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Primitive and totally primitive Fricke families with applications (II)
J. Math. Anal. Appl. 472 (2019) 432–446
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Primitive and totally primitive Fricke families with applications (II) ✩ Ho Yun Jung a , Ja Kyung Koo b , Dong Hwa Shin c,∗ a
Applied Algebra and Optimization Research Center, Sungkyunkwan University, Suwon-si, Gyeonggi-do 16419, Republic of Korea b Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea c Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do 17035, Republic of Korea
a r t i c l e
i n f o
Article history: Received 9 May 2018 Available online 13 November 2018 Submitted by M.J. Schlosser Keywords: Fricke families Modular functions
a b s t r a c t We find necessary and sufficient conditions for a Fricke family of level N (≥ 2) to be primitive or totally√primitive. Let √K be an imaginary quadratic field of discriminant dK other than Q( −1) and Q( −3). As applications of Fricke families, we show that if |dK | is sufficiently large, then the special values of a primitive Fricke family generate the ray class field K(N ) modulo N over K. Moreover, we construct a primitive generator of K(N ) over K in terms of the special values of classical Fricke functions for every K which would be a partial answer to a question of Hasse and Ramachandra. © 2018 Elsevier Inc. All rights reserved.
1. Introduction The modular group SL2 (Z) acts on the upper half-plane H = {τ ∈ C | Im(τ ) > 0} and H∗ = H ∪Q ∪{i∞} by fractional linear transformations. For each positive integer N , let Γ(N ) = {γ ∈ SL2 (Z) | γ ≡ I2 (mod N · M2 (Z))} be the principal congruence subgroup of SL2 (Z) of level N . Then, the orbit space X(N ) = Γ(N )\H∗ ✩ The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MIST) (2016R1A5A1008055 and 2017R1C1B2010652). The third (corresponding) author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2017R1A2B1006578), and by Hankuk University of Foreign Studies Research Fund of 2018. * Corresponding author. E-mail addresses: [email protected] (H.Y. Jung), [email protected] (J.K. Koo), [email protected] (D.H. Shin).
https://doi.org/10.1016/j.jmaa.2018.11.033 0022-247X/© 2018 Elsevier Inc. All rights reserved.
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can be given the structure of a compact Riemann surface, called the modular curve of level N ([13, §1.5]). We denote by C(X(N )) the field of meromorphic functions on X(N ), which is a Galois extension of C(X(1)) with Galois group isomorphic to SL2 (Z)/ ±Γ(N ) SL2 (Z/N Z)/{±I2 } ([11, Theorem 2 in Chapter 6]). Note that every function in C(X(N )) has Fourier expansion with respect to q 1/N , where q = e2πiτ with τ ∈ H. Let FN be the subfield of C(X(N )) consisting of functions with Fourier coefficients in the N th cyclotomic field Q(ζN ), where ζN = e2πi/N . As is well known, FN is also a Galois extension of F1 with Gal(FN /F1 ) GL2 (Z/N Z)/{±I2 }. Refer to §2 for details on the action of GL2 (Z/N Z)/{±I2 } on FN . Now, let N ≥ 2 and VN = {v ∈ Q2 | N is the least positive integer so that N v ∈ Z2 }.
(1)
We call a family {hv }v∈VN of functions in FN a Fricke family of level N if it satisfies the following three axioms: (F1) Every hv is weakly holomorphic, that is, it is holomorphic on H. (F2) hu = hv whenever u ≡ ±v (mod Z2 ). T (F3) hα v = hαT v for α ∈ GL2 (Z/N Z)/{±I2 }, where α stands for the transpose of α. As for a Fricke family, Kubert and Lang first introduced the definition without the axiom (F1) ([8, pp. 32–33]), and Koo and Yoon further classified all Fricke families of level N when N is divisible by 4, 5, 6, 7 or 9 ([10]). We say that a Fricke family {hv }v∈VN is primitive if it satisfies (F2 ) hu = hv if and only if u ≡ ±v (mod Z2 ) instead of (F2). Moreover, the family is said to be totally primitive if {hnv }v∈VN is primitive for every positive integer n. In a recent paper [6] the same authors of this paper presented a couple of examples of Fricke families which are primitive or totally primitive, in terms of Fricke functions and Siegel functions (Proposition 2.2). Motivated by these concrete examples, we shall develop in this paper necessary and sufficient conditions for a Fricke family to be primitive or totally primitive (Theorems 3.2, 3.5, 3.7, Corollary 3.6 and Remark 3.8). √ √ Let K be an imaginary quadratic field of discriminant dK other than Q( −1) and Q( −3), and set τK =
√ (−1 + dK )/2 √ dK /2
if dK ≡ 1 (mod 4), if dK ≡ 0 (mod 4).
(2)
The authors proved in [5, Theorem 3.5 and Remark 3.6] that if {gv12N }v∈VN is the Fricke family consisting of the 12N th power of Siegel functions which is primitive (see §2), then g 0 (τK )12N generates the ray class 1/N
field K(N ) modulo N over K. In general, let {hv }v∈VN be a primitive Fricke family of level N . They also showed in [6, Theorem 5.2] that if |dK | is sufficiently large, then h 0 (τK ) generates K(N ) over the Hilbert 1/N
class field HK of K. In this paper, we shall further improve this result by pulling HK down to the base field K with finitely many generators (Theorem 5.6). In particular, we shall construct finitely many generators and further a primitive generator of K(N ) over K in terms of the special values of classical Fricke functions, without the j-function and the assumption that |dK | is sufficiently large (Theorem 6.3 and Corollary 6.7). This would be a partial answer to a question of Hasse and Ramachandra (Remark 6.8) and also certain improvement of Shimura’s work ([13, Proposition 6.33]) from Hasse–Ramachandra’s point of view.
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2. Galois actions on modular function fields We shall briefly review the Galois action of GL2 (Z/N Z)/{±I2 } on the modular function field FN and further introduce a couple of examples of (totally) primitive Fricke families, given in [6]. Let N be a positive integer. The group GL2 (Z/N Z)/{±I2 } can be decomposed into GL2 (Z/N Z)/{±I2 } = GN · SL2 (Z/N Z)/{±I2 }, where GN =
1 0 0 d
| d ∈ (Z/N Z)
∗
.
Regarded as the Galois group of FN over F1 , GL2 (Z/N Z)/{±I2 } acts on the field FN as follows: Proposition 2.1. Let f ∈ FN with Fourier expansion f=
cn q n/N
(cn ∈ Q(ζN )).
n−∞
1 (A1) The matrix 0
0 in GN acts on f as d
f
1 0 0 d
=
cσnd q n/N ,
n−∞ σd d where σd is the automorphism of Q(ζN ) determined by ζN = ζN . (A2) The matrix γ in SL2 (Z/N Z)/{±I2 } acts on f by
fγ = f ◦ γ
, where γ
is any preimage of γ under the reduction SL2 (Z) → SL2 (Z/N Z)/{±I2 }. Proof. See [11, Theorem 3 in Chapter 6]. 2 Let Λ be a lattice in C. We let 1 1 1 ℘(z; Λ) = 2 + − 2 z (z − λ)2 λ
(z ∈ C)
Λ−{0}
be the Weierstrass ℘-function relative to Λ, and let g2 (Λ) = 60
λ∈Λ−{0}
1 , λ4
g3 (Λ) = 140
λ∈Λ−{0}
1 λ6
and Δ(Λ) = g2 (Λ)3 − 27g3 (Λ)2 .
We define σ(z; Λ) = z
λ∈Λ−{0}
and
1−
z (z/λ)+(1/2)(z/λ)2 e λ
(z ∈ C)
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1 ζ(z; Λ) = + z
λ∈Λ−{0}
1 z 1 + + 2 z−λ λ λ
435
(z ∈ Z).
