Kartik Prasanna' generals
May 7, 1999
Committee : Katz (chair), Yu , Forni
Subjects : Algebraic Number Theory, Algebraic Geometry

Prove the fund. thm. of algebra using cmplx analysis. Give an example of a
doubly connected region and a holomorphic function on it that doesn't
extend to a bounded component of the complement. Why are two annuli
conformally equivalent ? What about a squarish annular region and an
annulus ? Suppose you had a conformal map from a simply connected region
to the unit disc. What can you say about continuous extension to the
boundary ? Give conditions on the boundary for this to exist.

What do you know about convergence of Fourier series ? Cesaro
summability? Dirichlet kernel ? When is convergence uniform ? What
is the Riemann-Lebesgue lemma ? What conditions on the function
would ensure rapid decay of the Fourier coefficients ? Suppose you
had a harmonic function on the unit disc. What can you say about
continuous extension to the boundary ? When is it a Poisson
integral ? How is this related to Fourier series ?

Compute the Fourier transform of sin x / x . What is the image of L^1
under the Fourier transform. Suppose you had a C^\infty function with
compact support. What can you say about its Fourier transform ? Does it
extend to an analytic function ? What can you say about the rate of growth
of this analytic function ? ( - followed by some general questions about
rates of growth of analytic functions - why are people interested in such
stuff ? why were Hadamard ... interested in such things ? do you know the
prime number theorem ? )  How is this related to factorising an entire
function into an infinite product ? What can you say about the Fourier
transform of a radially symmetric function ? Prove it is radially
symmetric. Do you know what Bessel functions are ?

Let's do some measure theory. Construct a measurable set that is not
Borel. Construct a non-measurable set. Suppose you had a sequence of sets
A_n \subset W where W has finite measure and \mu (A_n) \geq 1/2, what can
you say about the set of x which lie in infinitely many of the A_n 's ?

All these questions were asked by Katz and Forni. Now Katz suggested that
Yu ask me an Algebra question. (Whew ! analysis was over !) Most of the
questions that follow were asked by Katz, some by Yu.

What is the ring of adeles? Topology on the adeles ? Ideles ? How are
these related ? Do you know any compactness statements regarding them ?
What classical facts are these equivalent to ? ( finiteness of class
number and finite generation of the group of units.) I was asked to prove
these facts. Does this give a "different" proof of these classical
theorems ?

Prove Dirichlet's theorem on primes in A.P. (To show L(1,\chi) is not 0
for nontrivial \chi , I used the fact that the product of all L(1,\chi) =
the zeta function of the cyclotomic field up to a finite number of factors;
consequently: ) Dirichlet didn't know this, so how did he prove L(1,\chi)
is not 0 ?

Talk about generalisations of Dirichlet's thm. (primes equidistributed in
generalised ideal classes.) How do you prove the corresponding L-function
is analytic at 1 ? Suppose one knew a priori that the zeta function had a
pole of order 1 at 1. Could one use this fact to show finiteness of the
ideal class group and Dirichlet's unit theorem ? Then I was asked some
question that lead me to speak about Artin reciprocity,  how this implied
that L-functions coming from characters of the ideles were essentially the
same as those coming from generalised ideal classes, and how this
explained the fact about the product of the L(1,\chi) 's stated above.

What do you know about non-Abelian analogues of these L-functions ? What
is the corresponding formula for the zeta function ?

Can you use all this to produce extensions of Q with Galois group = full
symmetric group ? I said "Oh, reduction mod p", but Katz made me work out
the details: Take a polynomial f with integer coefficients. What can you
say about the Galois group of the splitting field, when you know how this
polynomial splits mod some prime p ? (You get an element with cycle
structure (r_1,r_2,....) where the r_i 's are the degrees of the
irreducible factors of f mod p. So to complete the argument, one must
find an f such that for each possible cycle structure in S_n, there is a
prime p such that the reduction mod p of f yields an element with that
cycle structure. ) What fact from finite group theory are you using here ?
(I couldn't see the point, so Katz explained that I was using the
following fact: A subgroup H of a finite group G that meets every
conjugacy class is in fact = G. Why is that true ?) Is there any
obstruction to producing such polynomials mod p ? Now that you have all
these polynomials how do you complete the argument ? ( I said "you have to
solve lots of congruences) What's one word for that ? ("Chinese remainder
theorem" and Katz was happy !)

Let's do algebraic geometry ! Give me all defns. of the genus of a curve
that you know. Why is geometric genus = arithmetic genus ? (Serre duality)
Does the genus determine the curve completely ? For what genus is this
true (over an algebraically closed field )? what's the genus of a plane
curve of degree d ? Write down a basis for the differentials of the first
kind on the Fermat curve. What is an elliptic curve ? Does an elliptic
curve over a field K necessarily have a K-rational point ? (No) Give an
example. What about a finite field K ? I said yes and mentioned the Hasse
bound, so was asked to prove it. (I started by claiming that the F_q
rational points are the fixed points of the Frobenius morphism \phi, so
form the kernel of 1-\phi; Katz said that wasn't kosher since this assumed
that E has at least one F_q rational point, namely the origin ! indeed the
Hasse bound holds only if E has at least one F_q rational point.) What can
you say about higher genus curves over finite fields ? ( I had forgotten
the "Weil bound" so Katz joked that it was amazing people had forgotten
Weil's work less than a year after his death !!)

Why is a conic over an algebraically closed field isomorphic to P^1 ? Can
you prove this using a picture ? What about a non-algebraically closed
field ? What if K is a finite field ? Why does every conic over a finite
field have a K-rational point ? (Do you know what the Brauer group is ?
What defn. of the Brauer group would help you solve this problem? What are
the automorphisms of P^n ? Why is H^1( Gal(L/K), GL(n+1,L)) = 0 ? I said
you could prove it the same way as you prove H^1( G, L^*) = 0. What's a
generic name for such results - "Hilbert 90")

Let's go back to Serre duality. How might an ordinary person understand it
for the Fermat curve ? (I suggested Cech cohomology.) Katz made me write
down a long exact cohomology sequence to show that H^1(X, O_X) is
isomorphic to H^2(P^2, O(-d)). Can you "see" now that this space has the
same dimension as the space of differentials you wrote down before ?

End ! lasted 3 hours 15 min, was very pleasant throughout.