Langlands reciprocity for the even dimensional noncommutative tori

We conjecture an explicit formula for the higher dimensional Dirichlet character; the formula is based on the K-theory of the so-called noncommutative tori. It is proved, that our conjecture is true for the two-dimensional and one-dimensional (degenerate) noncommutative tori; in the second case, one gets a noncommutative analog of the Artin reciprocity law.


Introduction
The aim of the underlying note is to bring some evidence in favor of the following analog of the Langlands reciprocity [5]: For the notation and terminology we refer the reader to sections 1 and 3; the noncommutative torus A 2n RM can be regarded as a substitute of the "automorphic cuspidal representation π σ n+1 of the group GL(n + 1)" in terms of the Langlands theory. Roughly speaking, conjecture 1 says, that the Galois extensions of the field of rational numbers come from the even dimensional noncommutative tori with real multiplication. Note, that the noncommutative tori are intrinsic to the problem, since they classify the irreducible (infinite-dimensional) representations of the Lie group GL(n + 1) [11]; such representations are the heart of the Langlands program [5]. Our conjecture is supported by the following evidence.
Theorem 1 Conjecture 1 is true for n = 1 (resp., n = 0) and K abelian extension of an imaginary quadratic field k (resp., the rational field Q).
The structure of the note is as follows. A minimal necessary notation is introduced in section 1 and a brief summary of the Teichmüller functor(s) is given in section 3. Theorem 1 is proved in section 2.

Noncommutative tori
A. The k-dimensional noncommutative tori ( [4], [12]). A noncommutative k-torus is the universal C * -algebra generated by k unitary operators u 1 , . . . , u k ; the operators do not commute with each other, but their commutators u i u j u −1 i u −1 j are fixed scalar multiples exp (2πiθ ij ), θ ij ∈ R of the identity operator. The k-dimensional noncommutative torus, A k Θ , is defined by a skew symmetric real matrix Θ = (θ ij ), 1 ≤ i, j ≤ k. Further, we think of the A k Θ as a noncommutative topological space, whose algebraic K-theory is a Z-module, whose generators τ = (τ i ) are polynomials in θ ij . (More precisely, τ = exp(Θ), where the exterior algebra of θ ij is nilpotent.) Recall, that the C * -algebras A and A ′ are said to be stably isomorphic (Morita equivalent), if A ⊗ K ∼ = A ′ ⊗ K for the C * -algebra K of compact operators; such an isomorphism indicates, that the C * -algebras are homeomorphic as noncommutative topological spaces. By a result of Rieffel and Schwarz [13], the noncommutative tori A k Θ and A k Θ ′ are stably isomorphic, if the matrices Θ and Θ ′ belong to the same orbit of a subgroup SO(k, k | Z) of the group GL 2k (Z), which acts on Θ by the formula Θ ′ = (AΘ + B) / (CΘ + D), where (A, B, C, D) ∈ GL 2k (Z) and the matrices A, B, C, D ∈ GL k (Z) satisfy the conditions: (Here I is the unit matrix and t at the upper right of a matrix means a transpose of the matrix.) The group SO(k, k | Z) can be equivalently defined as a subgroup of the group SO(k, k | R) consisting of linear transformations of the space R 2k , which preserve the quadratic form B. The even dimensional normal tori. Further, we restrict to the case k = 2n (the even dimensional noncommutative tori). It is known, that by the orthogonal linear transformations every (generic) real even dimensional skew symmetric matrix can be brought to the normal form: where θ i > 0 are linearly independent over Q. We shall consider the noncommutative tori A 2n Θ 0 , given by the matrix (2); we refer to the family as a normal family. Recall, that any noncommutative torus has a canonical trace τ , which defines a homomorphism from K 0 (A k Θ ) ∼ = Z 2 k−1 to R; it follows from [4], that the image of K 0 (A 2n Θ 0 ) under the homomorphism has a basis, given by the formula τ (K 0 (A 2n Θ 0 )) = Z + θ 1 Z + . . .
C. The real multiplication ( [8]). The noncommutative torus A k Θ is said to have a real multiplication, if the endomorphism ring End (τ (K 0 (A k Θ ))) exceeds the trivial ring Z. Since any endomorphism of the Z-module τ (K 0 (A k Θ )) is the multiplication by a real number, it is easy to deduce, that all the entries of Θ = (θ ij ) are algebraic integers. (Indeed, the endomorphism is described by an integer matrix, which defines a polynomial equation involving θ ij .) Thus, the noncommutative tori with real multiplication is a countable subset of all k-dimensional tori; any element of the set we shall denote by A k RM . Notice, that for the even dimensional normal tori with real multiplication, the polynomials p i (θ) produce the algebraic integers in the extension of Q by θ i ; any such an integer is a linear combination (over Z) of the θ i . Thus, the trace formula reduces to τ (K 0 (A 2n RM )) = Z + θ 1 Z + . . . + θ n Z.

