On some extension of Gauss' work and applications

Let $K$ be an imaginary quadratic field of discriminant $d_K$, and let $\mathfrak{n}$ be a nontrivial integral ideal of $K$ in which $N$ is the smallest positive integer. Let $\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ whose leading coefficients are relatively prime to $N$. We adopt an equivalence relation $\sim_\mathfrak{n}$ on $\mathcal{Q}_N(d_K)$ so that the set of equivalence classes $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ can be regarded as a group isomorphic to the ray class group of $K$ modulo $\mathfrak{n}$. We further present an explicit isomorphism of $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ onto $\mathrm{Gal}(K_\mathfrak{n}/K)$ in terms of Fricke invariants, where $K_\mathfrak{n}$ is the ray class field of $K$ modulo $\mathfrak{n}$. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.


Introduction
For a negative integer D such that D ≡ 0 or 1 (mod 4), let Q(D) be the set of primitive positive definite binary quadratic forms Q(x, y) = ax 2 +bxy +cy 2 ∈ Z[x, y] of discriminant b 2 −4ac = D. The modular group SL 2 (Z) (or PSL 2 (Z)) acts on the set Q(D) from the right and defines the proper equivalence ∼ as Q ∼ Q ′ ⇐⇒ Q ′ = Q γ = Q γ x y for some γ ∈ SL 2 (Z).
In his celebrated work Disquisitiones Arithmeticae of 1801 ( [6]), Gauss introduced the beautiful law of composition of integral binary quadratic forms. And, it seems that he first understood the set of equivalence classes C(D) = Q(D)/ ∼ as a group. However, his original proof of the group structure is long and complicated to work in practice. After 93 years later Dirichlet ([4]) presented a different approach to the study of composition and genus theory, which seemed to be influenced by Legendre. (See [3, §3].) On the other hand, in 2004 Bhargava ([1]) derived a wonderful general law of composition on 2 × 2 × 2 cubes of integers, from which he was able to obtain Gauss' composition law on binary quadratic forms as a simple special case. Now, in this paper we will make use of Dirichlet's composition law to proceed the arguments. Given the order O of discriminant D in the imaginary quadratic field K = Q( √ D), let I(O) be the group of proper fractional O-ideals and P (O) be its subgroup of nonzero principal Oideals. When Q = ax 2 + bxy + cy 2 is an element of Q(D), let ω Q be the zero of the quadratic polynomial Q(x, 1) in H = {τ ∈ C | Im(τ ) > 0}, namely .
On the other hand, if we let H O be the ring class field of order O and j be the elliptic modular function on lattices in C, then we attain the isomorphism by the theory of complex multiplication ([3, Theorem 11.1 and Corollary 11.37] or [10, Theorem 5 in Chapter 10]). Thus, composing two isomorphisms given in (2) and (3) yields the isomorphism Now, let K be an imaginary quadratic field of discriminant d K and O K be its ring of integers. If we set then we get O K = [τ K , 1]. For a positive integer N and n = N O K , let I K (n) be the group of fractional ideals of K relatively prime to n and P K (n) be its subgroup of principal fractional ideals. Furthermore, let P K, Z (n) = {νO K | ν ∈ K * such that ν ≡ * m (mod n) for some integer m prime to N }, which are subgroups of P K (n). As for the multiplicative congruence ≡ * modulo n, we refer to [7,§IV.1]. Then the ring class field H O of order O with conductor N in K and the ray class field K n modulo n are defined to be the unique abelian extensions of K for which the Artin map modulo n induces the isomorphisms respectively ([3, §8 and §9] and [7, Chapter V]). And, for a congruence subgroup Γ of level N in SL 2 (Z), let F Γ, Q be the field of meromorphic modular functions for Γ whose Fourier expansions with respect to q 1/N = e 2πiτ /N have rational coefficients and let Then it is a subfield of the maximal abelian extension K ab of K ([17, Theorem 6.31 (i)]). In particular, for the congruence subgroups Observe that Γ may not act on Q N (d K ). Here, by Q γ we mean the action of γ as an element of SL 2 (Z). For a subgroup P of I K (n) with P K, 1 (n) ⊆ P ⊆ P K (n), let K P be the abelian extension of K so that I K (n)/P ≃ Gal(K P /K). In this paper, motivated by (4) and (6) we shall present several pairs of P and Γ for which (Propositions 4.2, 5.3 and Theorems 2. 5, 5.4). This result would be certain extension of Gauss' original work. We shall also develop an algorithm of finding distinct form classes in Q N (d K )/ ∼ Γ and give a concrete example (Proposition 6.2 and Example 6.3). To this end, we shall apply Shimura's theory which links the class field theory for imaginary quadratic fields and the theory of modular functions ( [17,Chapter 6]). And, we shall not only use but also improve the ideas of our previous work [5]. See Remarks 5.5.

