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Ulam Seminar, November 1, 2006

Ulam Seminar, November 1, 2006. Example 2. Galois Extension of Non-normal Cubic field ring prime ideal finite field. K=F(e 2 p i/3 ) O K P O K / P. O F p O F / p 3 Q Z p Z Z /p Z

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Ulam Seminar, November 1, 2006

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  1. Ulam Seminar, November 1, 2006

  2. Example 2. Galois Extension of Non-normal Cubic field ring prime ideal finite field K=F(e2pi/3) OKP OK/P OFp OF/p 3 Q Z pZZ/pZ More details are in Stark’s article in From Number Theory to Physics, edited by Waldschmidt et al 2 Splitting of Rational Primes inOF - Type 1. Primes that Split Completely:pOF=p1p2p3 , with distinctpiof degree 1(p=31 is 1st example) Frobenius of primePabovepihas order 1 density 1/6 by Tchebotarev There are also 2 other types of unramified primes.

  3. Exercise. Do a function field analog of the preceding example in the spirit of E. Artin’s thesis. Artin’s thesis did quadratic extensions of the field of rational functions of 1 indeterminate with coefficients in a finite field.

  4. 3(2) 1(2) 4(1) 4(2) 2(2) 2(4) 1(1) 3(1) Y6 2(3) 2(1) 1(3) 4(3) 1(4) 3(5) 4(4) 3(3) Galois Cover of Non-Normal Cubic 2(5) x=1,2,3 a(x),a(x+3) a(x) 3(4) 1(5) 3(6) 4(6) 1(6) 4(5) 2(6) 3’’ 4’’ 1’’ G=S3, H={(1),(23)} fixes Y3. a(1)=(a,(1)), a(2)=(a,(13)), a(3)=(a,(132), a(4)=(a,(23)),a(5)=(a,(123)),a(6)=(a,(12)). Here we use standard cycle notation for elements of the symmetric group. 2’’ 2’’’ Y3 4’’’ 1’’’ a(x) a 3’ 3’’’ 4’ 1’ 2’ 4 X 3 2 1

  5. 4 2 3 1 • Prime Splitting Completely • path in X (list vertices) 14312412431 • f=1, g=33 lifts to Y3 • 1’4’3’’’1’’’2’’’4’’1’’2’’4’’’3’1’ • 1’’4’’3’’1’’2’’4’’’1’’’2’’’4’’3’’1’’ • 1’’’4’’’3’1’2’4’1’2’4’3’’’1’’’ • Frobenius trivial  density 1/6 page 60 of my manuscript on my website gives examples of primes with other splitting behavior in the non-Galois cubic extension

  6. Suppose Y/X a normal cover. Given a path p of X, there is a unique lift of p to a path p on Y starting on sheet 1 and having the same length as p. Here you may take p to be an edge if you like. (x,h) C Y (x,g) D a prime in Y above C a prime of X consists of f lifts of C. Frob(D) = = hg-1 G=Gal(Y/X) where hg-1 maps sheet g to sheet h  X xC Frobenius Automorphism Normalized Frobenius : start lift of C on sheet 1, end on sheet h: (C)=h

  7. Properties of Frobenius 1) (p1p2) = (p1)(p2) 2) If D begins on sheet 1, [Y/X,D] = (C). 3) Replace D with different prime above C and see that Conjugacy class of Frob(D)  Gal(Y/X) unchanged. 4) Varying x = start of D or C does not change Frob(D). So [Y/X,[D]] makes sense. Exercise: Prove these properties. See page 55 of my “book.”

  8. 5) Order of Frob(D) is f=residual degree. 6) Define decomposition group as Z(D)={gG|[gD]=[D]}. Then Z(D)=cyclic subgroup of G of order f generated by [Y/X,D].

  9. Artin L-functions involve (irreducible) representations  of the Galois group. If you do not know about this subject, assume the Galois group G is abelian, or even better cyclic. In that case a representation is a group homomorphism c:GT T=group under multiplication of complex numbers of absolute value 1. Moreover c is given by complex exponentials: G=Z/nZunder + and c(x mod n) = exp(2 iax/n) Here a is a fixed element of G. As you run through all n elements of the cyclic group G you get all the representations of G. A reference for representations of finite groups is my book, Fourier Analysis on Finite Groups and Applications.

  10. Artin L-Function  = representation of G=Gal(Y/X), uC, |u| small [C]=primes of X (C)=length C, D a prime in Y over C

  11. L(u,1,Y/X) = (u,X) = Ihara zeta function of X (our analogue of the Dedekind zeta function, also Selberg zeta) • 2) • product over all irreducible reps of G, d=degree Properties of Artin L-Functions Copy from Lang, Algebraic Number Theory Proofs of 1) and 2) require basic facts about reps of finite groups. See A. T., Fourier Analysis on Finite Groups and Applications.

