Class fields from the fundamental Thompson series of level $N=o(g)$

CLASS FIELDS FROM THE FUNDAMENTAL THOMPSON SERIES OF LEVEL N = o(g)

So Young Choi and Ja Kyung Koo

Abstract. Thompson series is a Hauptmodul for a genus zero group which lies between

(N ) and its normalizer in P SL

(R) ([1]). We construct explicit ring class elds over an imaginary quadratic eld

(α) (Theorem 4), which would be an extension of [3], Theorem 3.7.5 (2) by using the Shimura theory and the standard results of complex multiplica- tion. Also we construct various class elds over

+ ζ

), and over a eld

). Furthermore, we nd an explicit formula for the conjugates of T

(α) to calculate its minimal polynomial where α(∈ H) is the quotient of a basis of an integral ideal in K.

1. Introduction

The main purpose of this paper is to study the class elds generated by singular values of Thompson series T

at imaginary quadratic argu- ments in the complex upper half plane H, over K, CM-eld K(ζ

+ ζ

) and K(ζ

). To this end, we recall the classical results on singular mod- uli of the elliptic modular function j for SL

( Z) evaluated at imaginary quadratic arguments ([2], [5], [9], [11]). Let K be an imaginary quadratic eld over Q of discriminant d

and O be an order of K of conductor f , discriminant f

d

and class number h( O). Let α ∈ H ∩ O be an imaginary quadratic argument. Then a singular modulus j(α) generates the ring class eld L of O (the Hilbert class eld if O is the maximal order of K).

Helling showed in [6] that the group

(N )

generated by {

(N ), (

N 0

) } has a genus zero exactly for N = 1 ∼ 21, 23 ∼ 27, 29, 31, 32, 35, 36, 39, 41, 47, 49, 50, 59, 71. Moreover, for all such N but 49 and 50,

Received August 6, 2003.

2000 Mathematics Subject Classication: 11F11, 11R04, 11R37.

Key words and phrases: modular functions, Thompson series, class elds.

This work was Supported by KOSEF Research Grant 98-0701-01-01-3.

0

(N )

has a fundamental Thompson series T

(α) corresponding to itself, that is, T

(α) is dened by an element of order N of the Monster group ([4], Table 2). Throughout this paper, we denote α to be a root in H of a quadratic equation az

+ bz + c = 0 with (a, b, c) = 1 and a > 0, and K as an imaginary quadratic eld Q(α). Chen-Yui ([3], Theorem 3.7.5 (2)) showed by using the Shimura reciprocity law that when (a, N ) = 1, T

(α) generates a ring class eld over K.

In §2, we rst construct some sort of class elds with

(N )

by means of Shimura’s ideas.

Next in §3, we generate a ring class eld K(T

(α)) over K under the condition (a, b, N ) ̸= N or (

, N ) ̸= 1. This further generalizes Chen-Yui’s result under the assumption (a, N ) = 1.

On the other hand, it has been known that there exists a funda- mental Thompson series T

of level N = o(g) exactly when N = 1 ∼ 36, 38, 39, 41, 42, 44 ∼ 47, 49, 50, 51, 54 ∼ 56, 59, 60, 62, 66, 69 ∼ 71, 78, 87, 92, 94, 95, 105, 110, 119 ([4], table 2) where o(g) is the order of el- ement g of the Monster group. In §4, we will construct, from such T

, ζ

+ ζ

or ζ

, not only various class elds over K which are nei- ther ray class elds of conductor N nor ring class elds of order O with discriminant N

d

, but also ray class elds of conductor N by applying Chen-Yui’s method. We also demonstrate what sort of class eld T

(α) generates over a CM-eld K(ζ

+ ζ

) and over a eld K(ζ

).

In §5, we explore an explicit formula for the conjugates of T

(α) to calculate its minimal polynomial.

Throughout the article we adopt the following notations:

• = SL

( Z)

•

(N ) = { (

c d

) ∈ SL

( Z)|c ≡ 0 mod N}

•

(N )

= ⟨

{

(N ), (

N 0

) } ⟩

• Z

the ring of p-adic integers

• Q

the eld of p-adic numbers

• H upper half complex plane

• ζ

= e

• i = √

−1

• T

a fundamental Thompson series for a genus zero group

(N )

• T

a fundamental Thompson series of level N = o(g)

• x ∼

SL2(Z)

y means that x = γy for some γ ∈ SL

( Z)

• G

the stabilizer of α for a group G

2. Class elds obtained by applying Shimura’s method Let be a Fuchsian group of the rst kind. Then X() = \H

is a compact Riemann surface. Hence there exists a projective nonsingular algebraic curve V , dened over C, biregularly isomorphic to \H

. We specify a -invariant holomorphic map φ of H

to V which gives a biregular isomorphism of \H

to V . In that situation, we call (V , φ ) a model of \H

. Through this article we always assume that the genus of \H

is zero. Then it’s function eld K(X()) is equal to C(J

) for some J

∈ K(X()) and the pair ( P

( C), J

) is a model of \H

([7], Lemma 14).