L-function of noncommutative tori
We consider even dimensional normal tori with real multiplication. Denote by A a positive integer matrix, whose (normalized) Perron-Frobenius eigenvector coincides with the vector θ = (1, θ 1 , . . . , θ n ) and A is not a power of a positive integer matrix; in other words, Aθ = λ A θ, where A ∈ GL n+1 (Z) and λ A is the corresponding eigenvalue. (Explicitly, A can be obtained from vector θ as the matrix of minimal period of the Jacobi-Perron continued fraction of θ [2].) Let p be a prime number; take the matrix A p and consider its characteristic polynomial char (A p ) = x n+1 + a 1 x n + . . . + a n x + 1. We introduce the following notation: A local zeta function of the A 2n RM is defined as the reciprocal of det (I n+1 − L n+1 p z); in other words, An L-function of A 2n RM is a product of the local zetas over all, but a finite number, of primes L(A 2n Remark 1 It will be shown, that for n = 0 and n = 1 formula (4)  Each one-dimensional abelian variety is a non-singular elliptic curve; choose this curve to have complex multiplication by (an order in) the imaginary quadratic field k and denote such a curve by E CM . Then, by theory of complex multiplication, the (maximal) abelian extension of k coincides with the minimal field of definition of the curve E CM , i.e. E CM ∼ = E(K) [14]. The Teichmüller functor F := F 1 maps E(K) into a two-dimensional noncommutative torus with real multiplication (section 3); we shall denote the torus by A 2 RM . To calculate the corresponding L-function L(A 2 RM , s), let A be a 2 × 2 positive integer matrix, whose normalized Perron-Frobenius eigenvenctor is (1, θ 1 ). For a prime p, the characteristic polynomial of the matrix A p writes as char (A p ) = x 2 + tr (A p )x + 1 and the matrix L 2 p takes the form: The corresponding local zeta function is the local zeta function for the elliptic curve E CM ; the proof will be arranged into a series of lemmas 1-5.

Lemma 2 The matrix
Proof. We shall use the following criterion: the (integer) matrices A and B are similar, if and only if the characteristic matrices xI − A and xI − B have the same Smith normal form. The calculation for the matrix xI − A gives: where ∼ are the elementary operations between the rows (columns) of the matrix. Similarly, a calculation for the matrix xI − A t gives: Thus, (xI − A) ∼ (xI − A t ) and lemma 2 follows. Let E CM be elliptic curve with complex multiplication by an order R in the ring of integers of the imaginary quadratic field k. Then A RM = F (E CM ) is a noncommutative torus with real multiplication by the order R in the ring of integers of a real quadratic field k (section 3). Let tr (α) = α +ᾱ be the trace function of a (quadratic) algebraic number field.
Note, that R * = {±ε k | k ∈ Z}, where ε > 1 is a fundamental unit of the order R ⊆ O k ; here O k means the ring of integers of a real quadratic field k = Q(θ). Choosing a sign in front of ε k , the following index map is defined Let α ∈ R and deg (α) = −n. To calculate the ι(α), recall some notation from Hasse [6], §16.5.C. Let Z/nZ be a cyclic group of order n. For brevity, let I = Z + Zθ be a pseudo-lattice and I n = Z + (nθ)Z its sub-lattice of index n; the fundamental units of I and I n are ε and ε n , respectively. By G n one understands a subgroup of Z/nZ of prime residue classes mod n. The g n ⊂ G n is a subgroup of the non-zero divisors of the G n . Finally, let g n be the smallest number, such that it divides |G n /g n | and ε gn ∈ I n . (The notation drastically simplifies in the case n = p is a prime number.) Proof. Notice, that deg (ω) = −deg (α) = n, where ω = F (α). Then the map ρ defines I and I n ; one can now apply the calculation of [6], pp 296-300. Namely, Theorem XIII ′ on p. 298 yields the required result. (We kept the notation of the original.) Corollary 3 ι(ψ E(K) (P)) = p.
Proof. It is known, that deg (ψ E(K) (P)) = −p, where ψ E(K) (P) ∈ R is the Grössencharacter. To calculate the g n in the case n = p, notice that the G p ∼ = Z/pZ and g p is trivial. Thus, |G p /g p | = p is divisible only by 1 or p. Since ε 1 is not in I n , one concludes that g p = p. The corollary follows.
Proof. It is not hard to see, that A is a hyperbolic matrix with the eigenvector (1, θ); the corresponding (Perron-Frobenius) eigenvalue is a fundamental unit ε > 1 of the pseudo-lattice Z + Zθ. In other words, A is a matrix form of the algebraic number ε. It is immediate, that A p is the matrix form for the ε p and tr (A p ) = tr (ε p ). In view of lemma 3 and corollary 3, tr (α) = tr (F (α)) = tr (ρ(F (α))) for ∀α ∈ R. In particular, if α = ψ E(K) (P) then, by corollary 3, one gets ρ(F (ψ E(K) (P))) = ε p . Taking traces in the last equation, we obtain the conclusion of lemma 5.
The fact ζ p (A 2 RM , z) = ζ p (E CM , z) follows from lemma 5, since the trace of the Grössencharacter coincides with such for the matrix A p . Let E CM be an elliptic curve with complex multiplication by an order in the imaginary quadratic field k and K the minimal field of definition of the E CM .

