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.