Since ζ (z; Λ) = −℘(z; Λ) is periodic with respect to Λ, corresponding to each λ ∈ Λ there is a constant η(λ; Λ) such that ζ(z + λ; Λ) − ζ(z; Λ) = η(λ; Λ) (z ∈ C). v1 ∈ Q2 − Z2 . We define the Fricke function fv by Now, let v = v2
fv (τ ) = −27 35
g2 (τ )g3 (τ ) ℘(v1 τ + v2 ; [τ, 1])) Δ(τ )
(τ ∈ H),
where g2 (τ ) = g2 ([τ, 1]), g3 (τ ) = g3 ([τ, 1]) Δ(τ ) = Δ([τ, 1]). Furthermore, we define the Siegel function gv by gv (τ ) = e−(1/2)(v1 η(τ ; [τ, 1])+v2 η(1; [τ, 1]))(v1 τ +v2 ) σ(v1 τ + v2 ; [τ, 1])η(τ )2
(τ ∈ H),
where η(τ ) is a 24th root of Δ(τ ) given by η(τ ) =
∞ √ 2πζ8 q 1/24 (1 − q n ) n=1
([11, Theorem 5 in Chapter 18]). It has the following infinite product expansion gv = −eπiv2 (v1 −1) q (1/2)B2 (v1 ) (1 − q v1 e2πiv2 )
∞
(1 − q n+v1 e2πiv2 )(1 − q n−v1 e−2πiv2 ),
(3)
n=1
where B2 (x) = x2 − x + 1/6 is the Bernoulli second polynomial ([8, K 4 in p. 29]). Proposition 2.2. Let N ≥ 2. (i) {fv }v∈VN is a primitive Fricke family of level N . Moreover, if N ≥ 7 and gcd(6, N ) = 1, then {fv }v∈VN is totally primitive. (ii) {gv12N }v∈VN is a totally primitive Fricke family of level N . Proof. (i) See [1, Lemma 10.4], [11, §6.2–6.3] and [6, Example 3.3]. (ii) See [8, Proposition 1.3 in Chapter 2] and [6, Example 3.1]. 2 Remark 2.3. We call a function in FN a modular unit of level N if it has a cuspidal divisor. One can see from (3) that gv12N is a modular unit of level N for every v ∈ VN . 3. Primitivity of Fricke families In this section, we shall present several necessary and sufficient conditions for a Fricke family to be primitive or totally primitive. For a positive integer N , let 1 0 Γ (N ) = γ ∈ SL2 (Z) | γ ≡ (mod N · M2 (Z)) ∗ 1 1
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which is a congruence subgroup of level N . Let X 1 (N ) = Γ1 (N )\H∗ be the modular curve for Γ1 (N ), and let C(X 1 (N )) be its field of meromorphic functions. Since the natural map X 1 (N ) → X(1) is a covering of degree [SL2 (Z) : ±Γ1 (N )], we have [C(X 1 (N )) : C(X(1))] = [SL2 (Z) : ±Γ1 (N )]
(4)
([13, §1.5]). Now, let N ≥ 2. We define an equivalence relation ∼ on the set VN stated in (1) as follows: u ∼ v ⇐⇒ u ≡ ±v (mod Z2 )
(u, v ∈ VN ).
Lemma 3.1. We get |VN / ∼ | = [SL2 (Z) : ±Γ1 (N )]. Proof. The group SL2 (Z/N Z) acts on the set VN / ∼ by (v ∈ VN / ∼, γ ∈ SL2 (Z/N Z)).
vγ = γ T v
Following the reduction SL2 (Z) → SL2 (Z/N Z), the group SL 2 (Z)also acts on the set VN / ∼. One can 1/N is the whole set VN / ∼. Furthermore, readily show that this action is transitive, and so the orbit of 0 1/N the isotropy subgroup of is ±Γ1 (N ). This proves the lemma. 2 0 For a field F and an element ν of the algebraic closure of F , we denote by min(ν, F ) the minimal polynomial of ν over F . Theorem 3.2. Let {hv }v∈VN be a Fricke family of level N . Then, it is primitive if and only if . C(X (N )) = C(X(1)) h 1/N 1
(5)
0
Proof. Note by (4) and Lemma 3.1 that the equality (5) holds if and only if deg min(h 1/N , C(X(1))) = [C(X 1 (N )) : C(X(1))] = [SL2 (Z) : ±Γ1 (N )] = |VN / ∼ |. 0
We obtain the theorem by (F2) and (F3). 2 Lemma 3.3. Let {hv }v∈VN be a primitive Fricke family of level N , and let n be a positive integer. Then, {hnv }v∈VN is not primitive if and only if n is even and
h
1/N 0
⎧ ⎪ ⎨ −h 1/4 =
if N = 4,
1/2
⎪ ⎩ −h a/N
if N = 4
0
for some integer a such that gcd(a, N ) = 1 and a ≡ ±1 (mod N ).
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Proof. The “if” is straightforward. Conversely, assume that {hnv }v∈VN is not primitive. Then we have hnu = hnv for some u, v ∈ VN such that u = ±v (mod Z2 ). By acting an element of SL2 (Z/N Z)/{±I2 } on both sides, if necessary, we may assume by (F2) and (F3) that hn 1/N = hn v1 v2
0
for some
v1 v 1/N ∈ VN such that 1 ≡ ± (mod Z2 ). It then follows that v2 v2 0 h 1/N = ζh v1
for some nth root of unity ζ.
(6)
v2
0
1 Here, we observe that ζ = 1 because {hv }v∈VN is primitive. By applying the composition of 1 to both sides of (6), we get by (F3) and (A2) that
0 ∈ SL2 (Z) 1
h 1/N = ζh v1 +v2 .
(7)
v2
0
We then obtain by (6) and (7) that h v1 = h v1 +v2 . v2
v2
Since {hv }v∈VN is primitive, we must have
v1 v + v2 ≡± 1 (mod Z2 ), v2 v2
and hence v2 ∈ Z Moreover, since
v1 v2
or 2v1 + v2 , 2v2 ∈ Z.
has primitive denominator N and
v1 1/N ± ≡ (mod Z2 ), we find that 0 v2
⎧ 1/4 3/4 ⎪ ⎪ ⎨ or (mod Z2 ) 1/2 1/2 v1 ≡ v2 ⎪ ⎪ ⎩ a/N (mod Z2 ) 0
if N = 4, if N = 4
for some integer a such that gcd(a, N ) = 1 and a = N ). Here, we observe by (F2) and (F3) that ±1 (mod 1 0 of GN , which implies by (A2) in Proposition 2.1 both h 1/N and h v1 are fixed by every element 0 d v2 0 that they have rational Fourier coefficients. Thus we attain by (6) that ζ = −1 and n is even. 2 Remark 3.4. Now we see from Lemma 3.3 that if a Fricke family {hv }v∈VN is primitive, then so is {hnv }v∈VN for every positive odd integer n.
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Theorem 3.5. Let {hv }v∈VN be a Fricke family of level N . Then, it is totally primitive if and only if {h2v}v∈VN is primitive. Proof. The “only if” follows from definition. To prove the “if” part of the theorem, assume that {hv }v∈VN is not totally primitive. If {hv }v∈VN is not primitive, then {h2v }v∈VN is not primitive. And, if {hv }v∈VN is primitive, then
h2
1/N 0
⎧ 2 ⎪ ⎨ h 1/4 =
if N = 4,
1/2
⎪ ⎩ h2 a/N
if N = 4
0
for some integer a such that gcd(a, N ) = 1 and a ≡ ±1 (mod N ) by Lemma 3.3 and (F2); and hence {h2v }v∈VN is not primitive. This completes the proof. 2 Corollary 3.6. Let {hv }v∈VN be a primitive Fricke family of level N . For each v ∈ VN , let cv be the constant term of the Fourier expansion of hv , and let S = {(1/2)(ca + cb ) | a, b ∈ VN / ∼} which is a finite subset of Q(ζN ). If c ∈ Q − S, then the Fricke family {hv − c}v∈VN is totally primitive. Proof. Suppose on the contrary that {hv − c}v∈VN is not totally primitive. Then we have (ha − c)2 = (hb − c)2 for some a, b ∈ VN such that a ≡ ±b (mod Z2 ). Since {hv − c}v∈VN is primitive, we must derive ha − c = −(hb − c). This yields ha + hb = 2c, which contradicts c ∈ / S. Therefore, {hv − c}v∈VN is totally primitive.
2
Let QN = {a ∈ Z | 1 < a ≤ N/2, a ≡ ±1 (mod N ) and a2 ≡ ±1 (mod N )}. Theorem 3.7. Let {hv }v∈VN be a primitive Fricke family of level N = 4. Then, it is not totally primitive if and only if there is a primitive Fricke family {tv }v∈VN of level N and a ∈ QN so that hv = tv − tav
(v ∈ VN ).
Proof. Assume that {hv }v∈VN is not totally primitive. Since N = 4, we deduce by Lemma 3.3 that h 1/N = −h a/N 0
(8)
0
for some integer a such that gcd(a, N )= 1 and a ≡ ±1 (mod N ). Here we may further assume by (F2) a 0 ∈ GL2 (Z/N Z)/{±I2 } on both sides of (8) yields that 1 < a ≤ N/2. The action of 0 a
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h a/N = −h a2 /N .