Case n = 0
When n = 0, one gets a one-dimensional (degenerate) noncommutative torus; such an object, A Q , can be obtained from the 2-dimensional torus A 2 θ by forcing θ = p/q ∈ Q be a rational number (hence our notation). One can always assume θ = 0 and, thus, τ (K 0 (A Q )) = Z. To calculate matrix L 1 p , notice that the group of automorphisms of the Z-module τ (K 0 (A Q )) = Z is trivial, i.e. is a multiplication by ±1; hence our 1 × 1 (real) matrix A is either 1 or −1. Since A must be positive, we conclude, that A = 1. However, A = 1 is not a prime matrix, if one allows the complex entries; indeed, for any N > 1 matrix A ′ = ζ N gives us A = (A ′ ) N , where ζ N = e 2πi N is the N-th root of unity. Therefore, A = ζ N and L 1 p = tr (A p ) = A p = ζ p N . A degenerate noncommutative torus, corresponding to the matrix A = ζ N , we shall write as A N Q ; in turn, such a torus is the image (under the Teichmüller functor) of a zero-dimensional abelian variety, which we denote by V N 0 . Suppose that Gal (K|Q) is abelian and let σ : Gal (K|Q) → C × be a homomorphism. Then, by the Artin reciprocity [5], there exists an integer N σ and a Dirichlet character χ σ : (Z/N σ Z) × → C × , such that σ(F r p ) = χ σ (p); choose our zerodimensional variety be V Nσ 0 . In view of the notation, L 1 p = ζ p Nσ ; on the other hand, it is verified directly, that ζ p Nσ = e 2πi Nσ p = χ σ (p). Thus, L 1 p = χ σ (p). To obtain a local zeta function, we substitute a 1 = L 1 p into the formula (4) and get where χ σ (p) is the Dirichlet character. Therefore, L(A Nσ Q , s) ≡ L(s, χ σ ) is the Dirichlet L-series; such a series, by construction, coincides with the Artin L-series of the representation σ : Gal (K|Q) → C × . Case n = 0 of theorem 1 follows.

Teichmüller functors
Denote by Λ a lattice of rank 2n; recall, that an n-dimensional (principally polarized) abelian variety, V n , is the complex torus C n /Λ, which admits an embedding into a projective space [9].

Abelian varieties of dimension n = 1
A. Basic example. Let n = 1 and consider the complex torus V 1 ∼ = C/(Z + τ Z); it always embeds (via the Weierstrass ℘ function) into a projective space as a non-singular elliptic curve. Let H = {τ = x + iy ∈ C | y > 0} be the upper half-plane and ∂H = {θ ∈ R | y = 0} its (topological) boundary. We identify V 1 (τ ) with the points of H and A 2 θ with the points of ∂H. Let us show, that the boundary is natural; the latter means, that the action of the modular group SL 2 (Z) extends to the boundary, where it coincides with the stable isomorphisms of tori. Indeed, conditions (1) are equivalent to where ad−bc = 1, a, b, c, d ∈ Z and Θ ′ = (AΘ+B)/(CΘ+D) = (0, aθ+b cθ+d , − aθ+b cθ+d , 0). Therefore, θ ′ = (aθ + b)(cθ + d) −1 for a matrix (a, b, c, d) ∈ SL 2 (Z). Thus, the action of SL 2 (Z) extends to the boundary ∂H, where it induces stable isomorphisms of the noncommutative tori.
B. The Teichmüller functor ( [10]). There exists a continuous map F 1 : H → ∂H, which sends isomorphic complex tori to the stably isomorphic noncommutative tori. An exact result is this. Let φ be a closed form on the torus, whose trajectories define a measured foliation; according to the Hubbard-Masur theorem (applied to the complex tori), this foliation corresponds to a point τ ∈ H. The map F 1 : H → ∂H is defined by the formula τ → θ = γ 2 φ/ γ 1 φ, where γ 1 and γ 2 are generators of the first homology of the torus. The following is true: (i) H = ∂H × (0, ∞) is a trivial fiber bundle, whose projection map coincides with F 1 ; (ii) F 1 is a functor, which sends isomorphic complex tori to the stably isomorphic noncommutative tori. We shall refer to F 1 as the Teichmüller functor. Recall, that the complex torus C/(Z + τ Z) is said to have a complex multiplication, if the endomorphism ring of the lattice Λ = Z + τ Z exceeds the trivial ring Z; the complex multiplication happens if and only if τ is an algebraic number in an imaginary quadratic field. The following is true: is a torus with complex multiplication.

