Singular moduli of rth Roots of modular functions

1 Introduction and statement of results

Singular moduli are finite values of modular functions at imaginary quadratic arguments in the complex upper half plane H . It is well known that the finite value of every modular function for any congruence subgroup at an imaginary quadratic argument lies in a finite abelian extension of an imaginary quadratic field [1, (3.7.2)]. Many authors studied singular moduli of modular functions that generate ray class fields or ring class fields of imaginary quadratic fields [1–6]. We refer [7–12] to the reader for several results related to class fields.

Let K be an imaginary quadratic field of discriminant d K < − 4 with the ring of integers Z [ α ] . Suppose that α ∈ H is a root of a quadratic equation z 2 + B z + C = 0 , where B and C are rational integers, and d K = B 2 − 4 C . The classical elliptic modular function j ( z ) is defined as follows:

The first main theorem of complex multiplication says that the singular modulus j ( α ) generates the Hilbert class field of K [4, Theorem 1]. The classical elliptic modular function j ( z ) can be written as follows:

where g 2 ( z ) and g 3 ( z ) are normalized Eisenstein series of weights 4 and 6 on SL 2 ( Z ) , respectively, and η ( z ) = q 1 ∕ 24 ∏ n = 1 ∞ ( 1 − q n ) is the Dedekind η -function. Gee [4] considered holomorphic roots j ( z ) 3 and j ( z ) − 1 2 3 . The resulting Weber functions

are modular functions on Γ ( 3 ) and Γ ( 2 ) , respectively. Gee [4, Theorem 10] showed that γ 3 ( α ) generates the Hilbert class field of K if d K is odd, and e 2 B π i 3 γ 2 ( α ) generates the Hilbert class field of K if d K is not divisible by 3.

For an imaginary quadratic order O of discriminant N 2 d K , we call O the order in K of conductor N . Let T g be a fundamental Thompson series of level N [1, Section 1], which has shape

where N , r , a , b , … are positive integers. Let t g ( z ) be an rth root of T g ( z ) . Chen and Yui [1, (3.7.5) Theorem] proved that the singular modulus T g ( α ) generates the ring class field of the order in K of conductor N , and further gave a conjecture in [1, (6.1.1) Conjecture (3)] that the singular modulus t g ( α ) generates the same ring class field as does the singular modulus T g ( α ) . But, in Remark 4.2, we give an example of a negative answer to this conjecture. In this article, when singular moduli of Hauptmodules generate ring class fields (resp. ray class fields) of imaginary quadratic fields, by generalizing the main idea in [4] given by Gee and using the theory of Shimura reciprocity law, we determine a necessary and sufficient condition for singular moduli of rth roots of the Hauptmodules to generate the same ring class fields (resp. ray class fields) as do the singular moduli of the Hauptmodules.

Through this article, we adopt the following notations:

1. A k ( Γ ) is the C -vector space of meromorphic modular forms of weight k on a congruence subgroup Γ .

2. A k ( Γ , R ) is the set of f ∈ A k ( Γ ) whose Fourier expansion has coefficients in a ring R .

3. N is a positive integer.

4. Γ 0 ( N ) ≔ { a b c d ∈ SL 2 ( Z ) ∣ c ≡ 0 ( mod N ) }

5. Γ 1 ( N ) ≔ { γ ∈ SL 2 ( Z ) ∣ γ ≡ 1 ∗ 0 1 ( mod N ) }

6. Γ ( N ) ≔ { γ ∈ SL 2 ( Z ) ∣ γ ≡ 1 0 0 1 ( mod N ) }

7. ζ N ≔ e 2 π i ∕ N .

8. F N ≔ A 0 ( Γ ( N ) , Q ( ζ N ) ) .

9. F ≔ ∪ N ∈ N F N the automorphic function field.

For a genus zero congruence subgroup Γ of SL 2 ( Z ) , we define a Hauptmodul of Γ by a meromorphic modular function f on Γ satisfying A 0 ( Γ ) = C ( f ) , and f has a unique simple pole of residue 1 at the cusp i ∞ . It is known that Γ 0 ( N ) has genus zero only for 1 ≤ N ≤ 10 and N ∈ { 12 , 13 , 16 , 18 , 25 } , Γ 1 ( N ) has genus zero only for 1 ≤ N ≤ 10 and N = 12 , and Γ ( N ) has genus zero only for 1 ≤ N ≤ 5 . The Hauptmoduls for Γ ∈ { Γ 0 ( N ) , Γ 1 ( N ) , Γ ( N ) } are known ([13, Remark 1.3], [14, Table 3], [15, p. 713], [16, p. 370], [17, Table 3], [18, (4.8)]). We record them as a reference for the reader (Tables 1, 2, 3):