(9)
0
0
Thus we get by (8) and (9) h 1/N = h a2 /N , 0
0
from which it follows that a2 ≡ ±1 (mod N ) because {hv }v∈VN is primitive. Now, let tv =
1 hv 2
(v ∈ VN )
which form a primitive Fricke family {tv }v∈VN of level N . Since the action of SL2 (Z) on the set VN / ∼ is transitive, we obtain by (8) hv = −hav
(v ∈ VN ),
and hence tv − tav =
1 1 1 1 hv − hav = hv + hv = hv 2 2 2 2
(v ∈ VN ).
Conversely, assume that there is a primitive Fricke family {tv }v∈VN of level N and a ∈ QN satisfying hv = tv − tav
(v ∈ VN ).
We then see that hav = tav − ta2 v = tav − tv = −hv
(v ∈ VN )
by the fact a2 ≡ ±1 (mod N ) and (F2). And, we attain h2av = h2v
(v ∈ VN ).
Since a ≡ ±1 (mod N ), we have av ≡ ±v (mod Z2 ) for every v ∈ VN . Therefore, {h2v }v∈VN is not primitive, and so {hv }v∈VN is not totally primitive. 2 Remark 3.8. The set QN can be identified with the subset of the group (Z/N Z)∗ /{±1} consisting of elements of order 2. Now, consider the case where N = pe or 2pe for a prime p such that p ≡ 3 (mod 4) and a positive integer e. Since |(Z/N Z)∗ /{±1}| = pe−1 (p − 1)/2 which is odd by the assumption p ≡ 3 (mod 4), the group (Z/N Z)∗ /{±1} contains no element of order 2. Thus in this case the set QN is empty; and hence every primitive Fricke family {hv }v∈VN of level N is totally primitive by Theorem 3.7.
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4. Class fields over imaginary quadratic fields As for applications of Fricke families to constructing class fields, we shall review basic facts about Galois groups of class fields over imaginary quadratic fields, well known as consequences of the main theorem of the theory of complex multiplication. Let K be an imaginary quadratic field of discriminant dK (< 0), and let OK be its ring of integers. We set τK as in (2) so that τK ∈ H and OK = Z[τK ]. Let HK be the Hilbert class field of K and hK be the class number of K, that is, hK = [HK : K]. And, for a positive integer N , let K(N ) be the ray class field of K modulo (N ) = N OK . Proposition 4.1. Let K be an imaginary quadratic field and N ≥ 2. Then (i) HK = K(j(τK )). (ii) K(N ) = K(f (τK ) | f ∈ FN is finite at τK ). √ √ (iii) If K is different from Q( −1) and Q( −3), then K(N ) = HK (f
0 1/N
(τ
K )).
Proof. (i) See [11, Theorem 1 in Chapter 10]. (ii) See [11, Corollary to Theorem 2 in Chapter 10]. (iii) See [11, Corollary to Theorem 7 in Chapter 10]. See also [4] and [13, Chapter 6]. 2 Remark 4.2. Since j has Fourier expansion with respect to q = e2πiτ with rational Fourier coefficients, the singular value j(τK ) is a real number. Hence, min(j(τK ), K) has rational coefficients ([9, Remark 4.9]). Let C(dK ) be the group of reduced primitive positive definite quadratic forms Q = ax2 + bxy + cy 2 ∈ Z[x, y] of discriminant dK such that (−a < b ≤ a < c or 0 ≤ b ≤ a = c),
gcd(a, b, c) = 1 and b2 − 4ac = dK .
For each Q = ax2 + bxy + cy 2 ∈ C(dK ), let √ −b + dK . τQ = 2a Proposition 4.3. If K is an imaginary quadratic field of discriminant dK , then the map C(dK ) → Gal(HK /K) Q → (j(τK ) → j(τQ )) is an isomorphism. Proof. See [1, Theorems 5.23 and 5.30]. 2 Let min(τK , Q) = x2 + bK x + cK ∈ Z[x], and set WK, N =
t − bK s s
which is a subgroup of GL2 (Z/N Z).
−cK s ∈ GL2 (Z/N Z) | t, s ∈ (Z/N Z) t
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√ Proposition 4.4 (Shimura’s reciprocity law). Let K be an imaginary quadratic field other than Q( −1) and √ Q( −3), and let N be a positive integer. Then, the map WK, N → Gal(K(N ) /HK ) α → (f (τK ) → f α (τK ) | f ∈ FN is finite at τK ) is a surjection with kernel {±I2 }. Proof. See [16, §3].
2
5. Special values of Fricke families Let N ≥ 2 be given and {hv }v∈VN be a primitive Fricke family of level N . We shall show that if K is an imaginary quadratic field with sufficiently large |dK |, then the family of special values {hv (τK )}v∈VN generates K(N ) over K. Lemma 5.1. We have the relation ⎞3
⎛ ⎝g12 j=
0 1/2
+ 16⎠
g12 0
.
1/2
Proof. See [1, Theorem 12.17]. 2 Lemma 5.2. Let K be an imaginary quadratic field with |dK | ≥ 20 and hK ≥ 2, and let Q be a nonidentity reduced form of discriminant dK . Let g = g12 0 . We have the following inequalities: 1/2
−π |dK | (i) |g(τK )| < 4097e < 0.0033. −(1/2)π |dK | (ii) 3630e < |g(τQ )| < 19.71. 1 < 1.21e−π |dK | . (iii) j(τK ) j(τQ ) < 12.6e−(1/2)π |dK | . (iv) j(τK )
Proof. (i) See [7, Lemma 5.2]. (ii) See [7, Lemma 6.3]. (iii) We see that 1 g(τK ) = j(τK ) (g(τK ) + 16)3 by Lemma 5.1 g(τK ) ≤ 3 (−|g(τK )| + 16)
4907e−π |dK | < (−0.0033 + 16)3 < 1.21e−π
|dK |
.
by (i)
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(iv) We derive that j(τQ ) g(τQ ) + 16 3 g(τK ) = j(τK ) g(τK ) + 16 g(τQ ) by Lemma 5.1 |g(τ )| + 16 3 g(τ ) K Q ≤ −|g(τK )| + 16 g(τQ ) 3 19.71 + 16 4097e−π |dK | < by (i) and (ii) −0.0033 + 16 3630e−(1/2)π dK | < 12.6e−(1/2)π
|dK |
.
2
Lemma 5.3. Let f (x) be a nonconstant polynomial in x over Q. If |dK | is sufficiently large, then f (j(τK )) generates HK over K. Proof. Let f (x) = cm xm + cm−1 xm−1 + · · · + c1 x + c0 with cm , cm−1 , . . . , c1 , c0 ∈ Q such that cm = 0. Take |dK | large enough so as to have hK > m, and then f (j(τK )) = 0. Let Q be a nonidentity reduced form of discriminant dK . We induce from Lemma 5.2 (iii) and (iv) that the ratio
j(τQ ) m + f (j(τQ )) j(τK ) f (j(τK )) =
m−1
j(τQ ) j(τQ ) cm−1 c1 1 1 + · · · + m−1 cm j(τK ) j(τK ) cm j(τK ) j(τK ) cm−1 c1 c0 1 1 1 + cm j(τK ) + · · · + cm j(τK )m−1 + cm j(τK1 )m
+
c0 1 cm j(τK )m
converges to 0 as |dK | approaches ∞, regardless of Q. This implies that if |dK | is sufficiently large, then f (j(τK )) is different from f (j(τQ )) for every nonidentity reduced form Q, which claims that f (j(τK )) generates HK over K by Proposition 4.3. 2 Lemma 5.4. Every weakly holomorphic function in F1 is a polynomial in j over Q. Proof. See [11, Theorem 2 in Chapter 5]. 2 Remark 5.5. Let h be a modular unit in F1 . Since h and 1/h are polynomials in j over Q by Lemma 5.4, h must be a nonzero constant. Theorem 5.6. Let {hv }v∈VN be a primitive Fricke family of level N . Let K be an imaginary quadratic field √ √ other than Q( −1) and Q( −3). If |dK | is sufficiently large, then K(N ) = K (hv (τK ) | v ∈ VN / ∼) . Proof. Let F = K (hv (τK ) | v ∈ VN / ∼) which is a subfield of K(N ) by Proposition 4.1 (ii). Let h = h
(10) 0 1/N
with min(h, F1 ) = pm xm + · · · + p1 x + p0 for some pm , . . . , p1 , p0 ∈ F1 such that pm = 0. Since every hv is weakly holomorphic by (F1), we get by (F2), (F3) and Lemma 5.4 that the symmetric polynomials pm , . . . , p1 , p0 belong to Q[j]. Moreover, since
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h is a nonconstant, pr is a nonconstant polynomial in j for some r ∈ {0, 1, . . . , m}. Thus, if we take |dK | sufficiently large, then pr (τK ) generates HK over K by Lemma 5.3; and so F ⊇ HK .
