Nikolaev IV^{*}
Department of Mathematical Sciences, The Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada
Received Date: December 22, 2015; Accepted Date: January 25, 2016; Published Date; January 28, 2016
Citation: Nikolaev IV (2016) Real Multiplication Revisited. J Generalized Lie Theory Appl S2: 007. doi:10.4172/17364337.S2007
Copyright: © 2016 Nikolaev IV. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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It is proved that the Hilbert class field of a real quadratic field Q 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.
Class field; Real multiplication
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 jinvariant, 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 [1]. 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 studies of Shimura [2]. 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 Lfunctions [3]. Based on an analogy with complex multiplication, Manin suggested to use the socalled “pseudolattices” in having nontrivial real multiplications to produce abelian extensions of real quadratic fields [4]. Similar to the case of complex multiplication, the endomorphism ring of pseudolattice is an order in the real quadratic field , where Ok is the ring of integers of k and f is the conductor of Manin calls these pseudolattices 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 and a symmetry of complex and real multiplication. To give an idea, let
(1)
be a congruence subgroup of level N ≥ 1 and be the Lobachevsky halfplane; let be the corresponding modular curve and S_{2}(Γ_{1}(N)) the space of all cusp forms on Γ_{1} (N) of weight 2. Let be elliptic curve with complex multiplication by an order in the field [5]. Denote by the Hilbert class field of k modulo conductor f ≥ 1 and let N = fD; let be the Jacobian of modular curve . There exists an abelian subvariety , such that its points of finite order generate ^{ab}(k), [2,6,7], Section 8. The ^{ab}(k) is a CMfield, i.e. a totally imaginary quadratic extension of the totally real field _{φ} generated by the Fourier coefficients of the Hecke eigenform [2]. In particular, there exists a holomorphic map , where is a Riemann surface such that ; we refer to the above as a modularity of complex multiplication.
Recall that (twisted homogeneous) coordinate ring of an elliptic curve ε ( ) is isomorphic to a Sklyanin algebra, [8]; the normclosure of a selfadjoint representation of the Sklyanin algebra by the linear operators on a Hilbert space is isomorphic to a noncommutative torus _{θ}, [9] for the definition.
Whenever elliptic curve has complex multiplication, the noncommutative torus _{θ} has real multiplication by an order in the field ; moreover, it is known that f = f^{m} for the minimal power m satisfying an isomorphism:
(2)
where Cl(R_{f}) and Cl(_{f}) are the ideal class groups of orders R_{f} and _{f}, respectively. We shall refer to (2) as a symmetry of complex and real multiplication. The noncommutative torus with real multiplication by _{f} will be denoted by .
Remark 1: The isomorphism (2) can be calculated using the wellknown formula for the class number of a nonmaximal order + fO_{K} of a quadratic field
(3)
where is the class number of the maximal order O_{K}, e_{f} is the index of the group of units of + fO_{K} in the group of units of O_{K}, p is a prime number and is the Legendre symbol [10,11].
The (twisted homogeneous) coordinate ring of the Riemann surface is an AFalgebra linked to a holomorphic differential on , see Section 2.2, Definition 1 and Remark 5 for the details; the Grothendieck semigroup is a pseudolattice in the number field _{φ}, where n equals the genus of . Moreover, a holomorphic map − induces the C*algebra homomorphism between the corresponding coordinate rings, so that the following diagram commutes:
But is a pseudolattice + θ in the field , such that End ( + θ) ≅ _{f} in other words, one can use the above diagram to control the arithmetic of the field 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 is true.
Theorem 1. The Hilbert class field of a real quadratic field modulo conductor f^{m} is an extension of k by the Fourier coefficients of the Hecke eigenform , where m is the smallest positive integer satisfying isomorphism (2).
Remark 2. Theorem 1 can be used to compute concrete extensions. For instance, theorem 1 says that for the quadratic field its Hilbert class field is isomorphic to and for such a field modulo conductor f = 8 is isomorphic to see section 4 for more examples.
The article is organized as follows. Section 2 covers basic facts on real multiplication and AFalgebras 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 studies of Murphy [12] and their Ktheory in theory of Blackadar [13]. The noncommutative tori are covered in literature of Rieffel [9] and real multiplication in studies of Manin [4]. For main ideas of noncommutative algebraic geometry, see the survey by Stafford and van den Bergh [8]. The AFalgebras are reviewed in studies of Effros [14]. For a general theory of modular forms we refer to literature of Diamond and Shurman [15].