Abelian varieties of dimension n ≥ 1
A. The Siegel upper half-space ( [9]). The space H n : | Im (τ i ) > 0} of symmetric n × n matrices with complex entries is called a Siegel upper half-space; the points of H n are one-to-one with the n-dimensional principally polarized abelian varieties. Let Sp(2n, R) be the symplectic group; it acts on H n by the linear fractional transformations τ → τ ′ = (aτ + b)/(cτ + d), where (a, b, c, d) ∈ Sp(2n, R) and a, b, c and d are the n × n matrices with real entries. The abelian varieties V n and V ′ n are isomorphic, if and only if, τ and τ ′ belong to the same orbit of the group Sp(2n, Z); the action is discontinuous on H n [9], Ch.2, §4. Denote by Σ 2n a space of the 2n-dimensional normal noncommutative tori. The following lemma is critical. Proof. (i) The group O(n, n | R) can be defined as a subgroup of GL 2 (R), which preserves the quadratic form f (x 1 , . . . , x 2n ) = x 1 x n+1 + x 2 x n+2 + . . . + x n x 2n [13]. We shall denote u i = x 1 for 1 ≤ i ≤ n and v i = x i for n + 1 ≤ i ≤ 2n. Consider the following skew symmetric bilinear form q(u, v) = u 1 v n+1 + . . . + u n v 2n − u n+1 v 1 − . . . − u 2n v n , where u, v ∈ R 2n . It is known, that each linear substitution g ∈ Sp(2n, R) preserves the form q(u, v). Since q(u, v) = f (x 1 , . . . , x 2n ) − u n+1 v 1 − . . . − u 2n v n , one concludes that g also preserves the form f (x 1 , . . . , x 2n ), i.e. g ∈ O(n, n|R). It is easy to see, that the inclusion is proper except the case n = 1, i.e. when Sp(2, R) ∼ = O(1, 1|R) ∼ = SL 2 (R). Lemma follows.
(ii) We wish to give a second proof of this important fact, which is based on the explicit formulas for the block matrices A, B, C and D. The fact that a symplectic linear transformation preserves the skew symmetric bilinear form q(u, v) can be written in a matrix form: where t is the transpose of a matrix. Performing the matrix multiplication, one gets the following matrix identities a t d−c t b = I, a t c−c t a = 0 = b t d−d t b.
Let us show, that these identities imply the Rieffel-Schwarz identities (1) imposed on the matrices A, B, C and D. Indeed, in view of the formulas (11), the Rieffel-Schwarz identities can be written as: A step by step matrix multiplication in (13) shows that the identities a t d − c t b = I, a t c − c t a = 0 = b t d − d t b imply the identities (13). (Beware: the operation is not commutative.) Thus, any symplectic transformation satisfies the Rieffel-Schwarz identities, i.e. belongs to the group O(n, n|R). Lemma 7 follows.
B. The generalized Teichmüller functors. By lemma 7, the action of Sp(2n, Z) on the H n extends to the Σ 2n , where it acts by stable isomorphisms of the noncommutative tori; thus, Σ 2n is a natural boundary of the Siegel upper half-space H n . However, unless n = 1, the Σ 2n is not a topological boundary of H n . Indeed, dim R (H n ) = n(n + 1) and dim R (∂H n ) = n 2 + n − 1, while dim R (Σ 2n ) = n. Thus, Σ 2n is an n-dimensional subspace of the topological boundary of H n ; this subspace is everywhere dense in ∂H n , since the Sp(2n, Z)-orbit of an element of Σ 2n is everywhere dense in ∂H n [13]. A (conjectural) continuous map F n : H n → Σ 2n , we shall call a generalized Teichmüller functor. The F n has the following properties: (i) it sends each pair of isomorphic abelian varieties to a pair of the stably isomorphic even dimensional normal tori; (ii) the range of F n on the abelian varieties with complex multiplication consists of the noncommutative tori with real multiplication.
As explained, such a functor has been constructed only in the case n = 1; the difficulties in higher dimensions are due to the lack (so far) of a proper Teichmüller theory for the abelian varieties of dimension n ≥ 2.