(11)
If [K(N ) : HK ] = 1, then it is obvious by (11) that F = K(N ) . Now, we assume that [K(N ) : HK ] ≥ 2. Let VN / ∼ = {v1 , v2 , . . . , vn } and set
p=
(hva − hvb )2 .
1≤a
Observe by (F2), (F3), Lemma 5.4 and the primitivity of {hv }v∈VN that p is a nonzero polynomial in j over Q, namely, p = ck j k + · · · + c1 j + c0 for some ck , . . . , c1 , c0 ∈ Q such take |dK | sufficiently large so that hK > k, and then that ck = 0. Again, t − bK s −cK s ∈ WK, N /{±I2 } leaves h(τK ) fixed. Since p(τK ) = 0 and p(τK ) = 0. Suppose that α = s t 0 = h(τK ) − h(τK )α = h
0 1/N
(τ
K)
−h
αT
0 1/N
(τ
K)
= h
0 1/N
(τ
K)
− h s/N (τK ) t/N
s/N 0 ≡± (mod Z2 ). Hence α is the identity by Proposition 4.4, (F2) and (F3), we should have t/N 1/N element of WK, N /{±I2 }, which implies by Galois theory that h(τK ) generates K(N ) over HK . Therefore we conclude by (11) that F = K(N ) , as desired. 2 Remark 5.7. Moreover, if [K(N ) : HK ] = |VN / ∼ |, then we see from Proposition 4.4, (F2) and (F3) that hv (τK ) are exactly the Galois conjugates of h
0 1/N
(τ
K)
over HK (v ∈ VN / ∼). In this case, the equality (10) becomes K(N ) = K
h
0 1/N
(τ
K)
.
6. Families of Fricke functions √ √ Let K be an imaginary quadratic field other than Q( −1) and Q( −3). In this last section, we shall show that for the family {fv }v∈VN of Fricke functions, Theorem 5.6 holds without the assumption that |dK | is sufficiently large. Lemma 6.1. Let K be an imaginary quadratic field with hK ≥ 2. If Q is a nonidentity reduced form of discriminant dK , then j(τQ )2 (j(τQ ) − 1728)3 j(τK )2 (j(τK ) − 1728)3 < 1.
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Proof. It is well known that ⇐⇒
hK = 1
dK = −3, −4, −7, −8, −11, −19, −43, −67, −163
([1, Theorem 12.34]). Thus, the least |dK | with hK ≥ 2 is 15. In this case, we have C(dK ) = {Q1 = x2 + xy + 4y 2 , Q2 = 2x2 + xy + 2y 2 } and τQ1 = τK =
−1 +
√
−15
and τQ2 =
2
−1 +
√
−15
4
.
By utilizing Lemma 5.1 and some numerical estimations (using Maple 2016.2), we get that min(j(τK ), K) = x2 + 191025x − 121287375 and j(τQ1 ) = −
191025 85995 √ 191025 85995 √ − + 5 and j(τQ2 ) = − 5. 2 2 2 2
See also [3, §6.1]. It then follows that j(τQ2 )2 (j(τQ2 ) − 1728)3 −12 < 1. j(τQ )2 (j(τQ ) − 1728)3 < 2 · 10 1 1 When |dK | ≥ 20, the proof is similar to Lemma 5.2 (iv). Refer also to [7, Lemma 6.3].
2
Lemma 6.2. Let u, v ∈ Q2 − Z2 such that u ≡ ±v (mod Z2 ). We have the relation (fu − fv )6 = 212 36 j 2 (j − 1728)3
6 6 gu+v gu−v . gu12 gv12
Proof. See [8, p. 51] and [11, Theorem 2 in Chapter 18]. 2 √ √ Theorem 6.3. Let K be an imaginary quadratic field other than Q( −1) and Q( −3), and let N ≥ 2. We achieve K(N ) = K (fv (τK ) | v ∈ VN / ∼) . Proof. If hK = 1, then the theorem is immediate from Proposition 4.1 (iii). Now, we assume that hK ≥ 2. Let F = K (fv (τK ) | v ∈ VN / ∼) which is a subfield of K(N ) by Proposition 4.1 (ii). Let VN / ∼ = {v1 , v2 , . . . , vn }, and set D=
(fva − fvb )12N
1≤a
which belongs to F1 by (F2) and (F3). We then see that N n(n−1) D = 212 36 j 2 (j − 1728)3
1≤a
= c{j 2 (j − 1728)3 }N n(n−1)
gv12N g 12N a +vb va −vb gv24N gv24N a b
by Lemma 6.2
for some nonzero c ∈ Q by Remarks 2.3 and 5.5.
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We get by Lemma 6.1 that D(τK ) = D(τQ ) for every nonidentity reduced form Q of discriminant dK , which implies that D(τK ) generates HK over K by Proposition 4.3. Since D(τK ) belongs to F , we attain HK ⊆ F . Therefore, we conclude by Proposition 4.1 (iii) that F = K(N ) , as desired. 2 Lemma 6.4. If N ≥ 2 and v ∈ VN , then N 2 fv is integral over Z[j]. Proof. See [9, Lemma 5.6]. 2 Lemma 6.5. If τ0 ∈ H is imaginary quadratic, then j(τ0 ) is an algebraic integer. Proof. See [11, Theorem 4 in Chapter 5]. 2 Lemma 6.6. Let L be an abelian extension of a number field K. Suppose that L = K(u, v) for some u, v ∈ L. Let = [L : K(u)]. Then we get L = K(u + v − TrL/K(u) (v)). Proof. See [14, Theorem 8.2]. 2 As a corollary of Theorem 6.3 we obtain a primitive generator of K(N ) over K in terms of the special values of {fv }v∈VN . √ √ Corollary 6.7. Let K be an imaginary quadratic field other than Q( −1) and Q( −3), and let N ≥ 2. Then, the special value N 24N
(fva (τK ) − fvb (τK ))12N + |WK, N /{±I2 }|N 2 f
1≤a
0 1/N
(τ
K)
− N2
α∈WK, N /{±I2 }
f
αT
0 1/N
(τ
K)
becomes a primitive generator of K(N ) over K as an algebraic integer, where VN / ∼ = {v1 , v2 , . . . , vn }. Proof. Let u = N 24N
(fva (τK ) − fvb (τK ))12N
and v = N 2 f
1≤a
0 1/N
(τ
K)
which are algebraic integers by Lemmas 6.4 and 6.5. By Proposition 4.1 (iii) and the proof of Theorem 6.3, we derive K(N ) = HK (v) and HK = K(u). Note by Proposition 4.4 that Gal(K(N ) /HK ) WK, N /{±I2 }. The result follows from Lemma 6.6. 2 Remark 6.8. In a letter to Hecke in 1920s, Hasse asked the question whether the abelian extensions of K are already generated by the division values of the Weber function without the singular values of the j-function ([2, p. 91]). Ramachandra also mentioned this question later in [12, p. 105]. Since the Fricke functions are essentially induced from the Weber function ([15, Example 5.5.2 in Chapter II]), Theorem 6.3 and Corollary 6.7 could be considered as partial answers to the question of Hasse and Ramachandra for the ray class fields of K.
