Class Numbers, Continued Fractions, and the Hilbert Modular Group Jordan Schettler University of California, Santa Barbara 11/8/2013
Outline 1 Motivation 2 The Hilbert Modular Group 3 Resolution of the Cusps 4 Signatures
Motivation
Minus Continued Fractions Let α R\Q. Then unique continued fraction expansion 1 α = [[b 0 ; b 1, b 2,...]] := b 0 1 b 1 b 2 where b 0 Z and b 1, b 2,... Z >1. (b 0, b 1,...) is eventually periodic α is algebraic of degree 2. Note: [[2; 2, 2,...]] = 1, so we must have b k 3 for ly many k
An Amusing Connection Let l > 3 be a prime such that l 3 (mod 4). Then l = [[b0 ; b 1,..., b m ]] where m = minimal period is even, b m = 2b 0 and (b 1, b 2,..., b m 1 ) = (b m 1, b m 2,..., b 1 ). Theorem (Hirzebruch) If Q( l) has class number 1, the class number of Q( l) is h( l) = 1 3 m (b k 3). k=1
The Hilbert Modular Group
Notation H = {z C : I(z) > 0} GL + 2 (R) acts on H via Möbius transformations: ( ) a b z = az + b c d cz + d The action induces an isomorphism between the group of biholomorphic maps H H and the group PL + 2 (R) = GL+ 2 (R)/ {( ) a 0 0 a : a R }
Notation Fix an integer n 1, and consider H n = H } H {{ H }. n times (PL + 2 (R))n acts on H n component-wise. SES 1 (PL + 2 (R))n A n S n 1 where A n = group of biholomorphic maps H n H n and S n = a symmetric group.
Notation F = number field of degree n over Q Assume F is totally real: n distinct embeddings F R: x x (j) for j = 1,..., n GL + 2 (F) = {A GL 2(F) : det(a) (j) > 0, j = 1,..., n} We use the embeddings to view PL + 2 (F) (PL+ 2 (R))n
Notation O F = ring of integers of F We define the Hilbert modular group G = SL 2 (O F )/{±1} PL + 2 (F) G is a discrete, irreducible subgroup of (PL + 2 (R))n More generally, let Γ denote any discrete, irreducible subgroup of (PL + 2 (R))n such that [G : Γ] := [G : G Γ] [Γ : G Γ] <
Notation With coordinates z j = x j + iy j on H n, Gauss-Bonnet form: ω = ( 1)n (2π) n dx 1 dy 1 y1 2 dx n dy n yn 2 Theorem (Siegel) ω = [G : Γ] 2ζ F ( 1) Q H n /Γ where ζ F (s) is the Dedekind zeta function of F.
Define the isotropy group at z H n by Γ z = {γ Γ : γ z = z} Let a m (Γ) = # of Γ-orbits of points z with Γ z = m Every Γ z is finite cyclic, and m 2 a m(γ) <. Theorem H n /Γ is a non-compact complex analytic space with finitely many quotient singularities and Euler characteristic χ(h n /Γ) = ω + a m (Γ) m 1 m m 2 H n /Γ
The n = 1 Case If n = 1, then F = Q and G = SL 2 (Z)/{±1} biholomorphism j : H/G C 1 = χ(c) = χ(h/g) = 2ζ( 1) + 1 2 + 2 3 = 1
The n = 1 Case Continued For n = 1 we could take Γ = Γ 0 (N), a congruence subgroup of level N non-compact Riemann surface H/Γ = Y 0 (N) compact Riemann surface X 0 (N) = Y 0 (N) {cusps} where {cusps} = ({ } Q)/Γ 0 (N)
Back to General Case If Γ G, we can compactify H n /Γ by adding cusps P 1 (K )/Γ where we view P 1 (K ) ({ } R) n = H n Theorem bijection {cusps of H n /G} class group C of F: orbit of [α, β] P 1 (K ) with α, β O F ideal class of (α, β)
From now on take n = 2, so F = Q( d) for a squarefree d > 1. O F = Z[ d] if d 2, 3 (mod 4) [ ] Z 1+ d 2 if d 1 (mod 4) O F = {±1} εz H 2 /G is a Hilbert modular surface. The # of cusps = the class number h(d) of Q( d).
