns out on both sides there is some subtle extra structure which gives a simple answer (in terms of the shape of the prime factorization of n) not on the structure of the class group H(n) itself, but on its maximal quotient of exponent 2, H(n)/2H(n).

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

  • Scott Ahlgren and Ken Ono, Arithmetic of singular moduli and class polynomials (pdf )
  • Imin Chen, On Siegel's modular curve of level 5 and the class number one problem (pdf )
  • Luther Claborn, Dedekind domains and rings of quotients (pdf)
  • Luther Claborn, Every abelian group is a class group (pdf)
  • Keith Conrad, Ideal Classes and SL2 (pdf)
  • Jordan Ellenberg and Akshay Venkatesh, Reflection Principles and Bounds for Class Group Torsion (pdf)
  • Alexandru Gica, Quadratic Residues of Certain Types (pdf )
  • Benedict Gross and Don Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191-220.
  • Michael Rosen, Elliptic curves and Dedekind domains (pdf)
  • Jo(h)n Voight, Quadratic forms that represent almost the same primes (pdf)

    Handouts: Note that handouts include homework problems interspersed throughout, so should be read!

  • Handout 1: Use of Elementary Algebraic Number Theory (13 pages) (pdf)

  • Handout 2: pre/almost/Dedekind domains (19 pages) (pdf)

  • Handout 3: Elementary Theory of Quadratic Forms (14 pages) (pdf)

  • Handout 4: Identification of C(D) with Pic(O(D)) (7 pages) (pdf)

  • Handout 5: Chebotarev Density; Global Class Field Theory (15 pages) (pdf)

  • Handout 6: Proof of the Main theorem (13 pages) (pdf)