Real multiplication revisited

It is proved that the Hilbert class field of a real quadratic field ${\Bbb Q}(\sqrt{D})$ modulo a power $m$ of the conductor $f$ is generated by the Fourier coefficients of the Hecke eigenform for a congruence subgroup of level $fD$.


Introduction
The Kronecker's Jugendtraum is a conjecture that the maximal unramified abelian extension (the Hilbert class field) of any algebraic number field is generated by the special values of modular functions attached to an abelian variety. The conjecture is true for the rational field and imaginary quadratic fields with the modular functions being an exponent and the j-invariant, respectively. In the case of an arbitrary number field, a description of the abelian extensions is given by class field theory, but an explicit formula for the generators of these abelian extensions, in the sense sought by Kronecker, is unknown even for the real quadratic fields.
The problem was first studied by [Hecke 1910] [8]. A description of abelian extensions of real quadratic number fields in terms of coordinates of points of finite order on abelian varieties associated with certain modular curves was obtained in [Shimura 1972] [16]. Stark formulated a number of conjectures on abelian extension of arbitrary number fields, which in the real quadratic case amount to specifying generators of these extensions using special values of Artin L-functions, see [Stark 1976] [19]. Based on an analogy with complex multiplication, Manin suggested to use the so-called "pseudo-lattices" Z + Zθ in R having non-trivial real multiplications to produce abelian extensions of real quadratic fields, see [Manin 2004] [10]. Similar to the case of complex multiplication, the endomorphism ring R f = Z + fO k of pseudo-lattice Z + Zθ is an order in the real quadratic field k = Q(θ), where O k is the ring of integers of k and f is the conductor of R f ; Manin calls these pseudo-lattices with real multiplication.
The aim of our note is a formula for generators of the Hilbert class field of real quadratic fields based on a modularity of complex multiplication and a symmetry of complex and real multiplication. To give an idea, let )) the Hilbert class field of k modulo conductor f ≥ 1 and let N = f D; let Jac (X 1 (f D)) be the Jacobian of modular curve X 1 (f D). There exists an abelian sub-variety A φ ⊂ Jac (X 1 (f D)), such that its points of finite order generate K ab (k), see [Hecke 1928] [9], [Shimura 1971] [15], Theorem 1 and [Shimura 1972] [16], Section 8. The K ab (k) is a CM-field, i.e. a totally imaginary quadratic extension of the totally real field K φ generated by the Fourier coefficients of the Hecke eigenform φ(z) ∈ S 2 (Γ 1 (f D)), see [Shimura 1972] [16], p. 137. In particular, there exists a holomorphic map X 0 we refer to the above as a modularity of complex multiplication.
Recall that (twisted homogeneous) coordinate ring of an elliptic curve E(C) is isomorphic to a Sklyanin algebra, see e.g. [Stafford & van den Bergh 2001] [18], Example 8.5; the norm-closure of a self-adjoint representation of the Sklyanin algebra by the linear operators on a Hilbert space H is isomorphic to a noncommutative torus A θ , see [Rieffel 1990] [14] for the definition.
has complex multiplication, the noncommutative torus A θ has real multiplication by an order R f = Z + fO k in the field k = Q( √ D); moreover, it is known that f = f m for the minimal power m satisfying an isomorphism: where Cl (R f ) and Cl (R f ) are the ideal class groups of orders R f and R f , respectively. We shall refer to (2) as a symmetry of complex and real multiplication. The noncommutative torus with real multiplication by R f will be denoted by A (2) can be calculated using the well-known formula for the class number of a non-maximal order The (twisted homogeneous) coordinate ring of the Riemann surface X 0 1 (f D) is an AF-algebra A φ 0 linked to a holomorphic differential φ 0 (z)dz on X 0 1 (f D), see Section 2.2, Definition 1 and Remark 4 for the details; the Grothendieck semigroup K + 0 (A φ 0 ) is a pseudo-lattice Z + Zθ 1 + . . . + Zθ n−1 in the number field K φ , where n equals the genus of X 0 1 (f D). Moreover, the modu- induces a natural C * -algebra homomorphism RM between the corresponding coordinate rings, so that the following diagram commutes: is a pseudo-lattice Z + Zθ in the field k, such that End (Z + Zθ) ∼ = R f ; in other words, one can use the above diagram to control the arithmetic of the field K φ by such of the real quadratic field k. Roughly speaking, this observation solves the Kronecker's Jugendtraum for the real quadratic fields; namely, the following result is proved.