Real multiplication
The noncommutative torus _{θ} is a universal C*algebra generated by the unitary operators u and v acting on a Hilbert space and satisfying the commutation relation vu = e^{2π iθ} uv , where θ is a real number. The C*algebra _{θ} is said to be stably isomorphic (Morita equivalent) to , whenever where is the C* algebra of all compact operators on ; the _{θ} is stably isomorphic to if and only if
for some matrix (4)
The Ktheory of _{θ} is twoperiodic and so that the Grothendieck semigroup corresponds to positive reals of the pseudolattice . The _{θ} is said to have real multiplication, if θ is a quadratic irrationality, i.e. irrational root of a quadratic polynomial in [x]. The real multiplication says that the endomorphism ring of pseudolattice + θ exceeds the ring corresponding to multiplication by m endomorphisms; similar to complex multiplication, it means that the endomorphism ring is isomorphic to an order of conductor f ≥ 1 in the real quadratic field k = (θ ), hence the name. If D > 0 is the discriminant of k, then by we denote torus θ with real multiplication by the order .
The Sklyanin algebra S_{α, β, γ} ( ) is a free Calgebra on four generators and six relations:
(5)
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 P^{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
(6)
where Coh is the category of quasicoherent sheaves on ε_{α, β, γ} ( ) , Mod the category of graded left modules over the graded ring ε_{α, β, γ} ( ) and Tors the full subcategory of Mod consisting of the torsion modules, [8].
If one sets , then there exists a selfadjoint representation of the Sklyanin *algebra S_{α,1,1}( ) by linear operators on a Hilbert space , such that its normclosure is isomorphic to θ ; namely, where is a dense subalgebra of θ and I_{μ} is an ideal generated by the “scaled unit” relations , where μ > 0 is a constant. Thus the algebra 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 ε () has complex multiplication by an order in a quadratic field , then θ has real multiplication by an order in the quadratic field , where f is the smallest integer satisfying an isomorphism , [16]; the isomorphism is a necessary and sufficient condition for to discern nonisomorphic elliptic curves having the same endomorphism ring R_{f} . For the constraint f =f^{m}, see remark 6.
AFalgebra of the Hecke eigenform
An AFalgebra (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 . Here the index n=(n_{1},...,n_{k}) represents the semisimple matrix algebra . The ascending sequence mentioned above can be written as ,here M_{i} are the finite dimensional C*algebras and M_{i} the homomorphisms between such algebras. The homomorphisms M_{i} can be arranged into a graph as follows. Let and be the semisimple C* algebras and the homomorphism. One has two sets of vertices and joined by b_{rs} edges whenever the summand contains b_{rs} copies of the summand is under the embedding ∅_{i}. As i varies, one obtains an infinite graph called the Bratteli diagram of the AFalgebra. The matrix B= (b_{rs} ) is known as a partial multiplicity matrix; an infinite sequence of B_{i} defines a unique AFalgebra. An AFalgebra is called stationary if B_{i} = Const = B, [14], when two nonsimilar matrices B and B’ have the same characteristic polynomial, the corresponding stationary AFalgebras will be called companion AFalgebras.
Let N ≥ 1 be a natural number and consider a (finite index) subgroup of the modular group given by the formula:
(7)
Let be the upper halfplane and let Γ_{1}(N) act on by the linear fractional transformations; consider an orbifold / Γ_{1}(N) . To compactify the orbifold at the cusps, one adds a boundary to , so that and the compact Riemann surface is called a modular curve. The meromorphic functions (z) on that vanish at the cusps and such that
(8)
are called cusp forms of weight two; the (complex linear) space of such forms will be denoted by . The formula defines an isomorphism , where is the space of all holomorphic differentials on the Riemann surface X_{1}(N). Note that , where g =g(N) is the genus of the surface X_{1}(N) . A Hecke operator, T_{n}, acts on by the formula , where and is the Fourier series of the cusp form Ï at . Further, T_{n} is a selfadjoint linear operator on the vector space endowed with the Petersson inner product; the algebra is a commutative algebra. Any cusp form 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 Ï are algebraic integers, and we denote by K_{φ} = Q(c(m)) an extension of the field Q by the Fourier coefficients of Ï. Then ïφ is a real algebraic number field of degree , where g is the genus of the surface X_{1}(N) [5], Proposition 6.6.4. Any embedding conjugates Ï by acting on its coefficients; we write the corresponding Hecke eigenform and call Ï^{σ} a conjugate of the Hecke eigenform Ï.