Extended form class groups as ideal class groups
Let K be an imaginary quadratic field of discriminant d K and τ K be as in (5). And, let N be a positive integer, n = N O K and P be a subgroup of I K (n) satisfying P K, 1 (n) ⊆ P ⊆ P K (n). Each subgroup Γ of SL 2 (Z) defines an equivalence relation ∼ Γ on the set Q N (d K ) described in (7) in the same manner as in (8). In this section, we shall present a necessary and sufficient condition for Γ in such a way that becomes a well-defined bijection with ω Q as in (1). As mentioned in §1, the lattice [ω Q , 1] = Zω Q + Z is a fractional ideal of K.
The modular group SL 2 (Z) acts on H from the left by fractional linear transformations. For each Q ∈ Q(d K ), let I ω Q denote the isotropy subgroup of the point ω Q in SL 2 (Z). In particular, if we let Q 0 be the principal form in Q(d K ) ([3, p. 31]), then we have ω Q 0 = τ K and where S = 0 −1 1 0 and T = 1 1 0 1 . Furthermore, we see that One can readily check that if Q ′ = Q γ , then Lemma 2.1. Let Q = ax 2 + bxy + cy 2 ∈ Q(d K ). Then N K/Q ([ω Q , 1]) = 1/a and Lemma 2.2. Let Q = ax 2 + bxy + cy 2 ∈ Q N (d K ). (ii) Since P K (n)/P is a finite group, one can take an integral ideal c in the class C ([7, Lemma 2.3 in Chapter IV]). Furthermore, since O K = [aω Q , 1], we may express c as If we set u = ka, then we attain (ii) by (i).
Proposition 2.3. If the map φ Γ is well defined, then it is surjective.
Proof. Let ρ : be the natural homomorphism. Since Proposition 1.5 in Chapter IV]), the homomorphism ρ is surjective. Here, we refer to the following commutative diagram: Take γ 1 , γ 2 , . . . , γ h ∈ SL 2 (Z) so that by the isomorphism given in (2) (when D = d K ) and Lemma 2.1. Moreover, since ρ is a surjection with Ker(ρ) = P K (n)/P , we obtain the decomposition Now, let C ∈ I K (n)/P . By the decomposition (12) and Lemma 2.2 (ii) we may express C as for some i ∈ {1, 2, . . . , h} and u, v ∈ Z not both zero with gcd(N, Q ′ i (v, −u)) = 1. Take any . We then derive that This prove that φ Γ is surjective.
Proposition 2.4. The map φ Γ is a well-defined injection if and only if Γ satisfies the following property: Proof. Assume first that φ Γ is a well-defined injection. Let Q ∈ Q N (d K ) and γ ∈ SL 2 (Z) such that Q γ −1 ∈ Q N (d K ). If we set Q ′ = Q γ −1 , then we have Q = Q ′γ and so And, we deduce that Hence Γ satisfies the property (14). Conversely, assume that Γ satisfies the property (14). To show that φ Γ is well defined, suppose that Then we attain Q = Q ′α for some α ∈ Γ so that Now that Q α −1 = Q ′ ∈ Q N (d K ) and α ∈ Γ ⊆ Γ · I ω Q , we achieve by the property (14) that j(α, ω Q )O K ∈ P . Thus we derive by Lemma 2.1 and (16) that On the other hand, in order to show that φ Γ is injective, assume that .
from which it follows that Q = Q ′γ for some γ ∈ SL 2 (Z) (18) by the isomorphism in (2) when D = d K . We then derive by (17) and (18) that and so λ/j(γ, ω Q ) ∈ O * K . Therefore we attain and hence γ ∈ Γ · I ω Q by the fact Q γ −1 = Q ′ ∈ Q N (d K ) and the property (14). If we write γ = αβ for some α ∈ Γ and β ∈ I ω Q , then we see by (18) that This shows that which proves the injectivity of φ Γ .
Theorem 2.5. The map φ Γ is a well-defined bijection if and only if Γ satisfies the property (14) stated in Proposition 2.4. In this case, we may regard the set Q N (d K )/ ∼ Γ as a group isomorphic to the ideal class group I K (n)/P .
Proof. We achieve the first assertion by Propositions 2.3 and 2.4. Thus, in this case, one can give a group structure on Q N (d K )/ ∼ Γ through the bijection φ Γ : Q N (d K )/ ∼ Γ → I K (n)/P . Remark 2.6. By using the isomorphism given in (2) (when D = d K ) and Theorem 2.5 we obtain the following commutative diagram: The natural map is indeed a surjective homomorphism, which shows that the group structure of Q N (d K )/ ∼ Γ is not far from that of the classical form class group C(d K ).