Let G

be the adelization of an algebraic group G = GL

dened over Q.

Put

G

= GL

( Q

) (p : rational prime), G

= GL

( R),

G

= {x ∈ G

| det(x) > 0}, G

= {x ∈ GL

( Q)| det(x) > 0}.

We dene the topology of G

by taking U = ∏

p

GL

(Z

) × G

to be an open subgroup of G

. Let K be an imaginary quadratic eld and ξ

be an embedding of K into M

( Q). We call ξ

normalized if it is dened by a (

) = ξ

(a) (

) for a ∈ K where z is the xed point of ξ

(K

) ( ⊂ G

) in H. Observe that the embedding ξ

denes a continuous homomorphism of K

into G

, which we denote again by ξ

. Here G

is the group G

G

with G

the non-archimedean part of G

and K

is the idele group of K. Let Z be the set of open subgroups S of G

containing Q

G

such that S/Q

G

is compact. For S ∈ Z, we see that det(S) is open in Q

. Therefore the subgroup Q

det(S) of Q

corresponds to a nite abelian extension of Q, which we write k

. Put

= S ∩ G

for S ∈ Z. As is well known ([11], Proposition 6.27),

/Q

is a Fuchsian group of the rst kind commensurable with P SL

( Z). Let

U

= {x = (x

) ∈ U|x

∈ U

for all nite p } and U

= U

∪U

( N ),

where U

= { (

c d

) ∈ GL

( Z

) |c ≡ 0 mod NZ

}, ( N ) = (x

) ∈ G

and x

= (

N 0

) .

Then we have

Lemma 1. (i) Q

U

∈ Z, (ii) k

= Q,

(iii)

= Q

(N )

if S = Q

U

.

Proof. Since Q

U

= Q

U

∪ Q

U

( N ) and Q

U

is an open subgroup in G

, Q

U

is also an open subgroup in G

. Observing that Q

U

/ Q

G

is compact, we obtain Q

U

∈ Z. As for (ii), we see that Q corresponds to the norm group Q

Q

with Q

= R

∏

p

Z

and det ( U

)

= N det(U

). But det(U

) = Q

and hence by the class eld theory k

= Q. Indeed, clearly det(U

) is contained in Q

. Conversely, for any element (α

) ∈ Q

, take y

= (

0 αp

) , then (y

) ∈ U

and det(y

) = (det y

) = (α

). Lastly, we readily get that

S

= Q

U

∩ G

= Q

(

U

∩ G

)

= Q

(N )

Theorem 2. Let K be an imaginary quadratic eld. For xed z ∈ K ∩ H, let ξ

be the normalized embedding. Then T

(z) belongs to the maximal abelian extension K

of K and K(T

(z)) is the class eld of K corresponding to the subgroup K

· ξ

( Q

U

) of K

.

Proof. We have k

= Q and

= Q

(N )

by Lemma 1. Since T

gives a model of X (

(N )

), the assertion follows from [11] Proposition 6.31 and Proposition 6.33.

3. Ring class elds generated by singular values of T

In this section, we obtain Theorem 4 which would be an extension of [3], 3.7.5 Theorem(2) by using Shimura’s method and the standard results of complex multiplication. To this end, we need the following fact.

Theorem 3. Let F

be the eld of modular functions of level N rational over Q(e

), and let K be an imaginary quadratic eld. Let O

be the maximal order of K and a be an O

-ideal such that a = [z

, z

] and α = z

/z

∈ H. Then the eld KF

(α) generated over K by all values f (α) with f ∈ F

and f dened at α, is the ray class eld K

over K with modulus N .

Proof. [9], Ch. 10, Corollary of Theorem 2.

By class eld theory, the reciprocity map induces an isomorphism

[ ·, K] : K

/K

U

−→ Gal(K

/K)

where U

is the subgroup of K

given by

U

= {s ∈ K

|s

∈ O

and s

≡ 1 mod NO

for all nite primes p}.