The number of quotient singularities of H 2 /G is related to the class numbers of imaginary quadratic fields. Theorem (Prestel) For d > 6 and (d, 6) = 1, 10h( d) if d 3 (mod 8) a 2 (G) = 4h( d) if d 7 (mod 8) h( d) if d 1 (mod 4) a 3 (G) = h( 3d), a m (G) = 0 for m > 3.
Example Suppose d 1 (mod 12). Then 2ζ F ( 1) = 1 15 1 b< d b odd where σ 1 (m) = sum of divisors of m. Thus 30χ(H 2 /G) = 2 1 b< d b odd ( ) d b 4 σ 1 4 ( ) d b 4 σ 1 + 15h( d) + 20h( 3d) 4
Resolution of the Cusps
Consider a cusp of H 2 /G with representative x P 1 (F). Translate the cusp to infinity: for some ρ (PL + 2 (R))2. ρx = = (, ) deleted closed neighborhoods for, x: W (r) = {(x 1 + iy 1, x 2 + iy 2 ) H 2 : y 1 y 2 r} U(r) = ρ 1 W (r)
We can choose r 0 so that U(r)/G = U(r)/G x W (r)/ρg x ρ 1 H 2 /ρg x ρ 1 There is a SES 1 M ρg x ρ 1 V 1 where M is a fractional ideal and V = {u 2 : u O F }. The narrow ideal class of M is uniquely determined by the cusp (indep. of ρ), and we may choose ρ such that ρg x ρ 1 = G(M, V ) = {( v m 0 1 ) : v V, m M}
The quotient space H 2 /G(M, V ) is a complex manifold. We can compactify H 2 /G(M, V ) = H 2 /G(M, V ) { } to obtain a complex analytic space with a singularity at. We now show how M, V are determined and how to resolve the singularity at.
The narrow class group C + = fractional ideals modulo strict equivalence: a b a = λb for some totally positive λ F For a fractional ideal a of F, a a 2 induces a homomorphism Sq : C C + where C + is the narrow class group of F Hence to each cusp corresponding to an ideal class a, we have an associated narrow ideal class C = Sq(a).
Every narrow ideal class C C + contains an ideal of the form M = Z + wz with w K and w > 1 > w > 0 (w = Galois conjugate). This implies w has a purely periodic continued fraction: w = [[b 0 ; b 1,..., b m 1 ]] where m = minimal period. (Note: all b k 2 and b j > 2 some j)
The cycle ((b 0, b 1,..., b m 1 )) (defined up to cyclic permutation) depends only on C C +. We define b k by extension using periodicity for all k Z. For each k Z take R k = C 2 with coordinates (u k, v k ). biholomorphism ϕ k : R k R k+1 : (u k, v k ) (u b k k v k, 1/u k ) where R k = R k\{u k = 0} and R k+1 = R k+1\{v k+1 = 0}.
Take the disjoint union k R k and identify R k with R k+1 via ϕ k. This gives a complex manifold Y of dimension 2 with charts ψ k : R k C 2 given by coordinates (u k, v k ) curves S k in Y given by u k+1 = 0 in R k+1 (and v k = 0 in R k ). By construction, S k S k+1 = 1 while S k S j = 0 for k < j + 1, and we can compute the self-intersections: S k S k = b k
M acts freely on C 2 via λ (z 1, z 2 ) = (z 1 + λ, z 2 + λ ) Note that Y k Z S k = {(u 0, v 0 ) : u 0 0 v 0 }, so the map 2πiz 1 = wlog(u 0 ) + log(v 0 ) 2πiz 2 = w log(u 0 ) + log(v 0 ) induces a biholomorphism Φ: Y S k C 2 /M k Z
Define A 0 = 1 and inductively A k+1 = w 1 k+1 A k where w k+1 = [[b k+1, b k+2,..., b k+m ]] Then A m generates the group U + of totally positive units, and A cm = A c m generates V = (O F )2 where c = [U + : V ] {1, 2}. The group V acts on Y + = Φ 1 (H 2 /M) k Z S k: (A cm ) n sends (u k, v k ) in the kth coordinate system to the point with the same coordinates in the (k + ncm)th coordinate system. Under the action, S k is mapped by (A cm ) n to the curve S k+ncm
Y + is an open submanifold of Y with a free and properly discontinuous action of V. Y ((b 0,..., b cm 1 )) = Y + /V is a complex manifold cycle of curves S 0, S 1,..., S cm 1 with intersection matrix or b 0 1 0 0 1 1 b 1 1 0 0 0 1 b 2 1 0 0 0 1 b cm 2 1 1 0 0 1 b cm 1 ( ) b0 2 2 b 1 for cm = 2, or ( b 0 + 2) for cm = 1.