Theorem 1
The Hilbert class field of a real quadratic field k = Q( √ D) modulo conductor f m is an extension of k by the Fourier coefficients of the Hecke eigenform φ(z) ∈ S 2 (Γ 1 (f D)), where m is the smallest positive integer satisfying isomorphism (2). The article is organized as follows. Section 2 covers basic facts on real multiplication and AF-algebras of the Hecke eigenforms. Theorem 1 is proved in Section 3. Section 4 contains numerical examples illustrating theorem 1.

Preliminaries
The reader can find basics of the C * -algebras in [Murphy 1990] [11] and their K-theory in [Blackadar 1986] [1]. The noncommutative tori are covered in [Rieffel 1990] [14] and real multiplication in [Manin 2004] [10]. For main ideas of non-commutative algebraic geometry, see the survey by [

Real multiplication
The noncommutative torus A θ is a universal C * -algebra generated by the unitary operators u and v acting on a Hilbert space H and satisfying the commutation relation vu = e 2πiθ uv, where θ is a real number. The C * -algebra A θ is said to be stably isomorphic (Morita equivalent) to A θ ′ , whenever The K-theory of A θ is two-periodic and The real multiplication says that the endomorphism ring of pseudo-lattice Z + Zθ exceeds the ring Z corresponding to multiplication by m endomorphisms; similar to complex multiplication, it means that the endomorphism ring is isomorphic to an order The Sklyanin algebra S α,β,γ (C) is a free C-algebra on four generators and six relations: where α + β + γ + αβγ = 0; such an algebra corresponds to a twisted homogeneous coordinate ring of an elliptic curve in the complex projective space CP 3 given by the intersection of two quadric surfaces of the form Being such a ring means that the algebra S α,β,γ satisfies an isomorphism where Coh is the category of quasi-coherent sheaves on E α,β,γ (C), Mod the category of graded left modules over the graded ring S α,β,γ (C) and then there exists a self-adjoint representation of the Sklyanin * -algebra S α,1,−1 (C) by linear operators on a Hilbert space H, such that its norm-closure is isomorphic to A θ ; namely, A 0 θ ∼ = S α,1,−1 (C)/I µ , where A 0 θ is a dense sub-algebra of A θ and I µ is an ideal generated by the "scaled unit" relations Thus the algebra A θ is a coordinate ring of elliptic curve E(C), such that isomorphic elliptic curves correspond to the stably isomorphic (Morita equivalent) noncommutative tori; this fact explains the modular transformation law in (4). In particular, if E(C) has complex multiplication by an order , then A θ has real multiplication by an order [13]; the isomorphism is a necessary and sufficient condition for A