Let be a holomorphic differential on a Riemann surface X. We shall denote by R(w) a closed form on X (the real part of w) and consider its periods against a basis γi in the (relative) homology group , where is the set of zeros of the form R(w). Assume and consider the vector with . The JacobiPerron continued fraction of θ is given by the formula:
(9)
where is a vector of nonnegative integers, I is the unit matrix and = (0,...,0,1)^{T} [17]. By _{Ï} we shall understand the AFalgebra given the Bratteli diagram with partial multiplicity matrices B_{i}. If is a Hecke eigenform, then the corresponding AFalgebra _{Ï} is stationary with the partial multiplicity matrices B_{i} = Const = B; moreover, each conjugate eigenform Ï^{σ} defines a companion AFalgebra . It is known that , where φ is an algebraic number field generated by the Fourier coefficients of Ï, [18].
Definition 1. Let be an abelian variety associated to the Hecke eigenform [15], Definition 6.6.3. By we shall understand the Riemann surface of genus g, such that
(10)
By we denote the image of the Hecke eigenform under the holomorphic map .
Remark 3. The surface is correctly dened. Indeed, since the abelian variety A_{φ} is the product of g copies of an elliptic curve with the complex multiplication, there exists a holomorphic map from A_{φ} to the elliptic curve. For a Riemann surface X of genus g covering the elliptic curve ε_{CM} by a holomorphic map (such a surface and a map always exist), one gets a period map X → A_{φ} by closing the arrows of a commutative diagram A_{φ} → ε_{CM} ← X. It is easy to see, that the Jacobian of X coincides with A_{φ} and we set .
Lemma 1.
Proof. By definition, abelian variety A_{φ} is the quotient of ^{n} by a lattice of periods of the Hecke eigenform and all its conjugates φ^{σ}(z) on the Riemann surface . These periods are complex algebraic numbers generating the Hilbert class field over imaginary quadratic field modulo conductor f, [2,6,7], Section 8. The number of linearly independent periods is equal to the total number of the conjugate eigenforms φ^{σ}(z) , i.e. . Since real dimension , we conclude that and, therefore, . But and one gets . Lemma 1 follows.
Corollary 1.
Proof. Because is the Hilbert class field over k modulo conductor f, we must have
(11)
where Gal is the Galois group of the extension and is the class group of ring Rf, [5]. But and by lemma 1 we have . In view of this and isomorphism (11), one gets . Corollary 1 follows.
Lemma 2.
Proof. It is known that [15], Proposition 6.6.4. But abelian variety and, therefore, , hence the lemma.
Corollary 2.
Proof. From lemma 2 and corollary 1 one gets deg In view of this and equality (2), one gets the conclusion of corollary 2.
Lemma 3. (Basic lemma)
Proof. Let us outline the proof. In view of lemma 2 and corollaries 12, we denote by h the single integer . Since deg , there exist conjugate Hecke eigenforms [15], Theorem 6.5.4; thus one gets h holomorphic forms on the Riemann surface . Let be the corresponding stationary AFalgebras; the are companion AFalgebras, see Section 1.2. Recall that the characteristic polynomial for the partial multiplicity matrices of companion AFalgebras is the same; it is a minimal polynomial of degree h and let be the roots of such a polynomial, compare with studies of Effros [14], Corollary 6.3. Since , the numbers λ_{i} are algebraic units of the field φ. Moreover, λ_{i} are algebraically conjugate and can be taken for generators of the extension ; since there exists a natural action of group on these generators. The action extends to automorphisms of the entire field φ preserving ; thus one gets the Galois group of extension and an isomorphism . Let us pass to a stepbystep argument.
(i) Let and let be the Hecke eigenform. It is known that there exists conjugate Hecke eigenforms, so that Ï (z) is one of them [15], Theorem 6.5.4. Let be the corresponding forms on the Riemann surface .
Remark 4. The forms can be taken for a basis in the space we leave it to the reader to verify, that abelian variety Aφ is isomorphic to the quotient of by the lattice of periods of holomorphic differentials on .
(ii) Let be the AFalgebra corresponding to holomorphic differential on , see Section 2.2; the set consists of the companion AFalgebras. It is known that each is a stationary AFalgebra, i.e. its partial multiplicity matrix is a constant; we shall denote such a matrix by .