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Acknowledgment The authors would like to thank the referee for carefully reading the manuscript and for giving detailed comments that have been helpful to improve the manuscript. References [1] D.A. Cox, Primes of the Form x2 + ny 2 : Fermat, Class Field Theory, and Complex Multiplication, 2nd edn., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2013. [2] G. Frei, F. Lemmermeyer, P.J. Roquette, Emil Artin and Helmut Hasse: The Correspondence 1923–1958, Contributions in Mathematical and Computational Sciences, vol. 5, Springer, Heidelberg, 2014. [3] J. Guàrdia, E. Torres, M. Vela, Stable models of elliptic curves, ring class fields, and complex multiplication, in: Algorithmic Number Theory, in: Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004, pp. 250–262. [4] H. Hasse, Neue Begründung der Komplexen Multiplikation I, II, J. Reine Angew. Math. 157, 165 (1927, 1931) 115–139, 64–88. [5] H.Y. Jung, J.K. Koo, D.H. Shin, Ray class invariants over imaginary quadratic fields, Tohoku Math. J. (2) 63 (3) (2011) 413–426. [6] H.Y. Jung, J.K. Koo, D.H. Shin, Primitive and totally primitive Fricke families with applications, Results Math. 71 (3) (2017) 841–858. [7] H.Y. Jung, J.K. Koo, D.H. Shin, Generation of ray class fields modulo 2, 3, 4 or 6 by using the Weber function, J. Korean Math. Soc. 55 (2) (2018) 343–372. [8] D. Kubert, S. Lang, Modular Units, Grundlehren der mathematischen Wissenschaften, vol. 244, Springer-Verlag, New York–Berlin, 1981. [9] J.K. Koo, D.H. Shin, Singular values of principal moduli, J. Number Theory 133 (2) (2013) 475–483. [10] J.K. Koo, D.S. Yoon, Generators of the ring of weakly holomorphic modular functions for Γ1 (N ), Ramanujan J. 42 (3) (2017) 583–599. [11] S. Lang, Elliptic Functions, 2nd edn., Grad. Texts in Math., vol. 112, Springer-Verlag, New York, 1987. [12] K. Ramachandra, Some applications of Kronecker’s limit formula, Ann. of Math. (2) 80 (1964) 104–148. [13] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, Princeton, NJ, 1971. [14] D.H. Shin, On some theta constants and class fields, J. Korean Math. Soc. 51 (6) (2014) 1269–1289. [15] J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 151, Springer-Verlag, New York, 1994. [16] P. Stevenhagen, Hilbert’s 12th problem, complex multiplication and Shimura reciprocity, in: Class Field Theory—Its Centenary and Prospect, Tokyo, 1998, in: Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 161–176.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Primitive and totally primitive Fricke families with applications (II) ✩ Ho Yun Jung a , Ja Kyung Koo b , Dong Hwa Shin c,∗ a
Applied Algebra and Optimization Research Center, Sungkyunkwan University, Suwon-si, Gyeonggi-do 16419, Republic of Korea b Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea c Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do 17035, Republic of Korea
a r t i c l e
i n f o
Article history: Received 9 May 2018 Available online 13 November 2018 Submitted by M.J. Schlosser Keywords: Fricke families Modular functions
a b s t r a c t We find necessary and sufficient conditions for a Fricke family of level N (≥ 2) to be primitive or totally√primitive. Let √K be an imaginary quadratic field of discriminant dK other than Q( −1) and Q( −3). As applications of Fricke families, we show that if |dK | is sufficiently large, then the special values of a primitive Fricke family generate the ray class field K(N ) modulo N over K. Moreover, we construct a primitive generator of K(N ) over K in terms of the special values of classical Fricke functions for every K which would be a partial answer to a question of Hasse and Ramachandra. © 2018 Elsevier Inc. All rights reserved.
1. Introduction The modular group SL2 (Z) acts on the upper half-plane H = {τ ∈ C | Im(τ ) > 0} and H∗ = H ∪Q ∪{i∞} by fractional linear transformations. For each positive integer N , let Γ(N ) = {γ ∈ SL2 (Z) | γ ≡ I2 (mod N · M2 (Z))} be the principal congruence subgroup of SL2 (Z) of level N . Then, the orbit space X(N ) = Γ(N )\H∗ ✩ The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MIST) (2016R1A5A1008055 and 2017R1C1B2010652). The third (corresponding) author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2017R1A2B1006578), and by Hankuk University of Foreign Studies Research Fund of 2018. * Corresponding author. E-mail addresses: [email protected] (H.Y. Jung), [email protected] (J.K. Koo), [email protected] (D.H. Shin).
https://doi.org/10.1016/j.jmaa.2018.11.033 0022-247X/© 2018 Elsevier Inc. All rights reserved.
H.Y. Jung et al. / J. Math. Anal. Appl. 472 (2019) 432–446
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can be given the structure of a compact Riemann surface, called the modular curve of level N ([13, §1.5]). We denote by C(X(N )) the field of meromorphic functions on X(N ), which is a Galois extension of C(X(1)) with Galois group isomorphic to SL2 (Z)/ ±Γ(N ) SL2 (Z/N Z)/{±I2 } ([11, Theorem 2 in Chapter 6]). Note that every function in C(X(N )) has Fourier expansion with respect to q 1/N , where q = e2πiτ with τ ∈ H. Let FN be the subfield of C(X(N )) consisting of functions with Fourier coefficients in the N th cyclotomic field Q(ζN ), where ζN = e2πi/N . As is well known, FN is also a Galois extension of F1 with Gal(FN /F1 ) GL2 (Z/N Z)/{±I2 }. Refer to §2 for details on the action of GL2 (Z/N Z)/{±I2 } on FN . Now, let N ≥ 2 and VN = {v ∈ Q2 | N is the least positive integer so that N v ∈ Z2 }.
(1)
We call a family {hv }v∈VN of functions in FN a Fricke family of level N if it satisfies the following three axioms: (F1) Every hv is weakly holomorphic, that is, it is holomorphic on H. (F2) hu = hv whenever u ≡ ±v (mod Z2 ). T (F3) hα v = hαT v for α ∈ GL2 (Z/N Z)/{±I2 }, where α stands for the transpose of α. As for a Fricke family, Kubert and Lang first introduced the definition without the axiom (F1) ([8, pp. 32–33]), and Koo and Yoon further classified all Fricke families of level N when N is divisible by 4, 5, 6, 7 or 9 ([10]). We say that a Fricke family {hv }v∈VN is primitive if it satisfies (F2 ) hu = hv if and only if u ≡ ±v (mod Z2 ) instead of (F2). Moreover, the family is said to be totally primitive if {hnv }v∈VN is primitive for every positive integer n. In a recent paper [6] the same authors of this paper presented a couple of examples of Fricke families which are primitive or totally primitive, in terms of Fricke functions and Siegel functions (Proposition 2.2). Motivated by these concrete examples, we shall develop in this paper necessary and sufficient conditions for a Fricke family to be primitive or totally primitive (Theorems 3.2, 3.5, 3.7, Corollary 3.6 and Remark 3.8). √ √ Let K be an imaginary quadratic field of discriminant dK other than Q( −1) and Q( −3), and set τK =
√ (−1 + dK )/2 √ dK /2
if dK ≡ 1 (mod 4), if dK ≡ 0 (mod 4).
(2)
The authors proved in [5, Theorem 3.5 and Remark 3.6] that if {gv12N }v∈VN is the Fricke family consisting of the 12N th power of Siegel functions which is primitive (see §2), then g 0 (τK )12N generates the ray class 1/N
field K(N ) modulo N over K. In general, let {hv }v∈VN be a primitive Fricke family of level N . They also showed in [6, Theorem 5.2] that if |dK | is sufficiently large, then h 0 (τK ) generates K(N ) over the Hilbert 1/N
class field HK of K. In this paper, we shall further improve this result by pulling HK down to the base field K with finitely many generators (Theorem 5.6). In particular, we shall construct finitely many generators and further a primitive generator of K(N ) over K in terms of the special values of classical Fricke functions, without the j-function and the assumption that |dK | is sufficiently large (Theorem 6.3 and Corollary 6.7). This would be a partial answer to a question of Hasse and Ramachandra (Remark 6.8) and also certain improvement of Shimura’s work ([13, Proposition 6.33]) from Hasse–Ramachandra’s point of view.
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2. Galois actions on modular function fields We shall briefly review the Galois action of GL2 (Z/N Z)/{±I2 } on the modular function field FN and further introduce a couple of examples of (totally) primitive Fricke families, given in [6]. Let N be a positive integer. The group GL2 (Z/N Z)/{±I2 } can be decomposed into GL2 (Z/N Z)/{±I2 } = GN · SL2 (Z/N Z)/{±I2 }, where GN =
1 0 0 d
| d ∈ (Z/N Z)
∗
.
Regarded as the Galois group of FN over F1 , GL2 (Z/N Z)/{±I2 } acts on the field FN as follows: Proposition 2.1. Let f ∈ FN with Fourier expansion f=
cn q n/N
(cn ∈ Q(ζN )).
n−∞
1 (A1) The matrix 0
0 in GN acts on f as d
f
1 0 0 d
=
cσnd q n/N ,
n−∞ σd d where σd is the automorphism of Q(ζN ) determined by ζN = ζN . (A2) The matrix γ in SL2 (Z/N Z)/{±I2 } acts on f by
fγ = f ◦ γ
, where γ
is any preimage of γ under the reduction SL2 (Z) → SL2 (Z/N Z)/{±I2 }. Proof. See [11, Theorem 3 in Chapter 6]. 2 Let Λ be a lattice in C. We let 1 1 1 ℘(z; Λ) = 2 + − 2 z (z − λ)2 λ
(z ∈ C)
Λ−{0}
be the Weierstrass ℘-function relative to Λ, and let g2 (Λ) = 60
λ∈Λ−{0}
1 , λ4
g3 (Λ) = 140
λ∈Λ−{0}
1 λ6
and Δ(Λ) = g2 (Λ)3 − 27g3 (Λ)2 .
