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Real Multiplication Revisited | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Real Multiplication Revisited

Nikolaev IV*

Department of Mathematical Sciences, The Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada

Corresponding Author:
Nikolaev IV
Professor, Department of Mathematical Sciences
The Fields Institute for Research in Mathematical Sciences
222 College Street, Toronto, Ontario, Canada
Tel: 416-348-9710
E-mail: ig[email protected]

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:10.4172/1736-4337.S2-007

Copyright: © 2016 Nikolaev IV. This is an open-access 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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Abstract

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.

Keywords

Class field; Real multiplication

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 [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 L-functions [3]. Based on an analogy with complex multiplication, Manin suggested to use the so-called “pseudo-latticesEquation in Equation having non-trivial real multiplications to produce abelian extensions of real quadratic fields [4]. Similar to the case of complex multiplication, the endomorphism ring Equation of pseudo-lattice Equationis an order in the real quadratic field Equation, where Ok is the ring of integers of k and f is the conductor of Equation 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 and a symmetry of complex and real multiplication. To give an idea, let

Equation(1)

be a congruence subgroup of level N ≥ 1 and Equation be the Lobachevsky half-plane; let Equation be the corresponding modular curve and S21(N)) the space of all cusp forms on Γ1 (N) of weight 2. Let Equation be elliptic curve with complex multiplication by an order Equation in the field Equation[5]. Denote by Equation the Hilbert class field of k modulo conductor f ≥ 1 and let N = fD; let Equation be the Jacobian of modular curve Equation. There exists an abelian subvariety Equation, such that its points of finite order generate Equationab(k), [2,6,7], Section 8. The Equationab(k) is a CM-field, i.e. a totally imaginary quadratic extension of the totally real field Equationφ generated by the Fourier coefficients of the Hecke eigenform Equation[2]. In particular, there exists a holomorphic map Equation, where Equationis a Riemann surface such that Equation; we refer to the above as a modularity of complex multiplication.

Recall that (twisted homogeneous) coordinate ring of an elliptic curve ε (Equation ) is isomorphic to a Sklyanin algebra, [8]; the norm-closure of a self-adjoint representation of the Sklyanin algebra by the linear operators on a Hilbert space Equation is isomorphic to a noncommutative torus Equationθ, [9] for the definition.

Whenever elliptic curve Equationhas complex multiplication, the noncommutative torus Equationθ has real multiplication by an order Equationin the field Equation; moreover, it is known that f = fm for the minimal power m satisfying an isomorphism:

Equation(2)

where Cl(Rf) and Cl(Equationf) are the ideal class groups of orders Rf and Equationf, respectively. We shall refer to (2) as a symmetry of complex and real multiplication. The noncommutative torus with real multiplication by Equationf will be denoted by Equation.

Remark 1: The isomorphism (2) can be calculated using the wellknown formula for the class number of a non-maximal order Equation + fOK of a quadratic field Equation

Equation(3)

where Equationis the class number of the maximal order OK, ef is the index of the group of units of Equation+ fOK in the group of units of OK, p is a prime number and Equation is the Legendre symbol [10,11].

The (twisted homogeneous) coordinate ring of the Riemann surface Equation is an AF-algebra Equation linked to a holomorphic differential Equation on Equation, see Section 2.2, Definition 1 and Remark 5 for the details; the Grothendieck semigroup Equationis a pseudo-lattice Equation in the number field Equationφ, where n equals the genus of Equation. Moreover, a holomorphic map Equation− induces the C*-algebra homomorphism Equationbetween the corresponding coordinate rings, so that the following diagram commutes:

Equation

But Equationis a pseudo-lattice Equation + Equationθ in the field Equation, such that End (Equation + Equationθ) ≅ Equationf in other words, one can use the above diagram to control the arithmetic of the field Equationφ 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 Equation modulo conductor fm is an extension of k by the Fourier coefficients of the Hecke eigenform Equation, 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 Equationits Hilbert class field is isomorphic to Equationand for Equation such a field modulo conductor f = 8 is isomorphic to Equation see section 4 for more examples.