For a subeld L of K

, let

L/K

: I

(N ) −→ Gal(L/K)

denote the Artin map. Then Ker(

) = P

(N ), where I

(N ) is the ideal group generated by all fractional ideals in K prime to N and P

(N ) denotes the subgroup of I

(N ) generated by the principal ideals β O

with β ∈ O

and β ≡ 1 mod NO

. Of course, O

is the ring of integers in K.

Theorem 4. Let T

be a fundamental Thompson series for a genus zero group

(N )

. Let α be a root in H of a quadradic equation az

+ bz + c = 0 such that a > 0, (a, b, c) = 1, and b

− 4ac = m

d

<

0 (m > 0). Let K = Q(α) and O (= Z[aα]) be an order in K of discrimi- nant m

d

. Assume that (a, b, N ) ̸= N or (

, N ) ̸= 1. Then K(T

(α)) is a ring class eld of an imaginary quadratic order O

of discriminant f

d

where f = mN/(a, N ) and d

is the discriminant of K.

Proof. First of all, we describe the action of arbitrary prime deal ℘ of O

on T

(α) under the Artin map

N(α))/K

which is guaranteed by Theorem 4. Take a rational prime p which does not divide 2abcmN and splits in K. Then p O

= ℘℘

, where ℘ = (p,

√d_K

2

), r

= m

d

mod p and r ∈ Z with b − r and b + r both even (since aα is a root of the polynomial z

+ bz + ac = 0 and p splits in K, there exists r ∈ Z such that r

= m

d

mod p, z

+ bz + ac = (z +

)(z +

b+r

2

) mod p and b − r, b + r ∈ 2Z, and aα +

=

+

=

±r+m√ dK

2

). We dene v =

( ∈ Z) and hence ord

v = 0.

We now take an idele s corresponding to an ideal ℘ to be s = (1, · · · , −r + m √

d

2 , 1, · · · ),

where we put 1 at all places except the place corresponding to p. Under the embedding ξ

: K

→ G

, s is sent to

ξ

(s) = (I

, · · · , (

2 −c

a ^b−r₂

) , · · · )

because

= aα +

and

α =

α − c. This implies

ξ

(s

) = ( I

, · · · , 1 pv

(

2 c

−a ^−r−b₂

)

, · · · ).

Let p

, k, p

, l ∈ Z be such that ak + pp

= 1 and N l + pp

= 1.

Then

ξ

(s

) = (

(

0 1

)

, · · · , 1 v

(

2 v

−a 0a

) (

0 1

) , · · · )

· (

p

(r+b)kN l 2p

0 1

)

= u · g.

Since 1 − kNla = p(akp

+ p

N l + pp

p

), u = (u

) ∈ U and g ∈ GL

(Q). Notice that u

≡ (

0 1

) mod N M

(Z

) for every nite prime q and dene

U f

= (

0 1

) and g A

= (

0 p⁻¹

) U f

∈ SL

( Z/NZ).

Lift g A

to a matrix A

in

(N ). By the Shimura reciprocity law, we get

T

(α)

= T

(α) = T

(α)

= T

(A

gα) = T

(gα)

= T

(

(

0 p

) α).

Let A be a matrix (

1 (r+b)kN l/2

0 p

)

and suppose that T

(α)

= T

(α). Then

A ∈

(N )

GL

( Q)

. If A is contained in

(N ) (

N 0

) GL

( Q)

, then A = γ (

N 0

) (

y

z wN

) for some x, y, z, w ∈ Z and γ ∈

(N ) with x

+

= (zα + w)α. We also see that

=

=

:= λ ∈ Z\{0} and (λ,N)=1 ( because if λ = 0, then w = ± √

p/N / ∈ Z. This implies that λ is not zero. We now set λ =

with , △ ∈ Z\{0}, (, △) = 1, then △|(a, b, c) = 1 and hence λ ∈ Z. Also, (N, p) = 1 and xw − yz=(Nw − bλ)w + cλz = p imply (λ, N ) = 1). Here, the fact that z=

is an integer means that N divides a. Moreover, since N divides x and bλ = N w − x, N divides b and hence (

, N ) = 1. These give a contradiction to our assumption (a, b, N ) ̸= N or (

, N ) ̸= 1. Therefore, A ∈

(N )M

( Z)

and we can take γ ∈

(N ) and d, λ ∈ Z such that

A = (

1 (r+b)kN l/2

0 p

)

= γ (

aλ d

)

and (d − bλ)α − cλ = α(aλ + d). Observing p = acλ

− bdλ + d

=

α

(d − aαλ − bλ)=α

α

where α

:= d + aαλ ∈ O(= Z[aα]) and α

∈ O

is the complex conjugate of α

, we obtain p O

=(α

)(α

) =℘℘

. That is,

℘ is a principal ideal generated by an element α

(or α

) in O .