We define σ(z; Λ) = z
λ∈Λ−{0}
and
1−
z (z/λ)+(1/2)(z/λ)2 e λ
(z ∈ C)
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1 ζ(z; Λ) = + z
λ∈Λ−{0}
1 z 1 + + 2 z−λ λ λ
435
(z ∈ Z).
Since ζ (z; Λ) = −℘(z; Λ) is periodic with respect to Λ, corresponding to each λ ∈ Λ there is a constant η(λ; Λ) such that ζ(z + λ; Λ) − ζ(z; Λ) = η(λ; Λ) (z ∈ C). v1 ∈ Q2 − Z2 . We define the Fricke function fv by Now, let v = v2
fv (τ ) = −27 35
g2 (τ )g3 (τ ) ℘(v1 τ + v2 ; [τ, 1])) Δ(τ )
(τ ∈ H),
where g2 (τ ) = g2 ([τ, 1]), g3 (τ ) = g3 ([τ, 1]) Δ(τ ) = Δ([τ, 1]). Furthermore, we define the Siegel function gv by gv (τ ) = e−(1/2)(v1 η(τ ; [τ, 1])+v2 η(1; [τ, 1]))(v1 τ +v2 ) σ(v1 τ + v2 ; [τ, 1])η(τ )2
(τ ∈ H),
where η(τ ) is a 24th root of Δ(τ ) given by η(τ ) =
∞ √ 2πζ8 q 1/24 (1 − q n ) n=1
([11, Theorem 5 in Chapter 18]). It has the following infinite product expansion gv = −eπiv2 (v1 −1) q (1/2)B2 (v1 ) (1 − q v1 e2πiv2 )
∞
(1 − q n+v1 e2πiv2 )(1 − q n−v1 e−2πiv2 ),
(3)
n=1
where B2 (x) = x2 − x + 1/6 is the Bernoulli second polynomial ([8, K 4 in p. 29]). Proposition 2.2. Let N ≥ 2. (i) {fv }v∈VN is a primitive Fricke family of level N . Moreover, if N ≥ 7 and gcd(6, N ) = 1, then {fv }v∈VN is totally primitive. (ii) {gv12N }v∈VN is a totally primitive Fricke family of level N . Proof. (i) See [1, Lemma 10.4], [11, §6.2–6.3] and [6, Example 3.3]. (ii) See [8, Proposition 1.3 in Chapter 2] and [6, Example 3.1]. 2 Remark 2.3. We call a function in FN a modular unit of level N if it has a cuspidal divisor. One can see from (3) that gv12N is a modular unit of level N for every v ∈ VN . 3. Primitivity of Fricke families In this section, we shall present several necessary and sufficient conditions for a Fricke family to be primitive or totally primitive. For a positive integer N , let 1 0 Γ (N ) = γ ∈ SL2 (Z) | γ ≡ (mod N · M2 (Z)) ∗ 1 1
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which is a congruence subgroup of level N . Let X 1 (N ) = Γ1 (N )\H∗ be the modular curve for Γ1 (N ), and let C(X 1 (N )) be its field of meromorphic functions. Since the natural map X 1 (N ) → X(1) is a covering of degree [SL2 (Z) : ±Γ1 (N )], we have [C(X 1 (N )) : C(X(1))] = [SL2 (Z) : ±Γ1 (N )]
(4)
([13, §1.5]). Now, let N ≥ 2. We define an equivalence relation ∼ on the set VN stated in (1) as follows: u ∼ v ⇐⇒ u ≡ ±v (mod Z2 )
(u, v ∈ VN ).
Lemma 3.1. We get |VN / ∼ | = [SL2 (Z) : ±Γ1 (N )]. Proof. The group SL2 (Z/N Z) acts on the set VN / ∼ by (v ∈ VN / ∼, γ ∈ SL2 (Z/N Z)).
vγ = γ T v
Following the reduction SL2 (Z) → SL2 (Z/N Z), the group SL 2 (Z)also acts on the set VN / ∼. One can 1/N is the whole set VN / ∼. Furthermore, readily show that this action is transitive, and so the orbit of 0 1/N the isotropy subgroup of is ±Γ1 (N ). This proves the lemma. 2 0 For a field F and an element ν of the algebraic closure of F , we denote by min(ν, F ) the minimal polynomial of ν over F . Theorem 3.2. Let {hv }v∈VN be a Fricke family of level N . Then, it is primitive if and only if . C(X (N )) = C(X(1)) h 1/N 1
(5)
0
Proof. Note by (4) and Lemma 3.1 that the equality (5) holds if and only if deg min(h 1/N , C(X(1))) = [C(X 1 (N )) : C(X(1))] = [SL2 (Z) : ±Γ1 (N )] = |VN / ∼ |. 0
We obtain the theorem by (F2) and (F3). 2 Lemma 3.3. Let {hv }v∈VN be a primitive Fricke family of level N , and let n be a positive integer. Then, {hnv }v∈VN is not primitive if and only if n is even and
h
1/N 0
⎧ ⎪ ⎨ −h 1/4 =
if N = 4,
1/2
⎪ ⎩ −h a/N
if N = 4
0
for some integer a such that gcd(a, N ) = 1 and a ≡ ±1 (mod N ).
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437
Proof. The “if” is straightforward. Conversely, assume that {hnv }v∈VN is not primitive. Then we have hnu = hnv for some u, v ∈ VN such that u = ±v (mod Z2 ). By acting an element of SL2 (Z/N Z)/{±I2 } on both sides, if necessary, we may assume by (F2) and (F3) that hn 1/N = hn v1 v2
0
for some
v1 v 1/N ∈ VN such that 1 ≡ ± (mod Z2 ). It then follows that v2 v2 0 h 1/N = ζh v1
for some nth root of unity ζ.
(6)
v2
0
1 Here, we observe that ζ = 1 because {hv }v∈VN is primitive. By applying the composition of 1 to both sides of (6), we get by (F3) and (A2) that
0 ∈ SL2 (Z) 1
h 1/N = ζh v1 +v2 .
(7)
v2
0
We then obtain by (6) and (7) that h v1 = h v1 +v2 . v2
v2
Since {hv }v∈VN is primitive, we must have
v1 v + v2 ≡± 1 (mod Z2 ), v2 v2
and hence v2 ∈ Z Moreover, since
v1 v2
or 2v1 + v2 , 2v2 ∈ Z.
has primitive denominator N and
v1 1/N ± ≡ (mod Z2 ), we find that 0 v2
⎧ 1/4 3/4 ⎪ ⎪ ⎨ or (mod Z2 ) 1/2 1/2 v1 ≡ v2 ⎪ ⎪ ⎩ a/N (mod Z2 ) 0
if N = 4, if N = 4
for some integer a such that gcd(a, N ) = 1 and a = N ). Here, we observe by (F2) and (F3) that ±1 (mod 1 0 of GN , which implies by (A2) in Proposition 2.1 both h 1/N and h v1 are fixed by every element 0 d v2 0 that they have rational Fourier coefficients. Thus we attain by (6) that ζ = −1 and n is even. 2 Remark 3.4. Now we see from Lemma 3.3 that if a Fricke family {hv }v∈VN is primitive, then so is {hnv }v∈VN for every positive odd integer n.
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Theorem 3.5. Let {hv }v∈VN be a Fricke family of level N . Then, it is totally primitive if and only if {h2v}v∈VN is primitive. Proof. The “only if” follows from definition. To prove the “if” part of the theorem, assume that {hv }v∈VN is not totally primitive. If {hv }v∈VN is not primitive, then {h2v }v∈VN is not primitive. And, if {hv }v∈VN is primitive, then
h2
1/N 0
⎧ 2 ⎪ ⎨ h 1/4 =
if N = 4,
1/2
⎪ ⎩ h2 a/N
if N = 4
0
for some integer a such that gcd(a, N ) = 1 and a ≡ ±1 (mod N ) by Lemma 3.3 and (F2); and hence {h2v }v∈VN is not primitive. This completes the proof. 2 Corollary 3.6. Let {hv }v∈VN be a primitive Fricke family of level N . For each v ∈ VN , let cv be the constant term of the Fourier expansion of hv , and let S = {(1/2)(ca + cb ) | a, b ∈ VN / ∼} which is a finite subset of Q(ζN ). If c ∈ Q − S, then the Fricke family {hv − c}v∈VN is totally primitive. Proof. Suppose on the contrary that {hv − c}v∈VN is not totally primitive. Then we have (ha − c)2 = (hb − c)2 for some a, b ∈ VN such that a ≡ ±b (mod Z2 ). Since {hv − c}v∈VN is primitive, we must derive ha − c = −(hb − c). This yields ha + hb = 2c, which contradicts c ∈ / S. Therefore, {hv − c}v∈VN is totally primitive.
