Ko Honda's generals
Committee: Shimura (head), Katz, Nelson
Topics: Algebraic Geometry and Algebraic Number Theory
Talk about factorization of a prime in a polynomial ring. What is
irreducibility? For what rings R is it true that R[x_1,...,x_n] is
a unique factorization domain? (I said R UFD). Talk about PIDs. What is
wrong with unique factorization if we don't have a domain?
Now, PIDs are Noetherian, but are there UFDs which are not?
Define function of bounded variation. How can you associate a
measure to a monotonically increasing function which is bounded?
Take f entire. What are the conditions for the existence of a
square root function? Look at the case where the region is bounded
and simply connected. What are the conditions there? How would you
extend a locally constructed square root to all of the region? Talk
about analytic continuation on simply connected regions. Next, you
can do the same problem by using Weierstrass factorization....
Talk about Fourier transform. What is the image of L^1? Talk about
the L^2 theory of Fourier transforms. (Since I used the Schwartz
space to show that the image of L^1 is C_0,) what is the dual of the
Schwartz space? Do you know anything about Fourier analysis on L^p
(L^p <-> L^q duality, interpolation theorem)?
In connection with the taking the square root function, can you take
the square root of f(z)=z(z-1)(z-2). (I talked about the Riemann
surface associated to f(z), forgetting about the Riemann surface not
sitting in projective space.) What genus does the Riemann surface
have? What is the genus? (I gave the dim H^1(C) definition, which
was part of my downfall.) For the original curve in C^2 contained in
CP^2, what is H^1(C)? It's the same as H^1(C). Now take
C'=C-{p_1,...,p_n}? What is H^1(C')? (Here I said, "I don't know too
much about these things," but Katz responded, "Well, you should know it
anyway.") Talk about residues of
meromorphic differentials. State the residue theorem. How can you
use residues to compute the homology of H^1(C') via integrating mero
differentials?
===========================================================================
(Since I could only give vague generalities,)
Talk about Euler characteristic. What is it on a surface? Look at
the simplicial decomposition and tell me the Euler char. in terms
of the k-simplices. Now use this to compute H^1(C'). Need
H^2(C')=0. How does one know this? Talk about Poincare Duality.
What are the conditions you need for Poincare Duality to hold?
What is the Fermat curve? Compute its genus. Give a basis of its
holomorphic differentials.
Talk about the Hilbert polynomial. (I gave a definition in terms of
H^i(F(m)) with F coherent.) What does this have to do with the
classical definition of the Hilbert polynomial? What is the
arithmetic genus? What are the degree, the leading coefficient, and the
last coefficient of the Hilbert polynomial?
What is an algebraic group? Give one (O(n)). Give another. (I
gave U(n), which was not a good choice.) Why is what you wrote down
wrong? Give some more algebraic groups. GL(n), SL(n)... Which of
them are rational varieties? (GL(n), SL(n) are.) Prove it. You
defined O(n) as xx^t=1. Give a more intrinsic definition of this,
via nondegenerate symmetric bilinear forms. Give intrinsic
definitions of Sp, U, etc. in this way. Which ones are connected?
Talk about Dirichlet's theorem on arithmetic progressions. Define
I(C)/P(C). (This is where I got majorly bogged down.) Talk about
the case where we have Q(z_m)/Q. So C=mv_infty in this case, but
what happens when C=m only? (....by this time my brain was complete
mush.)
Exam took 3 1/4 hours.