On the other hand, for each a ∈ I

(N ), we can take a prime ideal

℘ ∈ I

(2abcmN ) and a principal ideal (x) ∈ P

(N ) with a = (x)℘.

Indeed, if an ideal a in K is prime to N , then a = a

(β) for some a

∈ I

(2abcmN ) and (β) ∈ P

(N ) ([10], Corollary 3.16 and 3.18), and the ideal a

just given can be factored into (x)℘ for some prime ideal

℘ ∈ I

(2abcmN ) and (x) ∈ P

(2abcmN ) ([8] VIII §4 ,Corollary of Theorem 8). Consequently, ker(

N(α))/K

) is contained in a subgroup P

(N )(P ( O)∩I

(N )) of I

(N ), where P ( O) is the group of principal O-ideals. ( Of course, we consider elements in P (O) as O

-ideals.)

Let (β) be a principal ideal of O relatively prime to N and write β = naα + l ∈ Zaα + Z(= O). We see from Lemma 5 below that the action of (β) on T

(α) is represented by a matrix A

∈ SL

( Z) whose image in SL

( Z/NZ) is equal to (

−bn+l −cn anN (β)⁻¹ lN (β)⁻¹

)

. Thus, we derive that (β) xes T

(α) if and only if A

∈

(N )

GL

(Q)

. First, suppose that N does not divide a. By similar arguments as we used to show that A ∈

(N )

GL

( Q)

implies A ∈

(N )M

( Z)

, we can verify that (β) xes T

(α) if and only if A

∈

(N )SL

( Z)

. But we know that SL

( Z)

is trivial unless α is SL

( Z)-equivalent to e

with r ∈ {3, 4}.

Assuming SL

( Z)

= {±1}, we see that A

∈ ±

(N ) if and only if N |an. Therefore the principal ideals in O which x T

(α) are of the form (β) with disc(β) dividing (mN/(a, N ))

d

. But these principal ideals are all in P ( O

). The cases α ∼

SL2(Z)

e

with r=3 or 4 can be treated similarly. These prove our assertion in the case that N does not divide a. Now suppose that N divides a. Then A

xes T

(α) and hence T

(α) generates the ring class eld of O

.

Lemma 5. Let f be a modular function of level N with rational Fourier coecients and (β) a principal ideal of O relatively prime to N . Write β = n(aα) + l ∈ Z(aα) + Z(= O). And let A

be a matrix in SL

( Z) whose image in SL

( Z/NZ) is equal to (

−bn+l −cn anN (β)⁻¹ lN (β)⁻¹

) . Then the action of (β) on f (α) is given by

f (α)

= f (A

· α) where [ · , K

/K] is the Artin map.

Proof. Let (β) correspond to an idele s = (1, . . . ; β, . . . ) ∈ K

where

we put 1 on each place dividing N and β on the other places. Under the

embedding ξ

: K

→ G

, s is sent to ξ

(s) = (

I

, (

an l

) , . . . )

because β · α = (naα + l)α = (−bn + l)α − cn and β · 1 = na · α + l. Then ξ

(s

) =

( I

, 1

N (β)

(

−an −bn+l

) , . . . )

= ((

an l

) , I

, . . . )

· 1 N (β)

(

−an −bn+l

)

= u · g where u = ((

an l

) , I

, . . . )

and g =

(

−an −bn+l

) ∈ G

. Since det (

an l

) = N (β) is relatively prime to N , u belongs to U . Write u = (u

). Note that u

is congruent to (

an l

) modulo N M

(Z

) for any nite prime p. Put f U

= (

an l

) ∈ GL

(Z/NZ) and A

=

(

0 det(gUN)⁻¹

) (

an l

) =

(

) ∈ SL

( Z/NZ).

We now lift A

to a matrix A

in SL

(Z). Then we derive by Shimura reciprocity law

f (α)

= f

(α) = f

(α)

= f ( A

· gα) since f has rational Fourier coecients

= f (A

α) since g = ξ

(β

).

4. Class elds over an imaginary quadratic eld K, CM- elds K(ζ

+ ζ

) and K(ζ

)

Theorem 6. Let T

be a fundamental Thompson series of level N = o(g). Let α be a root in H of a quadratic equation az

+ bz + c = 0 such that a > 0, (a, b, c) = 1, and b

− 4ac = d

< 0. Let K = Q(α) and O(N) be an order Z + NO

in K of discriminant N

d

and put K

be a CM-eld K(ζ

+ ζ

). Under the assumption (a, N ) = 1 the following assertions hold.