The Resolution holomorphic map such that: and σ : Y ((b 0,..., b cm 1 )) H 2 /G(M, V ) σ 1 ( ) = cm 1 k=0 S k cm 1 Y ((b 0,..., b cm 1 )) S k H 2 /G(M, V ) is a biholomorphism. k=0
Signatures
Definition Let M be a complex surface. symmetric bilinear form β : H 2 (M, R) H 2 (M, R) R given by the intersection of homology classes. We define the signature of M by sign(m) = b + b where b + (resp. b ) is the # of positive (resp. negative) eigenvalues of a matrix representing β.
Two Geometric Formulas Let M be a connected, complex surface. Theorem (Adjunction Formula) For a nonsingular, compact curve S on M, χ(s) = K S S S where K is a canonical divisor on M. Theorem (Signature Formula) If M is a compact manifold with no boundary, sign(m) = 1 (K K 2χ(M)) 3 where K is a canonical divisor on M.
a C we associate a compact manifold with boundary X a obtained by resolving the singularity (as above) of W (r)/g(m, V ) (U(r)/G) {x} H 2 /G where r 0 and x is the cusp corresponding to a. We define the signature deviation invariant δ(x a ) = 1 3 (K K 2χ(X a)) sign(x a ) where K is a canonical divisor on X a.
Computing δ(x a ) X a is constructed by blowing up the cusp x into a cycle of cm nonsingular curves S 0,..., S cm 1. The intersection matrix (as in the previous section) gives sign(x a ) = cm In fact, X a is homotopy equivalent to cm 1 k=0 S k, so χ(x a ) = 1 1 + cm = cm
Computing δ(x a ) For each k the adjunction formula gives 2 = χ(s k ) = K S k S k S k = K S k + b k, so and K = cm 1 k=0 S k, m 1 K K = c b k + 2cm. k=0
Computing δ(x a ) Therefore ( δ(x a ) = 1 3 c m 1 k=0 ) b k + 2cm 2cm ( cm) = c m 1 (b k 3) 3 k=0 Suppose F = Q( d) has no units of negative norm. (c = 2) Theorem (Curt Meyer) If ζ(s, C) = partial zeta function of C = Sq(a), ζ(0, C) = ζ(0, C ) = 1 m 1 ( (b k 3) = 1 ) 6 2c δ(x a) where a 2 = C C. k=0
Assume F = Q( l) where l > 3 is a prime with l 3 (mod 4). unique, character ψ on C + which is non-trivial and real-valued. Meyer s theorem implies sign(h 2 /G) = a C δ(x a ) = 4 C C + ψ(c)ζ(0, C) = 4L(0, ψ) = 4 h( l)h( 1) 2 = 2h( l)
If, additionally, F = Q( l) has class number 1, then C = {a} where a = [O F ] is the trivial ideal class. C = Sq(a) is the trivial narrow ideal class, so we may choose M = O F = Z + ( l + l)z with l = [[b0 ; b 1,..., b m ]] where m = minimal period, b 0 = l, b m = 2b 0, and whence l + l = [[2b 0 ; b 1,..., b m 1 ]].
Thus we recover the amusing connection ( ) 2h( l) = sign(h 2 /G) = δ(x a ) = 2 m 1 2b 0 + (b k 3), 3 k=1 or more simply h( l) = 1 3 m (b k 3) k=1 Note: We did NOT need ANY signatures or surfaces to derive the formula for the class number.