General Exam for Xin Wan
Special topics: Algebraic Number Theory & Algebraic Geometry.
Committee: Skinner(chair), Katz, Klartag.
April 25th, 2008, at Skinner's office.
Complex Analysis:
Maximal principle; Hadamard Three Circle theorem.
Real Analysis:
Construct a measurable set on [0, 1] interval such that its intersection with any
subinterval of [0, 1] has measure neither 0 nor equal to the measure of the whole
subinterval (Which I didn't quite remember). Then Klartag asked me to consider its
characteristic function. I thought he was giving some hints for the construction,
but actually he just wanted me to say that this function is not Riemann integrable.
I was totally confused at this point.
The questions above were asked by Klartag.
Algebra & Algebraic Number Theory:
S: How do you tell if the Galois group of an irreducible polynomial of degree n is
contained in A_n? (Discriminant) Can you give me an example?
S: How do you construct a polynomial/Q whose Galois group is S_n? (mod p)
Katz: Write down such an irreducible polynomial for n=7. (Here since 7 is a prime,
one only needs to require the Galois group to contain an element of the form (ab)
and a 7 permutation). How do you find a "universal" polynomial of degree p over F_p
that must be irreducible? (x^p-x-1, using Artin-Schrier theory, which I didn't quite
remember during that time)
S: How do you construct extensions of number fields? Talk about cyclotomic extensions
and Kummer extensions.
S: Do you know Chebatarev density theorem? Does it imply any classical results?
(Dirichlet) Why?
S: Talk about Galois groups over local fields. What's the maximal unramified and tamely
ramified extension?
S: Talk about class field theory, both local and global. How do they relate to each
other? What's the Artin symbol?
Katz: What's the common point in the proofs of CFT, density theorems and so on? I was
not quite sure about what he wanted. OK, let me make my question more directly. Is
there any proof of these theorems purely algebraic? (no)
S: What does CFT tell us about the L-series for Hecke characters? (They are actually
Artin L-series for 1 dimensional Galois representations)
Then we had a 10 minutes break.
Algebraic Geometry: (mostly by Katz)
Talk about smooth projective curves, anything you know. (genus, RR, Serre duality)
Define genus. Why are the geometric genus and arithmetic genus equal? State Serre
duality. What happens to this if the field of definition is not algebraically closed?
(nothing happens) Do you know the canonical isomorphism between H^1(X, \omega) and k
explicitly? (Residue map, which I couldn't answer at that time)
Talk about RR. What's it good for? (Group laws for elliptic curves) Why? Consider an
elliptic curve, what happens if the definition field is not algebraically closed?
Can you give me an example of an elliptic curve having no rational points? Compute
the genus of the curve: y^2=x^691-1. After I obtained 2g+2=692, Katz insisted to ask
me to write down g=345. What about the curve: x^691+y^691+z^691=0? It's smooth, so I
can use the genus formula. Then Katz said this time I don't want you to make the
multiplication.
Do you know deRham cohomology? Define it. I mentioned that for smooth projective
varieties/k algebraically closed, the first Hodge spectral sequence degenerates. Any
restrictions? Oops, of course only for char=0, otherwise only valid for small dimensions.
Then Katz asked me a question which seemed to be proving that the first deRham cohomology
of a curve is 2g, but probably I have misunderstood the question. After a while, we moved
on to other questions.
Do you know algebraic groups? Define it and give some examples. Are any of them rational?
(Here the tricky point is that the orthogonal groups and the Sp's are rational, using
Cayley transformation) Are abelian varieties rational? Why?
S: How do you compute the cohomology of projective spaces? (Here he wanted to hear the
words "Cech cohomology") What kind of covering are you using? (affine) Why?
Then they asked me to wait outside for the result. 1 minute later, Katz opened the door
and smiled: "you have done congratulations". Then they shook hands with me.
The process was very pleasant. Although sometimes I couldnt answer the question or made
some very stupid mistakes, they were very willing to give hints.
The exam lasted for about 2 hours 30 minutes.