(1) K

(T

(α)) is a class eld over K with

Gal(K

(T

(α))/K) ∼ = I

(N )/ e P

(N ) and

[K

(T

(α)) : K] = h( O(N))ψ(N)

2[ e P

(N ) : P

(N )]

where e P

(N ) is a subgroup of I

(N ) generated by all principal ideals β O

such that β = naα + l ∈ Zaα + Z(= O

), N |n and l

≡ ±1modN, and ψ(1) = ψ(2) = 2 and ψ is the Euler φ-function for N ≥ 3.

(2) T

(α) generates a class eld K

(T

(α)) over K

with Gal(K

(T

(α))/K

) ∼ = I

(N )/N

K^′/K

( e P

(N )) and

[K

(T

(α)) : K

] = h( O(N)) [ e P

(N ) : P

(N )]

where h( O(N)) is the class number of O(N) and I

(N ) is the ideal group generated by all fractional ideals in K

prime to N .

Proof. In the proof of Theorem 4, take m = 1, give a condition (N, a) = 1 and replace T

(α) by T

(α). Then the arguments from the beginning to the step of letting a matrix A = (

1 ^{(r+b)kN l}₂

0 p

)

in M

( Z) are exactly the same as those in Theorem 4. We assume that T

(α)

= T

(α). Then A ∈

(N )M

( Z)

. Indeed, if A ∈ W

GL

( Q)

, then A

= (

CN De

) (

)

for Ae

−BcN = e, (2, △) = 1, e|△, △ ̸= 1 and A, B, C, D, e, λ, △, 2 ∈ Z. A matrix (

CN De

)

A

implies that a

is an integer, and hence e divides a. This is absurd. Here W

is a non- trivial Atkin-Leher involution. Therefore, likewise as in the proof of The- orem 4, ker(

) is contained in a subgroup P

(N )(P (O

) ∩ I

(N )) of I

(N ).

Let (β) be a principal ideal of O

relatively prime to N and write β = naα + l ∈ Zaα + Z(= O

). Since we see from Lemma 5 that the action of (β) on T

(α) is represented by a matrix A

∈ SL

( Z) whose image in SL

( Z/NZ) is equal to (

−bn+l −cn anN (β)⁻¹lN (β)⁻¹

)

, we get that (β) xes T

(α) if and only if A

∈

(N )SL

(Z)

by replacing A by A

in the above argument. This implies that (β) ∈ ker(

) if and only if A

∈

(N )SL

( Z)

and N (β) = ±1 mod N. But we know that SL

( Z)

is trivial unless α is SL

( Z)-equivalent to e

with r ∈ {3, 4}.

Assuming SL

(Z)

= {±1}, we obtain that

(β) ∈ ker(

) if and only if N |an and N(β) ≡ ±1 mod N

if and only if N |n and l

≡ ±1 mod N

if and only if (β) ∈ e P

(N ).

The case where α is SL

( Z)-equivalent to e

with r ∈ {3, 4} can be treated similarly. These prove (1).

For any a ∈ I

(N ), [a, K

(T

(α))/K

] = [N

(a), K

(T

(α))/K]

and N

(I

(N )) ⊂ I

(N ) implies

Gal(K

(T

(α))/K

) ∼ = I

(N )/N

K^′/K

( e P

(N )).

Since P

(N ) is contained in e P

(N ), we obtain from the eld tower below

[K

(T

(α)) : K

] = ψ(N )h( O(N))

2[K

: K

(T

(α))][K

: K] = h( O(N)) [ e P

(N ) : P

(N )] .

K Q(ζ

+ ζ

) = K

K ∩ Q(ζ

+ ζ

) = Q

K ^ Q(ζ

+ ζ

)

  

Q Q Q Q Q

Q Q Q Q Q

   

Corollary 7. Under the same assumptions and notations as in Theorem 6, we have that if x ∈ Z and x

≡ ±1 mod N implies x ≡

±1 mod N, then

(1) K

(T

(α)) = K

is the ray class eld over K with modulus N and

[K

: K] = ψ(N )h(O(N))

2 ,

(2) T

(α) generates a class eld K

(T

(α)) over K

with Gal(K

(T

(α))/K

) ∼ = I

(N )/N

K^′/K

(P

(N )) and

[K

(T

(α)) : K

] = h( O(N)).

