One of my students came to me with the idea of having a reading course on “classic papers in number theory”. The idea is for everyone to spend the week reading a particular paper, and then have one student lead a discussion on the key details of the paper. The paper should ideally be a “classic”, but yet accessible enough that students can read it by themselves and at least be able to contibute useful thoughts/questions to the discussion section (it definitely should *not* end up as a lecture by me). The hope is, that by the end of the semester, the students have a good background knowledge of some of the “big ideas” in the subject. My question is: what are a good choice of papers to use for this seminar? Ideal is something in the 20-30 page range for which the key ideas can be absorbed in one to two weeks, for example, Ken’s paper on the converse to Herbrand. Some wag suggested the collected works of Tate, and truthfully, any mathematician who read and absorbed all the ideas of Tate would probably be a pretty strong mathematician. Here are some things I jotted down, along with some extra comments I’m adding now:
Ribet converse to Herbrand
Mazur-Wiles (too much like the previous paper?)
Mazur Eisenstein ideal (perhaps too long?)
Something on Iwasawa theory for class groups of cyclotomic fields (I haven’t read the original papers here myself)
Serre’s Duke paper on Serre’s Conjecture
Deligne?? (I’m not even sure what paper I meant here)
Tate p-divisible groups (could be a semester course)
Tate thesis
Oort-Tate
Gross-Zagier
Ravi Annals (on lifting Galois representations, I think)
Oort-Tate
Fontaine (not sure what paper I meant here either, could be AB/Q or the definition of B-cris)
Tunnell Congruence #
Faltings
Grothendieck Tohoku
Serre GAGA
I’m sure there are many great papers that I am missing, and some here that won’t work very well. One problem with this list is that it is a little too narrow (as I write this, I am thinking perhaps of adding a paper by Hardy on the circle method – every algebraic number theorist should know about the circle method!). Suggestions very much appreciated!
Dear GR,
Rather than Mazur Eisenstein (it’s inconceivable to me that students could read this in any unit of time at all approximating a week), what about Mazur–Tate on 13-torsion points? Some of the key ideas are there, it’s only four or five pages (though still tough going), and it’s possible to try to adapt it to the 11-torsion case as well, which would be a good (non-trivial) exercise.
How about Weil’s papers on the Weil conjectures. The Jacobi sums paper would be a place to start.
There are also various seminar reports of Serre, e.g. his report on mod p modular forms, but also his report from the late 60s on the possibility of Galois reps. attached to modular forms.
There is the Serre–Tate paper in good reduction of abelian varieties; it is beautifully written, and a student with a solid background in elliptic curves has a real chance of extracting useful ideas and understanding from it (and it points the way to a more theoretic understanding than one finds in Silverman’s treatment of elliptic curves).
Katz’s articles on Serre–Tate theory are more accessible than you might think, and beautiful.
The same is true for Gross’s Companion forms paper.
Tate’s paper on isogenies of ab. varieties over finite fields is a classic, and it wouldn’t make that much sense to study Faltings without studying this first.
If I had to choose one Deligne paper, I would take Weil I. It’s not easy, but it does provide
an avenue into sheaf theory etc., since most of the arguments about l-adic sheaves that are made have complex geometry analogues that are accessible to intuition.
By the way, it’s hard to imagine any of these papers being read in a single week in any meaningful way. Probably Serre–Tate’s ab. vars. paper and Ribet’s Herbrand criterion paper come closest (and perhaps Serre’s seminar report on Galois reps., and maybe Weil).
Best wishes,
Matt
Thanks. Anything slightly further outside the box, like the circle method? (Even there, I’m not sure what paper to choose)
I think perhaps one could make a good one-week segment out of some “classic algorithms.” e.g. on the more algebraic side, perhaps Lenstra’s article on elliptic curve factorization?
I had a student write a senior thesis on Lenstra’s article, that sounds like a good choice.
“Rational Isogenies of Prime Degree”?