2
Let QN = {a ∈ Z | 1 < a ≤ N/2, a ≡ ±1 (mod N ) and a2 ≡ ±1 (mod N )}. Theorem 3.7. Let {hv }v∈VN be a primitive Fricke family of level N = 4. Then, it is not totally primitive if and only if there is a primitive Fricke family {tv }v∈VN of level N and a ∈ QN so that hv = tv − tav
(v ∈ VN ).
Proof. Assume that {hv }v∈VN is not totally primitive. Since N = 4, we deduce by Lemma 3.3 that h 1/N = −h a/N 0
(8)
0
for some integer a such that gcd(a, N )= 1 and a ≡ ±1 (mod N ). Here we may further assume by (F2) a 0 ∈ GL2 (Z/N Z)/{±I2 } on both sides of (8) yields that 1 < a ≤ N/2. The action of 0 a
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h a/N = −h a2 /N .
(9)
0
0
Thus we get by (8) and (9) h 1/N = h a2 /N , 0
0
from which it follows that a2 ≡ ±1 (mod N ) because {hv }v∈VN is primitive. Now, let tv =
1 hv 2
(v ∈ VN )
which form a primitive Fricke family {tv }v∈VN of level N . Since the action of SL2 (Z) on the set VN / ∼ is transitive, we obtain by (8) hv = −hav
(v ∈ VN ),
and hence tv − tav =
1 1 1 1 hv − hav = hv + hv = hv 2 2 2 2
(v ∈ VN ).
Conversely, assume that there is a primitive Fricke family {tv }v∈VN of level N and a ∈ QN satisfying hv = tv − tav
(v ∈ VN ).
We then see that hav = tav − ta2 v = tav − tv = −hv
(v ∈ VN )
by the fact a2 ≡ ±1 (mod N ) and (F2). And, we attain h2av = h2v
(v ∈ VN ).
Since a ≡ ±1 (mod N ), we have av ≡ ±v (mod Z2 ) for every v ∈ VN . Therefore, {h2v }v∈VN is not primitive, and so {hv }v∈VN is not totally primitive. 2 Remark 3.8. The set QN can be identified with the subset of the group (Z/N Z)∗ /{±1} consisting of elements of order 2. Now, consider the case where N = pe or 2pe for a prime p such that p ≡ 3 (mod 4) and a positive integer e. Since |(Z/N Z)∗ /{±1}| = pe−1 (p − 1)/2 which is odd by the assumption p ≡ 3 (mod 4), the group (Z/N Z)∗ /{±1} contains no element of order 2. Thus in this case the set QN is empty; and hence every primitive Fricke family {hv }v∈VN of level N is totally primitive by Theorem 3.7.
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4. Class fields over imaginary quadratic fields As for applications of Fricke families to constructing class fields, we shall review basic facts about Galois groups of class fields over imaginary quadratic fields, well known as consequences of the main theorem of the theory of complex multiplication. Let K be an imaginary quadratic field of discriminant dK (< 0), and let OK be its ring of integers. We set τK as in (2) so that τK ∈ H and OK = Z[τK ]. Let HK be the Hilbert class field of K and hK be the class number of K, that is, hK = [HK : K]. And, for a positive integer N , let K(N ) be the ray class field of K modulo (N ) = N OK . Proposition 4.1. Let K be an imaginary quadratic field and N ≥ 2. Then (i) HK = K(j(τK )). (ii) K(N ) = K(f (τK ) | f ∈ FN is finite at τK ). √ √ (iii) If K is different from Q( −1) and Q( −3), then K(N ) = HK (f
0 1/N
(τ
K )).
Proof. (i) See [11, Theorem 1 in Chapter 10]. (ii) See [11, Corollary to Theorem 2 in Chapter 10]. (iii) See [11, Corollary to Theorem 7 in Chapter 10]. See also [4] and [13, Chapter 6]. 2 Remark 4.2. Since j has Fourier expansion with respect to q = e2πiτ with rational Fourier coefficients, the singular value j(τK ) is a real number. Hence, min(j(τK ), K) has rational coefficients ([9, Remark 4.9]). Let C(dK ) be the group of reduced primitive positive definite quadratic forms Q = ax2 + bxy + cy 2 ∈ Z[x, y] of discriminant dK such that (−a < b ≤ a < c or 0 ≤ b ≤ a = c),
gcd(a, b, c) = 1 and b2 − 4ac = dK .
For each Q = ax2 + bxy + cy 2 ∈ C(dK ), let √ −b + dK . τQ = 2a Proposition 4.3. If K is an imaginary quadratic field of discriminant dK , then the map C(dK ) → Gal(HK /K) Q → (j(τK ) → j(τQ )) is an isomorphism. Proof. See [1, Theorems 5.23 and 5.30]. 2 Let min(τK , Q) = x2 + bK x + cK ∈ Z[x], and set WK, N =
t − bK s s
which is a subgroup of GL2 (Z/N Z).
−cK s ∈ GL2 (Z/N Z) | t, s ∈ (Z/N Z) t
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√ Proposition 4.4 (Shimura’s reciprocity law). Let K be an imaginary quadratic field other than Q( −1) and √ Q( −3), and let N be a positive integer. Then, the map WK, N → Gal(K(N ) /HK ) α → (f (τK ) → f α (τK ) | f ∈ FN is finite at τK ) is a surjection with kernel {±I2 }. Proof. See [16, §3].
2
5. Special values of Fricke families Let N ≥ 2 be given and {hv }v∈VN be a primitive Fricke family of level N . We shall show that if K is an imaginary quadratic field with sufficiently large |dK |, then the family of special values {hv (τK )}v∈VN generates K(N ) over K. Lemma 5.1. We have the relation ⎞3
⎛ ⎝g12 j=
0 1/2
+ 16⎠
g12 0
.
1/2
Proof. See [1, Theorem 12.17]. 2 Lemma 5.2. Let K be an imaginary quadratic field with |dK | ≥ 20 and hK ≥ 2, and let Q be a nonidentity reduced form of discriminant dK . Let g = g12 0 . We have the following inequalities: 1/2
−π |dK | (i) |g(τK )| < 4097e < 0.0033. −(1/2)π |dK | (ii) 3630e < |g(τQ )| < 19.71. 1 < 1.21e−π |dK | . (iii) j(τK ) j(τQ ) < 12.6e−(1/2)π |dK | . (iv) j(τK )
Proof. (i) See [7, Lemma 5.2]. (ii) See [7, Lemma 6.3]. (iii) We see that 1 g(τK ) = j(τK ) (g(τK ) + 16)3 by Lemma 5.1 g(τK ) ≤ 3 (−|g(τK )| + 16)
4907e−π |dK | < (−0.0033 + 16)3 < 1.21e−π
|dK |
.
by (i)
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(iv) We derive that j(τQ ) g(τQ ) + 16 3 g(τK ) = j(τK ) g(τK ) + 16 g(τQ ) by Lemma 5.1 |g(τ )| + 16 3 g(τ ) K Q ≤ −|g(τK )| + 16 g(τQ ) 3 19.71 + 16 4097e−π |dK | < by (i) and (ii) −0.0033 + 16 3630e−(1/2)π dK | < 12.6e−(1/2)π
|dK |
.