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 studies of Murphy [12] and their K-theory 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 non-commutative 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 Equationθ is a universal C*-algebra generated by the unitary operators u and v acting on a Hilbert space Equation and satisfying the commutation relation vu = e2π iθ uv , where θ is a real number. The C*-algebra Equationθ is said to be stably isomorphic (Morita equivalent) to Equation, whenever Equationwhere Equation is the C*- algebra of all compact operators on Equation; the Equationθ is stably isomorphic to Equation if and only if

Equation for some matrix Equation(4)

The K-theory of Equationθ is two-periodic and Equationso that the Grothendieck semigroup Equationcorresponds to positive reals of the pseudo-lattice Equation . The Equationθ is said to have real multiplication, if θ is a quadratic irrationality, i.e. irrational root of a quadratic polynomial in Equation[x]. The real multiplication says that the endomorphism ring of pseudo-lattice Equation + Equationθ exceeds the ring Equation corresponding to multiplication by m endomorphisms; similar to complex multiplication, it means that the endomorphism ring is isomorphic to an order Equation of conductor f ≥ 1 in the real quadratic field k = Equation(θ ), hence the name. If D > 0 is the discriminant of k, then by Equationwe denote torus Equationθ with real multiplication by the order Equation.

The Sklyanin algebra Sα, β, γ (Equation ) is a free C-algebra on four generators and six relations:

Equation(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 EquationP3 given by the intersection of two quadric surfaces of the form

Equation

Being such a ring means that the algebra Sα, β, γ satisfies an isomorphism

Equation(6)

where Coh is the category of quasi-coherent sheaves on εα, β, γ ( Equation) , Mod the category of graded left modules over the graded ring εα, β, γ ( Equation) and Tors the full sub-category of Mod consisting of the torsion modules, [8].

If one sets Equation , then there exists a self-adjoint representation of the Sklyanin *-algebra Sα,1,-1( Equation) by linear operators on a Hilbert space Equation, such that its norm-closure is isomorphic to Equationθ ; namely, Equationwhere Equation is a dense sub-algebra of Equationθ and Iμ is an ideal generated by the “scaled unit” relations Equation, where μ > 0 is a constant. Thus the algebra Equationθ 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 (Equation) has complex multiplication by an order Equationin a quadratic field Equation , then Equationθ has real multiplication by an order Equation in the quadratic field Equation, where f is the smallest integer satisfying an isomorphism Equation, [16]; the isomorphism is a necessary and sufficient condition for Equationto discern non-isomorphic elliptic curves Equationhaving the same endomorphism ring Rf . For the constraint f =fm, see remark 6.

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 Mn, where Mn is the C*-algebra of the n × n matrices with entries in Equation. Here the index n=(n1,...,nk) represents the semi-simple matrix algebra Equation . The ascending sequence mentioned above can be written as Equation,here Mi are the finite dimensional C*-algebras and Mi the homomorphisms between such algebras. The homomorphisms Mi can be arranged into a graph as follows. Let Equationand Equationbe the semi-simple C*- algebras and Equationthe homomorphism. One has two sets of vertices Equationand Equationjoined by brs edges whenever the summand Equation contains brs copies of the summand is Equationunder the embedding ∅i. As i varies, one obtains an infinite graph called the Bratteli diagram of the AF-algebra. The matrix B= (brs ) is known as a partial multiplicity matrix; an infinite sequence of Bi defines a unique AF-algebra. An AF-algebra is called stationary if Bi = Const = B, [14], 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:

Equation(7)

Let Equationbe the upper half-plane and let Γ1(N) act on Equation by the linear fractional transformations; consider an orbifold Equation / Γ1(N) . To compactify the orbifold at the cusps, one adds a boundary to Equation, so that Equation and the compact Riemann surface Equationis called a modular curve. The meromorphic functions (z) on Equation that vanish at the cusps and such that

Equation(8)

are called cusp forms of weight two; the (complex linear) space of such forms will be denoted by Equation. The formula Equation defines an isomorphism Equation, where Equation is the space of all holomorphic differentials on the Riemann surface X1(N). Note that Equation, where g =g(N) is the genus of the surface X1(N) . A Hecke operator, Tn, acts on Equation by the formula Equation, where Equation and Equationis the Fourier series of the cusp form Ï at Equation . Further, Tn is a self-adjoint linear operator on the vector space Equation endowed with the Petersson inner product; the algebra Equationis a commutative algebra. Any cusp form Equation that is an eigenvector for one (and hence all) of Tn, is referred to as a Hecke eigenform. The Fourier coefficients c(m) of Ï are algebraic integers, and we denote by Kφ = 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 Equation, where g is the genus of the surface X1(N) [5], Proposition 6.6.4. Any embedding Equation conjugates Ï by acting on its coefficients; we write the corresponding Hecke eigenform Equation and call Ïσ a conjugate of the Hecke eigenform Ï.

