Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/ium/postscript/f10/conrad-exameng.pdf Дата изменения: Sat Dec 18 14:28:29 2010 Дата индексирования: Sun Feb 13 18:35:27 2011 Кодировка: Поисковые слова: arp 220

The Classical and Adelic Point of View in Number Theory Exam K. Conrad Due on 12/22/2010 at 9 PM Instructions. · Only students who want a grade for the course should submit solutions to this exam. The exam due date and time is above. · Tell me by December 12th if you plan to take this exam, so I know who I will be receiving solutions from later. · Do not discuss your work on the exam with anyone else. You may write to me if you have questions. My email address is kconrad@math.uconn.edu. · If you will write your work by hand, make sure to write neatly (I do not want to read terrible handwriting). Give the final version to Alexey Zykin at the end of his algebraic number theory lecture (IUM, room 310, 21:00). He will email a copy of your solutions to me and I will write to you after I receive them. · If you will type your work, email the final the website http://www.pdfonline.com if format by yourself.) I will confirm by email not hear back from me then it means I did · Good luck! version to me as a .pdf file. (Use you can't convert your file into .pdf that I received the file, so if you do not get anything from you.


1. (Computations) (a) Let x Z5 be the solution of the equation x2 = -1 such that x 2 mod 5Z5 . Compute {x/25}5 . (b) Let a = (3, -1/3, -1/3, 1, 1, . . . ) AQ , where the unwritten terms are all 1. Compute (a), where is the standard character on AQ . (c) Let b = (10, 20, 30, 40, 1, 1, . . . ) JQ , where the unwritten terms are all 1. Using the isomorphism JQ Qв в R>0 в p Zв , write b in the form q tu, = p в в where q Q , t > 0, and u p Zp . 2. Let G be a locally compact abelian group, H be a compact open subgroup, and let f L1 (G). Fix a Haar measure µ on G to define the Fourier transform f : G C. a) If f (g ) = 0 for all g H , show f : G C is constant on H -cosets in G: f ( ) = f () if H . b) If f is constant on H -cosets in G (that is, f (g h) = f (g ) for all h H ), show f () = 0 for all H . (Hint for part b: Write f () as an iterated integral over H and G/H using Weil's formula.) 3. Show AQ and JQ are both -compact, i.e., they can each be written as a countable union of compact subsets. 4. Let be the standard character on Qp and dx be the standard Haar measure on Qp . a) For n Z, show 1/pn - 1/pn+1 , (xy ) dx = -1/pn+1 , 0,
pn Z

if y (1/pn )Zp , if y (1/pn+1 )Zp - (1/pn )Zp , otherwise. -
p
n+1

|x|p =1/p

n

(Hint: Write the integral as a difference

p

Zp

.)


b) For x Qp , set f (x) = |x|p Zp (x). Show
1 1+1/p

f (y ) =

, ,

if y Zp , if y Zp .

- 5. Let p be a prime.

p 1 |y |2 1+1/p p

a) In the topological group R в Qp , show Z (embedded diagonally) is discrete but not co-compact and Z[1/p] (embedded diagonally) is both discrete and cocompact. Here Z[1/p] = {a/pn : a Z, n 0} is the set of fractions with p-power denominator. b) Show counting measure on Z[1/p], the Haar measure dx в dxp on R в Qp where dxp is the Haar measure on Qp which assigns Zp measure 1, and the normalized Haar measure on (R в Qp )/Z[1/p] are Weil compatible. 6. Let p be a prime. The group R в Qp is self-dual. For (x, y ) R в Qp , define (x,y) R в Qp by (x,y) (u, v ) = e-2ixu e2i{yv}p . (Note the minus sign!) This is a self-duality on R в Qp . a) For every t Z[1/p], show t = {t}p in Q/Z. b) Relative to the self-duality of RвQp described above, show Z[1/p] = Z[1/p], where Z[1/p] is viewed inside R в Qp diagonally. 7. (Bonus) Using the ideas from the last two questions, for any finite subset S in VQ such that S , find an example of a lattice L (that is, a discrete and co-compact subgroup) inside vS Qv such that L = L relative to a suitable self-duality on the group vS Qv . Can you also find a lattice L such that L = L in the group which is constructed like the adeles where one factor Qp (p a prime) is not used?