  12. Definitions.ndnd matrices A, Q, I, n=|X| nxn matrix A(g), g  Gal(Y/X), has entry for,X given by (A(g)),=# { edges in Y from (,e) to (,g) }, e=identity  G. Q = diagonal matrix, jth diagonal entry = qj = (degree of jth vertex in X)-1, Q= QId , I = Ind = identity matrix. Ihara Theorem for L-Functions r = rank fundamental group of X = |E|-|V|+1 = representation of G = Gal(Y/X), d = d = degree 

  13. d'' b'' c" a d' b' a'' c' c b a d EXAMPLE Y=cube, X=tetrahedron: G = {e,g} representations of G are 1 and: (e) = 1, (g) = -1 A(e)u,v = #{ length 1 paths u to v in Y} A(g)u,v = #{ length 1 paths u to v in Y} (u,e)=u' (u,g)=u" A1 = A = adjacency matrix of X = A(e)+A(g)

  14. Zeta and L-Functions of Cube & Tetrahedron • (u,Y)-1 = L(u,,Y/X)-1 (u,X)-1 • L(u,,Y/X)-1 = (1-u2)(1+u) (1+2u) (1-u+2u2)3 • (u,X)-1 = (1-u2)2(1-u)(1-2u) (1+u+2u2)3 We give a Bass proof of the preceding Ihara type theorem by looking a edge Artin L-functions next time.

  15. 3(2) 1(2) 4(1) 4(2) 2(2) 2(4) 1(1) 3(1) Y6 2(3) 2(1) 1(3) 4(3) 1(4) 3(5) 4(4) 3(3) Galois Cover of Non-Normal Cubic 2(5) x=1,2,3 a(x),a(x+3) a(x) 3(4) 1(5) 3(6) 4(6) 1(6) 4(5) 2(6) 3’’ 4’’ 1’’ G=S3, H={(1),(23)} fixes Y3. a(1)=(a,(1)), a(2)=(a,(13)), a(3)=(a,(132), a(4)=(a,(23)),a(5)=(a,(123)),a(6)=(a,(12)). Here we use standard cycle notation for elements of the symmetric group. 2’’ 2’’’ Y3 4’’’ 1’’’ a(x) a 3’ 3’’’ 4’ 1’ 2’ 4 X 3 2 1

  16. Ihara Zeta Functions of S3 cover of K4 (u,X)-1 = (1-u2)(1-u)(1+u2)(1+u+2u2)(1-u2-2u3) (u,Y3)-1 = (u,X)-1 (1-u2)2(1-u-u3+2u4) (1+u+2u2+u3+2u4) (1-u+2u2-u3+2u4)(1+u+u3+2u4) (u,Y6)-1 = (u,Y3)-1 (1-u2)3 (1+u)(1+u2)(1-u+2u2)(1-u2+2u3) (1-u-u3+2u4) (1-u+2u2-u3+2u4) (1+u+u3+2u4)(1+u+2u2+u3+2u4) • It follows that, as in number theory • (u,X)2(u,Y6) = (u,Y2) (u,Y3)2 • Y2 is an intermediate quadratic extension between Y6 and X. See Stark & Terras, Adv. in Math., 154 (2000), Fig. 13, for more info.

  17. You can also consider edge and path Artin L-functions. We will do this next week. We will explain the construction of the graphs on the next slide.

  18. Application of Galois Theory of Graph Coverings. You can’t hear the shape of a graph. 2 connected regular graphs (without loops & multiple edges) which are isospectral but not isomorphic

  19. See A.T. & Stark in Adv. in Math., Vol. 154 (2000) for the details or my manuscript. • Method goes back to algebraic number theorists who found number fields Ki which are non isomorphic but have the same Dedekind zeta. See Perlis, J. Number Theory, 9 (1977). • Galois group is GL(3,F2), order 168 • appears in Buser,Geometry & Spectra of Compact Riemann Surfaces also Gordon, Webb, & Wolpert (isospectral non-isomorphic planar drums), Inventiones Math.(1992) • Kac question: Can you hear shape of drum? Audrey • Robert Perlis and Aubi Mellein have used the same methods to find many examples of isospectral non isomorphic graphs with multiple edges and components. 2 such are on the right. Harold

  20. There are also nice constructions of isospectal non-isomorphic graphs in paper of Lubotzky, Samuels and Vishne

  21. How to construct intermediate coverings using the Frobenius automorphism 1) If Y/X is normal with Galois group G and e1,…er the (directed) edges deleted from X to get a spanning tree (edges corresponding to generators of fundamental group of X), then the normalized Frob autos (e1),…,(er) generate G. Proof. In our construction(t)=1 for all edges of the spanning tree T of X. Thus for any path p on X,(p) = product of some(ej) and their inverses. By lifting paths of X to paths starting on sheet 1 of Y we can get to every sheet of Y and thus to all of G.

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