Let Equationbe 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 Equationagainst a basis γi in the (relative) homology group Equation, where Equation is the set of zeros of the form R(w). Assume Equation and consider the vector Equationwith Equation. The Jacobi-Perron continued fraction of θ is given by the formula:

Equation(9)

where Equationis a vector of non-negative integers, I is the unit matrix and Equation= (0,...,0,1)T [17]. By EquationÏ we shall understand the AF-algebra given the Bratteli diagram with partial multiplicity matrices Bi. If Equation is a Hecke eigenform, then the corresponding AF-algebra EquationÏ is stationary with the partial multiplicity matrices Bi = Const = B; moreover, each conjugate eigenform Ïσ defines a companion AF-algebra Equation . It is known that Equation, where Equationφ is an algebraic number field generated by the Fourier coefficients of Ï, [18].

Proof of Theorem 1

Definition 1. Let Equation be an abelian variety associated to the Hecke eigenform Equation[15], Definition 6.6.3. By Equation we shall understand the Riemann surface of genus g, such that

Equation(10)

By Equation we denote the image of the Hecke eigenform Equation under the holomorphic map Equation.

Remark 3. The surface Equationis correctly dened. Indeed, since the abelian variety Aφ is the product of g copies of an elliptic curve with the complex multiplication, there exists a holomorphic map from Aφ to the elliptic curve. For a Riemann surface X of genus g covering the elliptic curve εCM by a holomorphic map (such a surface and a map always exist), one gets a period map X → Aφ by closing the arrows of a commutative diagram Aφ → εCM ← X. It is easy to see, that the Jacobian of X coincides with Aφ and we set Equation.

Lemma 1. Equation

Proof. By definition, abelian variety Aφ is the quotient of Equationn by a lattice of periods of the Hecke eigenform Equation and all its conjugates φσ(z) on the Riemann surface Equation. These periods are complex algebraic numbers generating the Hilbert class field Equation over imaginary quadratic field Equation modulo conductor f, [2,6,7], Section 8. The number of linearly independent periods is equal to the total number of the conjugate eigenforms φσ(z) , i.e. Equation. Since real dimension Equation , we conclude that Equation and, therefore, Equation. But Equationand one gets Equation. Lemma 1 follows.

Corollary 1. Equation

Proof. Because Equationis the Hilbert class field over k modulo conductor f, we must have

Equation(11)

where Gal is the Galois group of the extension Equationand Equation is the class group of ring Rf, [5]. But Equation and by lemma 1 we have Equation. In view of this and isomorphism (11), one gets Equation. Corollary 1 follows.

Lemma 2. Equation

Proof. It is known that Equation[15], Proposition 6.6.4. But abelian variety Equationand, therefore, Equation, hence the lemma.

Corollary 2. Equation

Proof. From lemma 2 and corollary 1 one gets deg Equation In view of this and equality (2), one gets the conclusion of corollary 2.

Lemma 3. (Basic lemma) Equation

Proof. Let us outline the proof. In view of lemma 2 and corollaries 1-2, we denote by h the single integer Equation. Since deg Equation, there exist Equation conjugate Hecke eigenforms Equation [15], Theorem 6.5.4; thus one gets h holomorphic forms Equationon the Riemann surface Equation. Let Equation be the corresponding stationary AF-algebras; the Equationare companion AFalgebras, see Section 1.2. Recall that the characteristic polynomial for the partial multiplicity matrices Equationof companion AF-algebras Equation is the same; it is a minimal polynomial of degree h and let Equation be the roots of such a polynomial, compare with studies of Effros [14], Corollary 6.3. Since Equation, the numbers λi are algebraic units of the field Equationφ. Moreover, λi are algebraically conjugate and can be taken for generators of the extension Equation ; since Equation there exists a natural action of group Equationon these generators. The action extends to automorphisms of the entire field Equationφ preserving Equation; thus one gets the Galois group of extension Equation and an isomorphism Equation. Let us pass to a step-by-step argument.

(i) Let Equationand let Equation be the Hecke eigenform. It is known that there exists Equation conjugate Hecke eigenforms, so that Ï (z) is one of them [15], Theorem 6.5.4. Let Equation be the corresponding forms on the Riemann surface Equation.

Remark 4. The forms Equationcan be taken for a basis in the space Equation we leave it to the reader to verify, that abelian variety Aφ is isomorphic to the quotient of Equation by the lattice of periods of holomorphic differentials Equation on Equation.

(ii) Let Equation be the AF-algebra corresponding to holomorphic differential Equationon Equation, see Section 2.2; the set Equation consists of the companion AF-algebras. It is known that each Equation is a stationary AF-algebra, i.e. its partial multiplicity matrix is a constant; we shall denote such a matrix by Equation.

(iii) By definition, the matrices Equationof companion AF-algebras Equation have the same characteristic polynomial Equation; the matrices Equation itself are not pairwise similar and, therefore, the AF-algebras Equation are not pairwise isomorphic. The total number h of such matrices is equal to the class number of the endomorphism ring of pseudo-lattice Equation, [14].