Proof. These are clear from Theorem 6

Examples 1. For N = 2, 3, 4, 6, 7, 9, 11, 14, 18, 19, 22, 23, 27, 31, 38, 46, 47, 54, 59, 62, 71, 94, x ∈ Z and x

≡ ±1 mod N imply x ≡ ±1 mod N so that K(ζ

+ ζ

, T

(α)) = K

is the ray class eld over K with modulus N . On the other hand, observing that

P e

(13)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 13|n and l ≡ 1 or 5 (mod 13)} ⟩ , [ e P

(13) : P

(13)] = 2, and

[K(ζ

+ ζ

, T

(α)) : K(ζ

+ ζ

)] = h( O(13)) 2

we see that K(ζ

+ ζ

, T

(α)) is neither a ring class eld of an order Z + 13O

nor a ray class eld K

with moduluds 13. We also have similar examples:

P e

(15)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 15|n and l ≡ 1 or 4 mod 15}⟩ , P e

(16)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 16|n and l ≡ 1 or 7 mod 16}⟩ , P e

(17)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 17|n and l ≡ 1 or 4 mod 17}⟩ , P e

(20)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 20|n and l ≡ 1 or 9 mod 20}⟩ , P e

(21)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 21|n and l ≡ 1 or 8 mod 21}⟩ , P e

(25)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 25|n and l ≡ 1 or 7 mod 25}⟩ , P e

(26)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 26|n and l ≡ 1 or 5 mod 26}⟩ , P e

(28)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 28|n and l ≡ 1 or 13 mod 28}⟩ , P e

(29)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 29|n and l ≡ 1 or 12 mod 29}⟩ ,

P e

(30)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 30|n and l ≡ 1 or 11 mod 30}⟩ , P e

(32)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 32|n and l ≡ 1 or 15 mod 32}⟩ , P e

(33)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 33|n and l ≡ 1 or 10 mod 33}⟩ , P e

(34)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 34|n and l ≡ 1 or 13 mod 34}⟩ , P e

(35)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 35|n and l ≡ 1 or 6 mod 35}⟩ , P e

(36)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 36|n and l ≡ 1 or 17 mod 36}⟩ , P e

(39)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 39|n and l ≡ 1 or 14 mod 39}⟩ , P e

(41)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 41|n and l ≡ 1 or 9 mod 41}⟩ , P e

(42)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 42|n and l ≡ 1 or 13 mod 42}⟩ , P e

(44)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 44|n and l ≡ 1 or 21 mod 44}⟩ , P e

(45)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 45|n and l ≡ 1 or 19 mod 45}⟩ , P e

(50)

= ⟨{βO

| β = naα + l ∈ Zaα + Z, 50|n and l ≡ 1 or 7 mod 50}⟩ , P e

(51)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 51|n and l ≡ 1 or 16 mod 51}⟩ , P e

(55)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 55|n and l ≡ 1 or 21 mod 55}⟩ , P e

(66)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 66|n and l ≡ 1 or 23 mod 66}⟩ ,

P e

(69)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 69|n and l ≡ 1 or 22 mod 69}⟩ , P e

(70)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 70|n and l ≡ 1 or 29 mod 70}⟩ , P e

(78)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 78|n and l ≡ 1 or 25 mod 78}⟩ , P e

(87)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 87|n and l ≡ 1 or 28 mod 87}⟩ , P e

(92)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 92|n and l ≡ 1 or 45 mod 92}⟩ , P e

(95)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 95|n and l ≡ 1 or 39 mod 95}⟩ , P e

(110)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 110|n and l ≡ 1 or 21 mod 110}⟩ , P e

(119)

= ⟨{βO

|β = naα + l ∈ Zaα + Z, 119|n and l ≡ 1 or 50 mod 119}⟩ , [ e P

(N ) : P

(N )] = 2, and [K(ζ

+ ζ

, T

(α)) : K(ζ

+ ζ

)] =

h(O(N))

2

for N = 13, 15, 16, 17, 20, 21, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 39, 41, 42, 44, 45, 50, 51, 55, 66, 69, 70, 78, 87, 92, 95, 110, 119. Meanwhile

P e

(56) =< {βO

|β = naα + l ∈ Zaα + Z, 56|n and l ≡ 1 or 13 or 15 or 27 (mod 56) }t >,

P e

(60) =< {βO

|β = naα + l ∈ Zaα + Z, 60|n and l ≡ 1 or 11 or 19 or 29 (mod 60) } >,

P e

(105) =< {βO

|β = naα + l ∈ Zaα + Z, 105|n and l ≡ 1 or 29 or 34 or 41 (mod 105) } >,

[ e P

(N ) : P

(N )] = 4, and [K(ζ

+ ζ

, T

(α)) : K(ζ

+ ζ

)] =

h(O(N))

4

for N = 56, 60, 105.