Class field theory over imaginary quadratic fields
In this section, we shall briefly review the class field theory over imaginary quadratic fields established by Shimura.
For an imaginary quadratic field K, let I fin K be the group of finite ideles of K given by the restricted product Kp for all but finitely many p .
As for the topology on I fin K one can refer to [12, p. 78]. Then, the classical class field theory of K is explained by the exact sequence There is a one-to-one correspondence via the Artin map between closed subgroups J of K * of finite index containing K * and finite abelian extensions L of K such that Proof. See [12,Chapter IV].
Let N be a positive integer, n = N O K and s = (s p ) ∈ K * . For a prime p and a prime ideal p of O K lying above p, let n p (s) be a unique integer such that s p ∈ p np(s) O * Kp . We then regard sO K as the fractional ideal By the approximation theorem ([7, Chapter IV]) one can take an element ν s of K * such that Proposition 3.2. We get a well-defined surjective homomorphism with kernel Thus J n corresponds to the ray class field K n . Let F N be the field of meromorphic modular functions of level N whose Fourier expansions with respect to q 1/N have coefficients in the N th cyclotomic field Q(ζ N ) with ζ N = e 2πi/N . Then Let h(τ ) be an element of F N whose Fourier expansion is given by where γ is any element of SL 2 (Z) which maps to β through the reduction SL 2 (Z) Proof. See [17,Proposition 6.21].
F N , then we attain the exact sequence Chaper 7] or [17,Chapter 6]). Here, we note that where p runs over all rational primes by the relation By continuity, q ω can be extended to an embedding q ω, p : (K ⊗ Z Z p ) * → GL 2 (Q p ) for each prime p and hence to an embedding q ω : K * → GL 2 ( Q).
By utilizing the concept of canonical models of modular curves, Shimura achieved the following remarkable results.  (ii) CF S is the field of meromorphic modular functions for Γ S /Q * .

Construction of class invariants
Let K be an imaginary quadratic field, N be a positive integer and n = N O K . From now on, let T be a subgroup of (Z/N Z) * and P be a subgroup of P K (n) containing P K, 1 (n) given by Let Cl(P ) denote the ideal class group Cl(P ) = I K (n)/P and K P be its corresponding class field of K with Cl(P ) ≃ Gal(K P /K). Furthermore, let where t −1 stands for an integer such that tt −1 ≡ 1 (mod N ). In this section, for a given h ∈ F Γ, Q we shall define a class invariant h(C) for each class C ∈ I K (n)/P . Proof. We adopt the notations in Proposition 3.2. Given t ∈ T , let t −1 be an integer such that tt −1 ≡ 1 (mod N ). Let s = s(t) = (s p ) ∈ K * be given by Then one can take ν s = t so as to have (19), and hence Since P contains P K, 1 (n), we obtain K P ⊆ K n and Gal(K n /K P ) ≃ P/P K, 1 (n). Thus we achieve by Proposition 3.2 that the field K P corresponds to by the definitions of P K, 1 (n) and P = t∈T s(t)J n by (22) and the fact J n = Ker(φ n ) Proposition 4.2. We have K P = KF Γ, Q (τ K ).
Following the notations in Proposition 3.5 one can readily show that ) for some t ∈ T and det(W ) = Z * .
Since O K = [τ K , 1] ⊆ a −1 and ξ ∈ H, one can express We then attain by taking determinant and squaring Chapter III]). Hence, det(A) = N K/Q (a) which is relatively prime to N . For α ∈ M 2 (Z) with gcd(N, det(α)) = 1, we shall denote by α its reduction onto GL 2 (Z/N Z)/{±I 2 } (≃ Gal(F N /F 1 )).  Proof. Let a ′ be also an integral ideal in C. Take any ξ ′ 1 , ξ ′ 2 ∈ K * so that Since O K ⊆ a ′−1 and ξ ′ ∈ H, we may write Now that [a] = [a ′ ] = C, we have Then it follows that And, we obtain by (25) and (27) that and On the other hand, consider t as an integer whose reduction modulo N belongs to T . Since a, a ′ = λa ⊆ O K , we see that (λ − t)a is an integral ideal. Moreover, since λ ≡ * t (mod n) and a is relatively prime to n, we get (λ − t)a ⊆ n = N O K , and hence

