Hyperbolic volumes and zeta values An introduction

Transcription

1 Hyperbolic volumes and zeta values An introduction Matilde N. Laĺın University of Alberta Annual North/South Dialogue in Mathematics University of Calgary, Alberta May 2nd, 2008

2 The hyperbolic space Hyperbolic Geometry: Lobachevsky, Bolyai, Gauss ( 1830) Beltrami s Half-space model (1868) H n = {(x 1,..., x n 1, x n ) x i R, x n > 0}, ds 2 = dx dx 2 n x 2 n dv = dx 1... dx n xn, H n = {(x 1,..., x n 1, 0)}.,

3 The hyperbolic space Hyperbolic Geometry: Lobachevsky, Bolyai, Gauss ( 1830) Beltrami s Half-space model (1868) H n = {(x 1,..., x n 1, x n ) x i R, x n > 0}, ds 2 = dx dx 2 n x 2 n dv = dx 1... dx n xn, H n = {(x 1,..., x n 1, 0)}.,

4 Geodesics are given by vertical lines and semicircles whose endpoints lie in {x n = 0} and intersect it orthogonally. Poincaré (1882): Orientation preserving isometries of H 2 {( a b PSL(2, R) = c d ) } M(2, R) ad bc = 1 / ± I. z = x 1 + x 2 i az + b cz + d.

5 Geodesics are given by vertical lines and semicircles whose endpoints lie in {x n = 0} and intersect it orthogonally. Poincaré (1882): Orientation preserving isometries of H 2 {( a b PSL(2, R) = c d ) } M(2, R) ad bc = 1 / ± I. z = x 1 + x 2 i az + b cz + d.

6 Orientation preserving isometries of H 3 is PSL(2, C). H 3 = {z = x 1 + x 2 i + x 3 j x 3 > 0}, subspace of quaternions (i 2 = j 2 = k 2 = 1, ij = ji = k). z (az + b)(cz + d) 1 = (az + b)( z c + d) cz + d 2. Poincaré: study of discrete groups of hyperbolic isometries. Picard (1884): fundamental domain for PSL(2, Z[i]) in H 3 has a finite volume. Humbert (1919) extended this result.

7 Volumes in H 3 Lobachevsky function: where θ l(θ) = log 2 sin t dt. 0 l(θ) = 1 2 Im ( Li 2 (e 2iθ)), Li 2 (z) = Li 2 (z) = n=1 z z n, z 1. n2 0 log(1 x) dx x. (multivalued) analytic continuation to C \ [1, )

8 Let be an ideal tetrahedron (vertices in H 3 ). Theorem (Milnor, after Lobachevsky) The volume of an ideal tetrahedron with dihedral angles α, β, and γ is given by Vol( ) = l(α) + l(β) + l(γ). α β γ β γ γ β α α Move a vertex to and use baricentric subdivision to get six simplices with three right dihedral angles.

9 Triangle with angles α, β, γ, defined up to similarity. Let (z) be the tetrahedron determined up to transformations by any of z, 1 1 z, 1 1 z. 1 1 z z If ideal vertices are z 1, z 2, z 3, z 4, z 1 1 z 0 1 z = [z 1 : z 2 : z 3 : z 4 ] = (z 3 z 2 )(z 4 z 1 ) (z 3 z 1 )(z 4 z 2 ).

10 Bloch-Wigner dilogarithm D(z) = Im(Li 2 (z) + log z log(1 z)). Continuous in P 1 (C), real-analytic in P 1 (C) \ {0, 1, }. ( ) 1 D(z) = D(1 z) = D = D( z). z Vol( (z)) = D(z).

11 Five points in H 3 = P 1 (C), then the sum of the signed volumes of the five possible tetrahedra must be zero: 5 ( 1) i Vol([z 1 : : ẑ i : : z 5 ]) = 0. i=0 Five-term relation D(x) + D(1 xy) + D(y) + D ( ) ( ) 1 y 1 x + D =0. 1 xy 1 xy

12 Dedekind ζ-function F number field, [F : Q] = n = r 1 + 2r 2 τ 1,..., τ r1 real embeddings σ 1,..., σ r2 a set of complex embeddings (one for each pair of conjugate embeddings). ζ F (s) = N(A) = O F /A norm. Euler product A ideal 0 P prime 1, Re s > 1, N(A) s 1 1 N(P) s.

13 Theorem (Dirichlet s class number formula) ζ F (s) extends meromorphically to C with only one simple pole at s = 1 with where lim (s 1)ζ F (s) = 2r1 (2π) r2 h F reg F, s 1 ω F DF h F is the class number. ω F is the number of roots of unity in F. reg F is the regulator. lim s 0 s1 r 1 r 2 ζ F (s) = h F reg F ω F.