AF-algebra of the Hecke eigenform
An AF-algebra (Approximately Finite C * -algebra) is defined to be the norm closure of an ascending sequence of finite dimensional C * -algebras M n , where M n is the C * -algebra of the n × n matrices with entries in C. Here the index n = (n 1 , . . . , n k ) represents the semi-simple matrix algebra M n = M n 1 ⊕ . . . ⊕ M n k . The ascending sequence mentioned above can be written as M 1 where M i are the finite dimensional C * -algebras and ϕ i the homomorphisms between such algebras. The homomorphisms ϕ i can be arranged into a graph as follows.
One has two sets of vertices V i 1 , . . . , V i k and V i ′ 1 , . . . , V i ′ k joined by b rs edges whenever the summand M ir contains b rs copies of the summand M i ′ s under the embedding ϕ i . As i varies, one obtains an infinite graph called the Bratteli diagram of the AF-algebra. The matrix B = (b rs ) is known as a partial multiplicity matrix; an infinite sequence of B i defines a unique AF-algebra. An AF-algebra is called stationary if B i = Const = B, see [Effros 1981] [6], Chapter 6; when two non-similar matrices B and B ′ have the same characteristic polynomial, the corresponding stationary AFalgebras will be called companion AF-algebras.
Let N ≥ 1 be a natural number and consider a (finite index) subgroup of the modular group given by the formula: Let H = {z = x + iy ∈ C | y > 0} be the upper half-plane and let Γ 1 (N) act on H by the linear fractional transformations; consider an orbifold H/Γ 1 (N).
To compactify the orbifold at the cusps, one adds a boundary to H, so that H * = H ∪ Q ∪ {∞} and the compact Riemann surface X 1 (N) = H * /Γ 1 (N) is called a modular curve. The meromorphic functions φ(z) on H that vanish at the cusps and such that are called cusp forms of weight two; the (complex linear) space of such forms will be denoted by S 2 (Γ 1 (N)). The formula φ(z) → ω = φ(z)dz defines an isomorphism S 2 (Γ 1 (N)) ∼ = Ω hol (X 1 (N)), where Ω hol (X 1 (N)) is the space of all holomorphic differentials on the Riemann surface X 1 (N). Note that dim C (S 2 (Γ 1 (N)) = dim C (Ω hol (X 1 (N)) = g, where g = g(N) is the genus of the surface X 1 (N). A Hecke operator, T n , acts on S 2 (Γ 1 (N)) by the formula T n φ = m∈Z γ(m)q m , where γ(m) = a|GCD(m,n) ac mn/a 2 and φ(z) = m∈Z c(m)q m is the Fourier series of the cusp form φ at q = e 2πiz . Further, T n is a self-adjoint linear operator on the vector space S 2 (Γ 1 (N)) endowed with the Petersson inner product; the algebra T N := Z[T 1 , T 2 , . . .] is a commutative algebra. Any cusp form φ ∈ S 2 (Γ 1 (N)) that is an eigenvector for one (and hence all) of T n , is referred to as a Hecke eigenform. The Fourier coefficients c(m) of φ are algebraic integers, and we denote by K φ = Q(c(m)) an extension of the field Q by the Fourier coefficients of φ. Then K φ is a real algebraic number field of degree 1 ≤ deg (K φ |Q) ≤ g, where g is the genus of the surface X 1 (N), see e.g. [Diamond & Shurman 2005] [5], Proposition 6.6.4. Any embedding σ : K φ → C conjugates φ by acting on its coefficients; we write the corresponding Hecke eigenform φ σ (z) := m∈Z σ(c(m))q m and call φ σ a conjugate of the Hecke eigenform φ.
Proof. Because K ab (k) is the Hilbert class field over k modulo conductor f , we must have where Gal is the Galois group of the extension K ab (k)|k and Cl (R f ) is the class group of ring R f , see e.g. [Silverman 1994] [17], p.112. But |Gal (K ab (k)|k)| = deg (K ab (k)|k) and by lemma 1 we have deg (K ab (k)|k) = g(X 0 1 (f D)). In view of this and isomorphism (11), one gets |Cl (R f )| = |Gal (K ab |k)| = g(X 0 1 (f D)). Corollary 1 follows.
Proof. From lemma 2 and corollary 1 one gets deg (K φ |Q) = |Cl (R f )|. In view of this and equality (2), one gets the conclusion of corollary 2.

Lemma 3 (Basic lemma) Gal
Proof. Let us outline the proof. In view of lemma 2 and corollaries 1-2, we denote by h the single integer g(X 0 i are companion AF-algebras, see Section 1.2. Recall that the characteristic polynomial for the partial multiplicity matrices B φ 0 i of companion AF-algebras A φ 0 i is the same; it is a minimal polynomial of degree h and let {λ 1 , . . . , λ h } be the roots of such a polynomial, compare with [Effros 1981] [6], Corollary 6.3. Since det (B φ 0 i ) = 1, the numbers λ i are algebraic units of the field K φ . Moreover, λ i are algebraically conjugate and can be taken for generators of the extension K φ |Q; since deg (K φ |Q) = h = |Cl (R f )| there exists a natural action of group Cl (R f ) on these generators. The action extends to automorphisms of the entire field K φ preserving Q; thus one gets the Galois group of extension K φ |Q and an isomorphism Gal (K φ |Q) ∼ = Cl (R f ). Let us pass to a step-by-step argument.

Remark 3
The forms {φ 0 1 , . . . , φ 0 h } can be taken for a basis in the space Ω hol (X 0 1 (f D)); we leave it to the reader to verify, that abelian variety A φ is isomorphic to the quotient of C h by the lattice of periods of holomorphic differentials φ 0 i (z)dz on X 0 1 (f D).
(ii) Let A φ 0 i be the AF-algebra corresponding to holomorphic differential i is a stationary AF-algebra, i.e. its partial multiplicity matrix is a constant; we shall denote such a matrix by B φ 0 i . (iii) By definition, the matrices B φ 0 i of companion AF -algebras A φ i have the same characteristic polynomial p(x) ∈ Z[x]; the matrices B φ 0 i itself are not pairwise similar and, therefore, the AF-algebras A φ 0 i are not pairwise isomorphic. The total number h of such matrices is equal to the class number of the endomorphism ring of pseudo-lattice [Effros 1981] [6], Corollary 6.3.