2
Lemma 5.3. Let f (x) be a nonconstant polynomial in x over Q. If |dK | is sufficiently large, then f (j(τK )) generates HK over K. Proof. Let f (x) = cm xm + cm−1 xm−1 + · · · + c1 x + c0 with cm , cm−1 , . . . , c1 , c0 ∈ Q such that cm = 0. Take |dK | large enough so as to have hK > m, and then f (j(τK )) = 0. Let Q be a nonidentity reduced form of discriminant dK . We induce from Lemma 5.2 (iii) and (iv) that the ratio
j(τQ ) m + f (j(τQ )) j(τK ) f (j(τK )) =
m−1
j(τQ ) j(τQ ) cm−1 c1 1 1 + · · · + m−1 cm j(τK ) j(τK ) cm j(τK ) j(τK ) cm−1 c1 c0 1 1 1 + cm j(τK ) + · · · + cm j(τK )m−1 + cm j(τK1 )m
+
c0 1 cm j(τK )m
converges to 0 as |dK | approaches ∞, regardless of Q. This implies that if |dK | is sufficiently large, then f (j(τK )) is different from f (j(τQ )) for every nonidentity reduced form Q, which claims that f (j(τK )) generates HK over K by Proposition 4.3. 2 Lemma 5.4. Every weakly holomorphic function in F1 is a polynomial in j over Q. Proof. See [11, Theorem 2 in Chapter 5]. 2 Remark 5.5. Let h be a modular unit in F1 . Since h and 1/h are polynomials in j over Q by Lemma 5.4, h must be a nonzero constant. Theorem 5.6. Let {hv }v∈VN be a primitive Fricke family of level N . Let K be an imaginary quadratic field √ √ other than Q( −1) and Q( −3). If |dK | is sufficiently large, then K(N ) = K (hv (τK ) | v ∈ VN / ∼) . Proof. Let F = K (hv (τK ) | v ∈ VN / ∼) which is a subfield of K(N ) by Proposition 4.1 (ii). Let h = h
(10) 0 1/N
with min(h, F1 ) = pm xm + · · · + p1 x + p0 for some pm , . . . , p1 , p0 ∈ F1 such that pm = 0. Since every hv is weakly holomorphic by (F1), we get by (F2), (F3) and Lemma 5.4 that the symmetric polynomials pm , . . . , p1 , p0 belong to Q[j]. Moreover, since
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h is a nonconstant, pr is a nonconstant polynomial in j for some r ∈ {0, 1, . . . , m}. Thus, if we take |dK | sufficiently large, then pr (τK ) generates HK over K by Lemma 5.3; and so F ⊇ HK .
(11)
If [K(N ) : HK ] = 1, then it is obvious by (11) that F = K(N ) . Now, we assume that [K(N ) : HK ] ≥ 2. Let VN / ∼ = {v1 , v2 , . . . , vn } and set
p=
(hva − hvb )2 .
1≤a
Observe by (F2), (F3), Lemma 5.4 and the primitivity of {hv }v∈VN that p is a nonzero polynomial in j over Q, namely, p = ck j k + · · · + c1 j + c0 for some ck , . . . , c1 , c0 ∈ Q such take |dK | sufficiently large so that hK > k, and then that ck = 0. Again, t − bK s −cK s ∈ WK, N /{±I2 } leaves h(τK ) fixed. Since p(τK ) = 0 and p(τK ) = 0. Suppose that α = s t 0 = h(τK ) − h(τK )α = h
0 1/N
(τ
K)
−h
αT
0 1/N
(τ
K)
= h
0 1/N
(τ
K)
− h s/N (τK ) t/N
s/N 0 ≡± (mod Z2 ). Hence α is the identity by Proposition 4.4, (F2) and (F3), we should have t/N 1/N element of WK, N /{±I2 }, which implies by Galois theory that h(τK ) generates K(N ) over HK . Therefore we conclude by (11) that F = K(N ) , as desired. 2 Remark 5.7. Moreover, if [K(N ) : HK ] = |VN / ∼ |, then we see from Proposition 4.4, (F2) and (F3) that hv (τK ) are exactly the Galois conjugates of h
0 1/N
(τ
K)
over HK (v ∈ VN / ∼). In this case, the equality (10) becomes K(N ) = K
h
0 1/N
(τ
K)
.
6. Families of Fricke functions √ √ Let K be an imaginary quadratic field other than Q( −1) and Q( −3). In this last section, we shall show that for the family {fv }v∈VN of Fricke functions, Theorem 5.6 holds without the assumption that |dK | is sufficiently large. Lemma 6.1. Let K be an imaginary quadratic field with hK ≥ 2. If Q is a nonidentity reduced form of discriminant dK , then j(τQ )2 (j(τQ ) − 1728)3 j(τK )2 (j(τK ) − 1728)3 < 1.
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Proof. It is well known that ⇐⇒
hK = 1
dK = −3, −4, −7, −8, −11, −19, −43, −67, −163
([1, Theorem 12.34]). Thus, the least |dK | with hK ≥ 2 is 15. In this case, we have C(dK ) = {Q1 = x2 + xy + 4y 2 , Q2 = 2x2 + xy + 2y 2 } and τQ1 = τK =
−1 +
√
−15
and τQ2 =
2
−1 +
√
−15
4
.
By utilizing Lemma 5.1 and some numerical estimations (using Maple 2016.2), we get that min(j(τK ), K) = x2 + 191025x − 121287375 and j(τQ1 ) = −
191025 85995 √ 191025 85995 √ − + 5 and j(τQ2 ) = − 5. 2 2 2 2
See also [3, §6.1]. It then follows that j(τQ2 )2 (j(τQ2 ) − 1728)3 −12 < 1. j(τQ )2 (j(τQ ) − 1728)3 < 2 · 10 1 1 When |dK | ≥ 20, the proof is similar to Lemma 5.2 (iv). Refer also to [7, Lemma 6.3].
2
Lemma 6.2. Let u, v ∈ Q2 − Z2 such that u ≡ ±v (mod Z2 ). We have the relation (fu − fv )6 = 212 36 j 2 (j − 1728)3
6 6 gu+v gu−v . gu12 gv12
Proof. See [8, p. 51] and [11, Theorem 2 in Chapter 18]. 2 √ √ Theorem 6.3. Let K be an imaginary quadratic field other than Q( −1) and Q( −3), and let N ≥ 2. We achieve K(N ) = K (fv (τK ) | v ∈ VN / ∼) . Proof. If hK = 1, then the theorem is immediate from Proposition 4.1 (iii). Now, we assume that hK ≥ 2. Let F = K (fv (τK ) | v ∈ VN / ∼) which is a subfield of K(N ) by Proposition 4.1 (ii). Let VN / ∼ = {v1 , v2 , . . . , vn }, and set D=
(fva − fvb )12N
1≤a
which belongs to F1 by (F2) and (F3). We then see that N n(n−1) D = 212 36 j 2 (j − 1728)3
1≤a
= c{j 2 (j − 1728)3 }N n(n−1)
gv12N g 12N a +vb va −vb gv24N gv24N a b
by Lemma 6.2
for some nonzero c ∈ Q by Remarks 2.3 and 5.5.
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We get by Lemma 6.1 that D(τK ) = D(τQ ) for every nonidentity reduced form Q of discriminant dK , which implies that D(τK ) generates HK over K by Proposition 4.3. Since D(τK ) belongs to F , we attain HK ⊆ F . Therefore, we conclude by Proposition 4.1 (iii) that F = K(N ) , as desired. 2 Lemma 6.4. If N ≥ 2 and v ∈ VN , then N 2 fv is integral over Z[j]. Proof. See [9, Lemma 5.6]. 2 Lemma 6.5. If τ0 ∈ H is imaginary quadratic, then j(τ0 ) is an algebraic integer. Proof. See [11, Theorem 4 in Chapter 5]. 2 Lemma 6.6. Let L be an abelian extension of a number field K. Suppose that L = K(u, v) for some u, v ∈ L. Let = [L : K(u)]. Then we get L = K(u + v − TrL/K(u) (v)). Proof. See [14, Theorem 8.2]. 2 As a corollary of Theorem 6.3 we obtain a primitive generator of K(N ) over K in terms of the special values of {fv }v∈VN . √ √ Corollary 6.7. Let K be an imaginary quadratic field other than Q( −1) and Q( −3), and let N ≥ 2. Then, the special value N 24N
(fva (τK ) − fvb (τK ))12N + |WK, N /{±I2 }|N 2 f
1≤a
0 1/N
(τ
K)
− N2
α∈WK, N /{±I2 }
f
αT
0 1/N
(τ
K)
becomes a primitive generator of K(N ) over K as an algebraic integer, where VN / ∼ = {v1 , v2 , . . . , vn }. Proof. Let u = N 24N
(fva (τK ) − fvb (τK ))12N
and v = N 2 f
1≤a
0 1/N
(τ
K)
which are algebraic integers by Lemmas 6.4 and 6.5. By Proposition 4.1 (iii) and the proof of Theorem 6.3, we derive K(N ) = HK (v) and HK = K(u). Note by Proposition 4.4 that Gal(K(N ) /HK ) WK, N /{±I2 }. The result follows from Lemma 6.6. 2 Remark 6.8. In a letter to Hecke in 1920s, Hasse asked the question whether the abelian extensions of K are already generated by the division values of the Weber function without the singular values of the j-function ([2, p. 91]). Ramachandra also mentioned this question later in [12, p. 105]. Since the Fricke functions are essentially induced from the Weber function ([15, Example 5.5.2 in Chapter II]), Theorem 6.3 and Corollary 6.7 could be considered as partial answers to the question of Hasse and Ramachandra for the ray class fields of K.
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