(iii) By definition, the matrices of companion AFalgebras have the same characteristic polynomial ; the matrices itself are not pairwise similar and, therefore, the AFalgebras are not pairwise isomorphic. The total number h of such matrices is equal to the class number of the endomorphism ring of pseudolattice , [14].
Remark 5. Notice that there are {X_{1} ,…, X_{h}} pairwise nonisomorphic Riemann surfaces endowed with a holomorphic map X_{i}→ ε_{i} where, {ε_{1} ,…,_{h}} are pairwise nonisomorphic elliptic curves corresponding to elements of the group Cl(Rf). Thus the companion AFalgebras can be viewed as coordinate rings of {X_{1} ,…, X_{h}} the latter means that discern nonisomorphic Riemann surfaces and represents the moduli space of .
(iv) The polynomial p(x) is minimal and splits in the totally real field φ. Indeed, matrices generate the Hecke algebra on ; thus each is selfadjoint and, therefore, all eigenvalues are real of multiplicity one; since is integer, all roots of characteristic polynomial p(x) of belong to the field φ.
(v) Let . It is easy to see that λi are algebraic units of the field φ because ; note that numbers are algebraically conjugate. Since , the numbers λ_{i} can be taken for generators of the field φ, i.e. .
(vi) Finally, let us establish an explicit formula for the isomorphism
(12)
Since is an automorphism group of the field φ preserving , it will suffice to define the action of an element on the generators λ_{i} of φ. Let be the set of all elements of the group . For an element define an index function α by the formula . Then the action * of an element on the generators λ_{i} of the field φ is given by the formula:
(13)
It is easy to verify that formula (13) gives an isomorphism , which is independ of the choice of {a_{i}} and {λ_{i}}. This argument completes the proof of lemma 3.
Remark 6. The class field theory says that f = f^{m}, i.e. the extensions of elds k and 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 of k and k, which are relatively prime to the principal ideals ( f ) and (f), respectively. Since one gets an isomorphism If ≅ If, i.e. f = f^{m} for some positive integer m.
Corollary 3. The Hilbert class field of real quadratic field modulo conductor f ≥ 1 is isomorphic to the field generated by the Fourier coefficients of the Hecke eigenform .
Proof. As in the classical case of imaginary quadratic fields, notice that ; therefore corollary 3 is an implication of lemma 3 and isomorphism
Theorem 1 follows from corollary 3.
Along with the method of Stark’s units [19], 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 Table 1.
Example 1. Let D = 15. The class number of quadratic field is known to be 2; such a number for quadratic field is also equal to 2. Thus
(14)
and isomorphism (2) is trivially satised for each power m, i.e. one obtains an unramied extension. By theorem 1, the Hilbert class field of k is generated by the Fourier coefficients of the Hecke eigenform Ï(z) ∈ S_{2} (Γ_{1}(15)). Using the computer programme 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 φ = (c(2)) is a root of the biquadratic equation [(x^{2})^{2}+1]^{2} − 15 = 0; it is easy to see that . One concludes, that the field is the Hilbert class field of quadratic field .
Example 2. Let D = 14. It is known, that for the quadratic field we have while for the quadratic field it holds . However, for the ramified extensions one obtains the following isomorphism:
(15)
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 × 4)). Using the SAGE, one finds that the characteristic polynomial of the Hecke operator T_{p=3} on S_{2}(Γ_{1}(2 × 4)) has an irreducible factor p(x)=x^{4}+3x^{2}+ 9. Thus the Fourier coecient 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 Kφ = (c(3)) is a root of the polynomial equation (x^{4} + 3x^{2}+ 9)^{2} − 4 × 14 = 0. The biquadratic equation has discriminant and one finds a generator of φ to be .Thus the field is the Hilbert class over modulo conductor f = 8. Clearly, the extension is ramified over the prime ideal p = (2).
Remark 7. Table 1 above lists quadratic fields for some squarefree discriminants 2 ≤ D ≤ 101. The conductors f and f satisfying equation (2) were calculated using tables for the class number of nonmaximal 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 .
D  f  f  Hilbert class field of modulo conductor f 


2  1  trivial  1  
3  1  trivial  1  
7  1  trivial  1  
11  1  trivial  1  
14  2  8  
15  1  1  
19  1  trivial  1  
21  2  8  
35  1  1  
43  1  trivial  1  
51  1  1  
58  1  1  
67  1  trivial  1  
82  1  1  
91  1  1 
Table 1: Squarefree discriminants 2 ≤ D ≤ 101.
I thank Yu. I. Manin for helpful correspondence.
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