Remark 4
Notice that there are {X 1 , . . . , X h } pairwise non-isomorphic Riemann surfaces X : corresponding to elements of the group Cl (R f ). Thus the companion AFalgebras {A φ 0 1 , . . . , A φ 0 h } can be viewed as coordinate rings of {X 1 , . . . , X h }; the latter means that A φ 0 i discern non-isomorphic Riemann surfaces and (iv) The polynomial p(x) is minimal and splits in the totally real field K φ . Indeed, matrices B φ 0 i generate the Hecke algebra T N on S 2 (Γ 1 (N)); thus each B φ 0 i is self-adjoint and, therefore, all eigenvalues are real of multiplicity one; since B φ 0 i is integer, all roots of characteristic polynomial p(x) of B φ 0 i belong to the field K φ . ( It is easy to see that λ i are algebraic units of the field K φ because det (B φ 0 i ) = 1; note that numbers {λ 1 , . . . , λ h } are algebraically conjugate. Since deg (K φ |Q) = h, the numbers λ i can be taken for generators of the field K φ , i.e. K φ = Q(λ 1 , . . . , λ h ).
(vi) Finally, let us establish an explicit formula for the isomorphism Since Gal (K φ |Q) is an automorphism group of the field K φ preserving Q, it will suffice to define the action * of an element a ∈ Cl (R f ) on the generators λ i of K φ . Let {a 1 , . . . , a h } be the set of all elements of the group Cl (R f ). For an element a ∈ Cl (R f ) define an index function α by the formula a i a = a α(i) . Then the action * of an element a ∈ Cl (R f ) on the generators λ i of the field K φ is given by the formula: We leave it to the reader to verify that formula (13) gives an isomorphism Cl (R f ) → Gal (K φ |Q); this argument completes the proof of lemma 3.

Remark 5
The class field theory says that f = f m , i.e. the extensions of fields k and k must ramify over the same set of prime ideals. Indeed, consider the commutative diagram below, where I f and I f are groups of all ideals Artin homomorphism of k and k, which are relatively prime to the principal ideals (f ) and (f), respectively. Since Gal (K ab (k)|Q) ∼ = Gal (K φ |Q), one gets an isomorphism I f ∼ = I f , i.e. f = f m for some positive integer m.

Corollary 3
The Hilbert class field of real quadratic field k = Q( √ D) modulo conductor f ≥ 1 is isomorphic to the field k(K φ ) generated by the Fourier coefficients of the Hecke eigenform φ(z) ∈ S 2 (Γ 1 (f D)).

Examples
Along with the method of Stark's units [Cohen & Roblot 2000] [4], theorem 1 can be used in the computational number theory. For the sake of clarity, we shall consider the simplest examples; the rest can be found in Figure 1.
The class number of quadratic field k = Q( √ −15) is known to be 2; such a number for quadratic field k = Q( √ 15) is also equal to 2. Thus, and isomorphism (2) is trivially satisfied for each power m, i.e. one obtains the unramified extensions. By theorem 1, the Hilbert class field of k is generated by the Fourier coefficients of the Hecke eigenform φ(z) ∈ S 2 (Γ 1 (15)). Using the computer program SAGE created by William A. Stein, one finds an irreducible factor p(x) = x 2 − 4x + 5 of the characteristic polynomial of the Hecke operator T p=2 acting on the space S 2 (Γ 1 (15)). Therefore, the Fourier coefficient c(2) coincides with a root of equation p(x) = 0; in other words, we arrive at an extension of k by the polynomial p(x). The generator x of the field K φ = Q(c(2)) is a root of the bi-quadratic equation it holds |Cl (R f=1 )| = 1. However, for the ramified extensions one obtains the following isomorphism: where m = 3 is the smallest integer satisfying formula (2). By theorem 1, the Hilbert class field of k modulo f = 8 is generated by the Fourier coefficients of the Hecke eigenform φ(z) ∈ S 2 (Γ 1 (2 × 14)). Using the SAGE, one finds that the characteristic polynomial of the Hecke operator T p=3 on S 2 (Γ 1 (2 × 14)) has an irreducible factor p(x) = x 4 + 3x 2 + 9. Thus the Fourier coefficient c(3) is a root of the polynomial p(x) and one gets an extension of k by the polynomial p(x). In other words, generator x of the field K φ = Q(c (3) is the Hilbert class field over Q( √ 14) modulo conductor f = 8. Clearly, the extension is ramified over the prime ideal p = (2). Table 1 below lists quadratic fields for some square-free discriminants 2 ≤ D ≤ 101. The conductors f and f satisfying equation (2) were calculated using tables for the class number of non-maximal orders in quadratic fields posted at www.numbertheory.org; the site is maintained by Keith Matthews. We focused on small conductors; the interested reader can compute the higher conductors using a pocket calculator. In contrast, computation of generator x of the Hilbert class field require the online program SAGE created by William A. Stein. We write an explicit formula for x or its minimal polynomial p(x) over k.