(idea for this page copied from Kiran Kedlaya's excellent page on the same topic...)

If you are thinking about whether I would be a good Ph.D. advisor for you, here is some information that might be useful.

**What math you should know.** My area is arithmetic geometry, which
has the somewhat vexing feature of offering many problems which are
easy to state, but which require quite a bit of machinery to address
meaningfully. Precisely what machinery you will need depends on what
problem you work on, but here is a list of books with which anyone
planning to go into this area should be comfortable. You don't have
to know everything on this list when you start, but I would expect any student
of mine to have mastered this material by the third year.

- Algebraic number theory at the level of one of the standard
texts; for instance, the first two chapters of Neukirch's
*Algebraic Number Theory*. - Silverman,
*The Arithmetic of Elliptic Curves*. - Hartshorne,
*Algebraic Geometry*, the first four chapters. In this case, "comfortable" means "I have done all or most of the exercises." - Galois cohomology at the level of Serre's book; ideally, some etale cohomology at the level of Milne's book.

**What's the best way to learn all this stuff?** For some people, reading books and doing exercises is the best way.
For others (perhaps for most) it's better to make it a bit more
social: form a group of grad students, read a section a week, writing
down all your questions and working out examples to aid your reading.
Then convene weekly and trade questions and comments. Some of these
topics are covered by grad courses at Wisconsin, but these do not
always cover the right amount of material (e.g. the amount of
algebraic geometry you can learn in a one-semester course is not
enough) and they may not meet at a convenient time.

** What kind of math are you thinking about?** I have fairly broad
interests, but most things I think about involve, in one way or
another, the problem of studying rational points on algebraic
varieties over global fields. There are many points of view on this
part of mathematics: this set of lecture
notes and open problems exposes one of them. The lecture notes
from the Rings of Low Rank conference at Leiden represents another. I
also like modular forms and Galois representations, and am happy to
supervise students in those areas. I definitely do not require that you work on problems directly related to my own research.

**Where will I get my thesis problem?** The best way to get a
thesis problem is to find one yourself. The best way to find one
yourself is to expose yourself to lots of problems; for instance, by
coming to number theory and algebraic geometry seminars every week, by
looking over new papers posted on the arXiv, and by going to conferences
in number
theory and arithmetic
geometry. Make a resolution that whenever you go to a seminar
talk or look at a paper, you are going to formulate some question
about the topic of the paper. It might turn out to be trivial, might
turn out to be impossible, but it will get you in the habit of
thinking up questions; and it will give you something to talk about
with the speaker after the talk.

That said, I am happy to suggest potential problems or to refine and tweak ones that you bring to me.

**What is the schedule?**
As you know, UW does not guarantee funding beyond the fifth year. The
schedule is as follows: First year, take courses. Second year, learn
number theory and algebraic geometry and work on a "toy problem" which
will hopefully lead to a publication in addition to your thesis.
Second or third year, take your specialty exam, explaining what you've
done so far and fielding questions about the "essential" topics as
listed at the top of this page. Third and/or fourth year, solve your
thesis problem. Fourth or fifth year, apply for jobs, write up
thesis, get started on your next big project!

**What else should I read?** Guillermo Mantilla-Soler suggested
that I add some recommended books to this page. Some books that I
find I use all the time, besides those already mentioned above, are:
Serre's *Local Fields* and
*Linear Representations of Finite Groups* and *Oeuvres*;
*Neron Models* (Bosch, Lutkebohmert, Raynaud); *Arithmetic
Geometry* (Cornell, Silverman); *Cohomology of Number Fields*
and *Algebraic Number Theory* (Neukirch); *Modular Forms and
Fermat's Last Theorem* (Cornell, Stevens, Silverman);
*Diophantine Geometry* (Hindry, Silverman). Snap these up when
you see them on sale!

Back to Jordan Ellenberg's home page

Jordan Ellenberg * ellenber@math.wisc.edu * revised 21 Dec 2006