Remark 5. Notice that there are {X1 ,…, Xh} pairwise non-isomorphic Riemann surfaces Equationendowed with a holomorphic map Xi→ εi where, {ε1 ,…,h} are pairwise non-isomorphic elliptic curves Equation corresponding to elements of the group Cl(Rf). Thus the companion AF-algebras Equationcan be viewed as coordinate rings of {X1 ,…, Xh} the latter means that Equationdiscern non-isomorphic Riemann surfaces and Equationrepresents the moduli space of Equation.

(iv) The polynomial p(x) is minimal and splits in the totally real field Equationφ. Indeed, matrices Equation generate the Hecke algebra Equationon Equation; thus each Equation is self-adjoint and, therefore, all eigenvalues are real of multiplicity one; since Equationis integer, all roots of characteristic polynomial p(x) of Equation belong to the field Equationφ.

(v) Let Equation. It is easy to see that λi are algebraic units of the field Equationφ because Equation; note that numbers Equation are algebraically conjugate. Since Equation, the numbers λi can be taken for generators of the field Equationφ, i.e. Equation .

(vi) Finally, let us establish an explicit formula for the isomorphism

Equation(12)

Since Equationis an automorphism group of the field Equationφ preserving Equation, it will suffice to define the action of an element Equation on the generators λi of Equationφ. Let Equation be the set of all elements of the group Equation . For an element Equation define an index function α by the formula Equation. Then the action * of an element Equation on the generators λi of the field Equationφ is given by the formula:

Equation(13)

It is easy to verify that formula (13) gives an isomorphism Equation, which is independ of the choice of {ai} and {λi}. This argument completes the proof of lemma 3.

Remark 6. The class field theory says that f = fm, i.e. the extensions of elds k and Equation must ramify over the same set of prime ideals. Indeed, consider the commutative diagram below, where If and If are groups of all ideals of k and k, which are relatively prime to the principal ideals ( f ) and (f), respectively. Since Equation one gets an isomorphism If ≅ If, i.e. f = fm for some positive integer m.

Equation

Corollary 3. The Hilbert class field of real quadratic field Equation modulo conductor f ≥ 1 is isomorphic to the field Equation generated by the Fourier coefficients of the Hecke eigenform Equation.

Proof. As in the classical case of imaginary quadratic fields, notice that Equation ; therefore corollary 3 is an implication of lemma 3 and isomorphism Equation

Theorem 1 follows from corollary 3.

Examples

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 Equationis known to be 2; such a number for quadratic field Equationis also equal to 2. Thus

Equation(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) ∈ S21(15)). Using the computer programme SAGE created by William A. Stein, one finds an irreducible factor p(x) = x2 − 4x + 5 of the characteristic polynomial of the Hecke operator Tp=2 acting on the space S21(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 Equationφ = Equation(c(2)) is a root of the bi-quadratic equation [(x2)2+1]2 − 15 = 0; it is easy to see that Equation. One concludes, that the field Equationis the Hilbert class field of quadratic field Equation.

Example 2. Let D = 14. It is known, that for the quadratic field Equationwe have Equationwhile for the quadratic field Equationit holds Equation. However, for the ramified extensions one obtains the following isomorphism:

Equation(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) ∈ S21(2 × 4)). Using the SAGE, one finds that the characteristic polynomial of the Hecke operator Tp=3 on S21(2 × 4)) has an irreducible factor p(x)=x4+3x2+ 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 = Equation(c(3)) is a root of the polynomial equation (x4 + 3x2+ 9)2 − 4 × 14 = 0. The bi-quadratic equation Equation has discriminant Equationand one finds a generator of Equationφ to be Equation.Thus the field Equationis the Hilbert class over Equation 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 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 Equation.

D f Equation f Hilbert class field of Equation
modulo conductor f
2 1 trivial 1 Equation
3 1 trivial 1 Equation
7 1 trivial 1 Equation
11 1 trivial 1 Equation
14 2 Equation 8 Equation
15 1 Equation 1 Equation
19 1 trivial 1 Equation
21 2 Equation 8 Equation
35 1 Equation 1 Equation
43 1 trivial 1 Equation
51 1 Equation 1 Equation
58 1 Equation 1 Equation
67 1 trivial 1 Equation
82 1 Equation 1 Equation
91 1 Equation 1 Equation

Table 1: Square-free discriminants 2 ≤ D ≤ 101.

Acknowledgements

I thank Yu. I. Manin for helpful correspondence.

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