14 Regulator {u 1,..., u r1 +r 2 1} basis for O F modulo torsion L(u i ) :=(log τ 1 u i,..., log τ r1 u i, 2 log σ 1 u i,..., 2 log σ r2 1u i ) reg F is the determinant of the matrix. = (up to a sign) the volume of fundamental domain for L(O F ).

15 Euler: ζ(2m) = ( 1)m 1 (2π) 2m B m 2(2m)! Klingen, Siegel: F is totally real (r 2 = 0), where r(m) Q. ζ F (2m) = r(m) D F π 2mn, m > 0

16 Building manifolds Bianchi: F = Q ( d ) d 1 square-free Γ a torsion-free subgroup of PSL (2, O d ), [PSL (2, O d ) : Γ] <. Then H 3 /Γ is an oriented hyperbolic three-manifold. Example: d = 3, O 3 = Z[ω], ω = Riley: [PSL (2, O 3 ) : Γ] = 12 H 3 /Γ diffeomorphic to S 3 \ Fig 8.

17 Theorem (Essentially Humbert) Vol ( H 3 /PSL(2, O d ) ) = D d Dd 4π 2 ζ Q( d) (2). M hyperbolic 3-manifold { d d 3 mod 4, D d = 4d otherwise. Vol(M) = J D(z j ). j=1 ζ Q( d) (2) = D d Dd 2π 2 J D(z j ). j=1

18 Example: Vol(S 3 \ Fig 8) = π 2 ζ Q( 3) (2) ) ( ) = 3D (e 2iπ 3 = 2D e iπ 3.

19 Zagier (1986): [F : Q] = r Γ torsion free subgroup of finite index of the group of units of an order in a quaternion algebra B over F that is ramified at all real places. Vol(H 3 DF /Γ) Q π 2(n 1) ζ F (2). [F : Q] = r 1 + 2r 2, r 2 > 1 Γ discrete subgroup of PSL(2, C) r 2 ( (H 3 Vol ) ) r 2 /Γ Q such that DF π 2(r 1+r 2 ) ζ F (2). ( H 3 ) r 2 /Γ = (z 1 ) (z r2 )

20 The Bloch group then J Vol(M) = D(z j ), j=1 J z j (1 z j ) = 0 j=1 2 C. 2 C ={x y x x = 0, x 1 x 2 y = x 1 y + x 2 y} Vol(M) = D(ξ M ), where ξ M A( Q), and { } A(F ) = ni [z i ] Z[F ] ni z i (1 z i ) = 0.

21 Let C(F ) = { [ ] [ ] 1 y 1 x [x] + [1 xy] + [y] xy 1 xy Bloch group is x, y F, xy 1}, B(F ) = A(F )/C(F ). D : B(C) R well-defined function, Vol(M) = D(ξ M ) for some ξ M B( Q), independently of the triangulation. Then ζ F (2) = D F π 2(n 1) D(ξ M ) for r 2 = 1.

22 Theorem (Zagier, Bloch, Suslin) For a number field [F : Q] = r 1 + 2r 2, B(F ) is finitely generated of rank r 2. ξ 1,... ξ r2 Proof: Q-basis of B(F ) Q. Then ζ F (2) Q DF π 2(r 1+r 2 ) det {D (σ i (ξ j ))} 1 i,j r2. B(F ) is K 3 (F ) Borel s theorem.

23 Theorem (Zagier, Bloch, Suslin) For a number field [F : Q] = r 1 + 2r 2, B(F ) is finitely generated of rank r 2. ξ 1,... ξ r2 Proof: Q-basis of B(F ) Q. Then ζ F (2) Q DF π 2(r 1+r 2 ) det {D (σ i (ξ j ))} 1 i,j r2. B(F ) is K 3 (F ) Borel s theorem.

24 Conjecture Let F be a number field. Let n + = r 1 + r 2, n = r 2, and = ( 1) k 1. Then B k (F ) is finitely generated of rank n. ξ 1,... ξ n Q-basis of B k (F ) Q. Then ζ F (k) Q DF π kn ± det {L k (σ i (ξ j ))} 1 i,j n.

25 Example F = Q( { [ 5), r 1 = 2, r 2 = 0. ]} [1], basis for B 3 (F ). = L 3 (1) L 3 ( L 3 (1) L 3 ( ) ) 1 25 ζ(3) 10ζ(3) L(3, χ5 ) 1 25 ζ(3) 10ζ(3) 48 5L(3, χ5 ) = 25 5ζ(3)L(3, χ5 ) = 25 5ζF (3)

26 Application D Andrea, L. (2007) 1 (2πi) 3 log (1 x)(1 y) z T 3 1 xy dx x dy y dz z = 25 5L(3, χ5 ) π 2