Here,

1. G 2 ( z ) Eisenstein series of weight 2 on SL 2 ( Z )

2. E k ( z ) normalized Eisenstein series of weight k on SL 2 ( Z )

3. G 2 ( p ) ( z ) = G 2 ( z ) − p G 2 ( p z )

4. E 2 ( p ) = ( G 2 ( z ) − p G 2 ( p z ) ) ∕ ( 2 ζ ( 2 ) )

5. P N , ( a 1 , a 2 ) ( z ) = P L z ( ( a 1 z + a 2 ) ∕ N ) Nth division value of P , where L z = Z z + Z and P L ( z ) is the Weierstrass P -function (relative to a lattice L )

6. θ 2 ( z ) = ∑ n ∈ Z e π i ( n + 1 ∕ 2 ) 2 z

7. θ 3 ( z ) = ∑ n ∈ Z e π i n 2 z

8. θ 4 ( z ) = ∑ n ∈ Z ( − 1 ) n e π i n 2 z

(See Section 2 for the exact definitions of W N r , α , W N r , α O N , and W N r , α N .)

The remainder of this article is organized as follows: in Section 2, we review fields of modular functions and Shimura reciprocity law, which are needed in the sequel; Section 3 is devoted to the proofs of Theorems 1.1 and 1.2; and in Section 4, we provide examples.

2 Fields of modular functions and Shimura reciprocity law

Throughout this article, let K be an imaginary quadratic field of discriminant d K < − 4 with the ring of integers O = Z [ α ] . Suppose that α ∈ H is a root of a quadratic equation z 2 + B z + C = 0 , where B and C are rational integers, and d K = B 2 − 4 C . For a positive integer N , let O N be the order in K of conductor N . We denote by K ( N ) (resp. K O N ) the ray class field of K modulo N (resp. the ring class field of the order O N ). For each rational prime p , put K p ≔ K ⊗ Q Q p , O p ≔ O ⊗ Z Z p , and O N , p ≔ O N ⊗ Z Z p . Then, we have the group of finite idéles J K f ≔ ∏ p ′ K p ∗ of K . The restricted product is taken with respect to the subgroups O p ∗ ⊂ K p ∗ . Let [ ∼ , K ] denote the Artin map on J K f . Then, we have the following exact sequence of class field theory:

As in Section 1, let F N ≔ A 0 ( Γ ( N ) , Q ( ζ N ) ) and F ≔ ∪ ∈ N F N be the automorphic function field. For any subfield F ′ of F , we denote by K ( F ′ ( α ) ) the field extension of K obtained by adjoining all of the function values h ( α ) for which h ∈ F ′ is pole-free at α . From the complex multiplication theory, we have the following.

Let q α be the embedding of K into M 2 ( Q ) given by q α ( μ ) α 1 = μ α μ for any μ ∈ K . Then, q α defines an injective continuous homomorphism of J K f into ∏ p ′ G L 2 ( Q p ) (we denote it again q α ), which maps

The restricted product is taken with respect to the subgroups G L 2 ( Z p ) ⊂ G L 2 ( Q p ) . Then, q α − 1 ( ∏ p G L 2 ( Z p ) ) = ∏ p O p ∗ because α is an algebraic integer [4, p. 50].

To consider another exact sequence, we need the following: for z ∈ H let L = Z z + Z be a lattice in C , g 2 ( L ) = 60 ∑ w ∈ L − { 0 } 1 ∕ w 4 , g 3 ( L ) = 140 ∑ w ∈ L − { 0 } 1 ∕ w 6 , Δ ( L ) = g 2 ( L ) 3 − 27 g 3 ( L ) 2 , and P L ( u ) is the Weierstrass P -function for u ∈ C (relative to a lattice L ). Let N be a positive integer. For each a ∈ N − 1 Z 2 \ Z 2 , let

It is known that F N = Q ( j , f a ∣ a ∈ N − 1 Z 2 \ Z 2 ) (see [20, Proposition 6.1, p. 137]). Let u = ( u p ) ∈ ∏ p GL 2 ( Z p ) . For every positive integer N , there exists α N ∈ M 2 ( Z ) ∩ G L 2 + ( Q ) such that u p ≡ α N mod N M 2 ( Z p ) for all finite primes p of Q by the strong approximation theorem. For every a ∈ N − 1 Z 2 \ Z 2 , f a ↦ f a α N defines an element of Gal ( F N ∕ F 1 ) , hence an element of Gal ( F ∕ F 1 ) . Call it τ ( u ) [20, Proposition 6.21]. We then have an exact sequence

By combining equations (1) and (2), we obtain the diagram

Then, the following properties hold:

1. For any positive integer N , we have that ± U N is the pre-image of  Gal ( F ∕ F N ) in ∏ p G L 2 ( Z p ) with respect to τ (see [20, Proposition 6.21]).