Theorem 8. Under the same assumptions and notations as in The- orem 6, we let E = K(ζ

) = Q(α, ζ

). Then the followings hold : (1) E(T

(α)) is a class eld over K with

Gal(E(T

(α))/K) ∼ = I

(N )/ e P e

(N ) and

[E(T

(α)) : K] = h(O(N))ψ(N) 2[ e P e

(N ) : P

(N )]

,

where e P e

(N ) is a subgroup of I

(N ) generated by all principal ideal β O

such that β = naα + l ∈ Zaα + Z(= O

), N |n and l

≡ 1 mod N.

(2) T

(α) generates a class eld E(T

(α)) over E with Gal(E(T

(α))/E) ∼ = I

(N )/N

( e P e

(N )) and

[E(T

(α)) : E] =

 



 

h(O(N))

[ ePeK(N ):PK,1(N )]

if K ⊂ Q(ζ

),

h(O(N))

2[ ePeK(N ):PK,1(N )]

otherwise

where I

(N ) is the ideal group generated by all fractional ideals in E prime to N .

Proof. We can show (1) by the same arguments as in Theorem 6.

Considering the eld tower bellow we have (2).

E = K(ζ

)

K ∩ Q(ζ

)

K ^ Q(ζ

)

  

Q Q Q Q Q

Q Q Q Q Q

   

. Q

Corollary 9. Under the same assumptions and notations as in The- orem 8, We derive that if x

≡ 1 mod N implies x ≡ ±1 mod N then

(1) E(T

(α)) = K

is a ray class eld over K with modulus N and [K

: K] = ψ(N )h(O(N))

2 ,

(2) T

(α) generates a class eld E(T

(α)) over E with Gal(E(T

(α))/E) ∼ = I

(N )/N

(P

(N )) and

[E(T

(α)) : E] =

{ h(O(N)) if K ⊂ Q(ζ

),

h(O(N))

2

otherwise.

Proof. They are clear from Theorem 8.

Examples 2. For N = 5, 10, 13, 17, 25, 26, 29, 41, 50, x

≡ 1 mod N implies x ≡ ±1 mod N and hence K(ζ

, T

(α))= K

is the ray class eld over K with modulus N .

On the other hand, taking α as a root in H of the equation z

+ 2 = 0 we have

K = Q( √

−2)(̸∈ Q(ζ

)), [Q( √

−2, ζ

)(T

( √

−2)) : Q( √

−2, ζ

)]

= h( O(N))

2 and Q( √

−2, ζ

, T

( √

−2))

= Q( √

−2, ζ

+ ζ

, T

( √

−2)) = Q( √

−2)

.

Taking α as a root H of the equation z

+ 3 = 0 we also get that K = Q( √

−3)(∈ Q(ζ

)), [ Q( √

−3, ζ

)(T

( √

−3)) : Q( √

−3, ζ

)]

= h( O(N)) and Q( √

−3, ζ

, T

( √

−3))

= Q( √

−3, ζ

+ ζ

, T

( √

−3)) = Q( √

−3)

.

Remark. Under the same assumptions and notations as in Theorem 6,8 let β be a root in H of the equation a

z

+ b

z + c

= 0 such that a

> 0, (a

, b

, c

) = 1 and b

−4a

c

= m

d

. In fact, if (a

, N ) = 1, then K

(T

(α))=K

(T

(β)) and E(T

(α))=E(T

(β)).

By using the same arguments as in Theorem 6, 8 we obtain the fol- lowing two theorems.

Theorem 10. Under the same assumptions and notations as in The-

orem 4, let K

be a CM-eld K(ζ

+ ζ

). Then the following assertions

hold.

(1) K

(T

(α)) is a class eld over K with

Gal(K

(T

(α))/K) ∼ = I

(N )/P

(N ) e P

(N, α) and

[K

(T

(α)) : K] = h( O(N))ψ(N)

2[P

(N ) e P

(N, α) : P

(N )] ,

where e P

(N, α) is a subgroup of I

(N ) generated by all principal ideal βO

such that β = naα + l ∈ Zaα + Z(= O), N|an and l

− bnl ≡

±1 mod N, and ψ(1) = ψ(2) = 2 and ψ is the Euler φ-function for N ≥ 3.