2. If h ∈ F and s ∈ ∏ p O p ∗ , then h ( α ) [ s , K ] = h τ ( q α ( s ) − 1 ) ( α ) (see [20, Theorem 6.31]).

3. If h ∈ F and γ ∈ SL 2 ( Z ) , then h τ ( γ ) ( z ) = h ( γ z ) (see [20, Proposition 6.21]).

4. We define an element u = ( ⋯ , u p , ⋯ ) ∈ ∏ p G L 2 ( Z p ) by putting u p = 1 0 0 x p with x p ∈ Z p ∗ at all primes p . Then, for any h ∈ F having Fourier expansion with rational coefficients, we have h τ ( u ) ( z ) = h ( z ) (see [20, Exercise 6.26]).

5. Suppose that G ⊂ ∏ p G L 2 ( Z p ) is an open subgroup with fixed field F ′ ⊂ F with respect to τ . Then, the subgroup of ∏ p O p ∗ that acts trivially on K ( F ′ ( α ) ) is generated by O ∗ and q α − 1 ( G ) = { x ∈ ∏ p O p ∗ ∣ q α ( x ) ∈ G } (see [20, Proposition 6.33]).

In fact, U N ⊂ ∏ p G L 2 ( Z p ) is the kernel of the map

defined by u = ( ⋯ , u p , … ) ↦ γ such that γ ≡ u p mod N ⋅ M 2 ( Z p ) for all p . Indeed, by strong approximation theorem [21, page 67], such γ exists in G L 2 ( Z ∕ N Z ) . We denote by μ N the induced isomorphism as follows:

We note that U 0 , N (resp. { ± 1 } U N ) is an open subgroup of ∏ p G L 2 ( Z p ) with fixed field A 0 ( Γ 0 ( N ) , Q ) (resp. A 0 ( Γ ( N ) , Q ( ζ N ) ) with respect to τ . Indeed, let U = ∏ p G L 2 ( Z p ) × G L 2 + ( R ) . We have a continuous homomorphism τ ˜ extending τ defined before as follows:

for any α ∈ G L 2 + ( Q ) and h ∈ F , where h τ ˜ ( α ) is defined by h ∘ α . Since x = ( u , u ∞ ) α with u ∈ ∏ p G L 2 ( Z p ) , u ∞ ∈ G L 2 + ( R ) , and α ∈ G L 2 + ( Q ) , we put τ ˜ ( x ) = τ ( u ) τ ( α ) so that

where we define τ ( u ) to be the same as defined before. For this τ ˜ , note that for S = Q × ( U N × G L 2 + ( R ) ) (resp. S = Q × ( U 0 , N × G L 2 + ( R ) ) ), we obtain F N = { h ∈ F ∣ h τ ˜ ( x ) = h ∀ x ∈ S } (resp. A 0 ( Γ 0 ( N ) , Q ) = { h ∈ F ∣ h τ ˜ ( x ) = h ∀ x ∈ S } ) (see [20, p. 154–157]). Since Q × G L 2 + ( R ) fix every element of F (see [20, Theorem 6.23]), an open subgroup of U 0 , N (resp. { ± 1 } U N ) of ∏ p G L 2 ( Z p ) has the fixed field A 0 ( Γ 0 ( N ) , Q ) (resp. A 0 ( Γ ( N ) , Q ( ζ N ) ) ) with respect to τ .

For each positive integer M , the map q α induces a well-defined injection q α , M : q α − 1 ( U 1 ) ∕ q α − 1 ( U M ) → G L 2 ( Z ∕ M Z ) between the groups

Here, note U 1 = ∏ p G L 2 ( Z p ) . For a positive integer M with N ∣ M , let

Then, for each positive integer r (see [4, (6)]),

Note that

We consider the images of q α − 1 ( U 0 , N ) ∕ q α − 1 ( U N r ) and q α − 1 ( U N ) ∕ q α − 1 ( U N r ) under the map q α , N r .