(2) T

(α) generates a class eld K

(T

(α)) over K

with Gal(K

(T

(α))/K

) ∼ = I

(N )/N

K^′/K

(P

(N ) e P

(N, α)) and

[K

(T

(α)) : K

] = h( O(N))

[P

(N ) e P

(N, α) : P

(N )]

where h( O(N)) is the class number of O(N)(= Z + NO

) and I

(N ) is the ideal group generated by all fractional ideals in K

prime to N .

Theorem 11. Under the same assumptions and notations as in The- orem 4, let E= K(ζ

). Then the following assertions hold.

(1) E(T

(α)) is a class eld over K with

Gal(E(T

(α))/K) ∼ = I

(N )/P

(N ) e P e

(N, α),

where e P e

(N, α) is a subgroup of I

(N ) generated by all principal ideal β O

such that β = naα + l ∈ Zaα + Z(= O), N|an and l

− bnl ≡ 1 mod N ,

(2) T

(α) generates a class eld E(T

(α)) over E with Gal(E(T

(α))/E) ∼ = I

(N )/N

(P

(N ) e P e

(N, α)),

where I

(N ) is the ideal group generated by all fractional ideals in E prime to N .

Examples 3. Let α and β be roots in H of two equations 2z

+2z+1 = 0 and 4z

+ 8z + 5 = 0, respectively. Then K = Q(α) = Q(β) = Q(i).

(1) Observing that e P

(36, α)=< {γO

| γ = n2α + l, 18|n and l ≡

1 or 17 mod 36 } >, we see that P

(36) ⊂ e P

(36, α) = I

(36) ∩

P

(18) = P

(18). Hence by class eld theory, K(ζ

+ ζ

, T

(α))

is the ray class eld over K with modulus 18 O

.

(2) Notice that the similar result as in (1) may not be obtained and that the class eld K

(T

(α)) over K depends on the choice of α. In- deed, we obtain that e P

(26, β)=< {γO

| γ = n4β + l, 13|n and l ≡ 1 or 5 mod 26} > is contained in e P

(26, α) =< {γO

|γ = n2α + l, 13 |n and l ≡ 1 or 5 mod 26} > and P

(26) is contained in e P

(26, α) by denition. Since P e

(26, α) is not equal to I

(26) ∩P

(13) (because 5 O

∈ e P

(26, α) but 5 O

̸∈ I

(26) ∩ P

(13) ), K(ζ

+ ζ

, T

(α)) is not the ray class eld over K with modulus 13 O

.

Suppose that γ = n2α + l, 13|n and l ≡ 1 or 5 mod 26. Then γ

≡ l

≡

±1 mod 26 and hence γ

O

P

(26) = P

(26). Now, if 26 |n and l ≡ 5 mod 26, then γ O

P

(26) = (5)P

(26). Meanwhile, if n = 26k +13 for some integer k and l ≡ 5 mod 26, then γO

P

(26) = (13 · 2α + 5) O

P

(26). Lastly, if n = 26k + 13 for some integer k and l ≡ 1 mod 26, then γ O

P

(26) = (13 · 2α + 1)O

P

(26). These imply [ e P

(26, α) : P

(26)] = 4. Moreover [P

(26) e P

(26, β) : P

(26)] = 2 because 4β = 2(2α) − 2.

(3) We see that e P e

(26, α)=< {γO

| γ = n2α + l, 13|n and l ≡ 1 mod 26 } > is contained in e P

(26, α). But e P e

(26, α) = I

(26) ∩ P

(13). Indeed, if γ = n2α + l, 13 |n and l ≡ 1 mod 13, then N(γ) ≡ l

mod 2 and hence l must be an odd number (i.e. l ≡ 1 mod 26) for γ O

to be contained in I

(26). Therefore by class eld theory, K(ζ

, T

(α)) is the ray class eld over K with modulus 13 O

and [ e P e

(26, α) : P

(26)] = 2.

Notice that P

(26) e P

(26, β), e P e

(26, α) and e P

(26, α) are congru- ence groups with the same modulus 26 O

for K, but they are all distinct and correspond to dierent class elds over K.

5. Explicit calculation of minimal polynomials

We will nd an explicit formula for the conjugates of T

(α) permitting the numerical calculation of its minimal polynomial. Let Q

(N ) be the set of primitive quadratic forms [a

, b

, c

] having discriminant d

with the property that a

> 0 and (a

, N ) = 1. For γ ∈

(N ) and Q ∈ Q

(N ), Q ◦ γ again belongs to Q

(N ). Hence the quotient Q

(N )/

(N ) is well-dened.

Theorem 12. Under the same conditions and notations as in Theo-

rem 6, the following assertions hold.