Thus we attain by the facts
We then derive that (24) and (26) This yields A ′ B − tA ≡ O (mod N M 2 (Z)) and so Therefore we establish by Proposition 3.3 that (31) where · means the reduction onto GL 2 ( This prove that h(C) depends only on the class C.
Remark 4.5. If we let C 0 be the identity class in Cl(P ), then we have h(C 0 ) = h(τ K ).
Proposition 4.6. Let C ∈ Cl(P ) and h ∈ F Γ, Q . If h(C) is finite, then it belongs to K P and satisfies where σ : Cl(P ) → Gal(K P /K) is the isomorphism induced from the Artin map.
Proof. Let a be an integral ideal in C and ξ 1 , ξ 2 ∈ K * such that Then we have Furthermore, let a ′ be an integral ideal in C ′ and ξ ′′ 1 , ξ ′′ 2 ∈ K * such that Since a −1 ⊆ (aa ′ ) −1 and ξ ′′ ∈ H, we get and so it follows from (33) that Let s = (s p ) be an idele in K * satisfying where a ′ p = a ′ ⊗ Z Z p . Since a ′ is relatively prime to n = N O K , we obtain by (37) that Now, we see that which shows by (32) and (38) that q ξ, p (s −1 p ) is also a Z p -basis for (aa ′ ) −1 p by (34) and (35). Thus we achieve Letting γ = (γ p ) ∈ p GL 2 (Z p ) we get We then deduce that In particular, if we consider the case where C ′ = C −1 , then we derive that This implies that h(C) belongs to K P by Proposition 4.2. For each p ∤ N and p lying above p, we have by (37) that ord p s p = ord p a ′ , and hence [s, K]| K P = σ(C ′ ).

Extended form class groups as Galois groups
With P , K P and Γ as in §4, we shall prove our main theorem which asserts that Q N (d K )/ ∼ Γ can be regarded as a group isomorphic to Gal(K P /K) through the isomorphism described in (9).
Proof. We obtain from Q = Q γ that This claims that j(γ, ω Q ) is a unit in O K .
Remark 5.2. This lemma can be also justified by using (10), (11) and the property Proposition 5.3. For given P , the group Γ satisfies the property (14).
Here we observe that u ≡ 0 (mod N ) and v ≡ t (mod N ) for some t ∈ T.
Thus we attain And, it follows from the fact gcd(N, a) = 1 and (48) that This shows that ζ −1 j(γ, ω Q )O K ∈ P , and hence j(γ, ω Q )O K ∈ P . Therefore, the group Γ satisfies the property (14) for P .
Theorem 5.4. We have an isomorphism Proof. By Theorem 2.5 and Proposition 5.3 one may consider Q N (d K )/ ∼ Γ as a group isomorphic to I K (n)/P via the isomorphism φ Γ in §2. Let C ∈ Cl(P ) and so Note that C contains an integral ideal a = a ϕ(N ) [ω Q , 1], where ϕ is the Euler totient function. We establish by Lemma 2.2 and the definition (1) that ). We then derive by Proposition 3.3 that if h ∈ F Γ, Q is finite at τ K , then where · means the reduction onto GL 2 ( where a −1 is an integer such that aa −1 ≡ 1 (mod N ) Now, the isomorphism φ Γ followed by the isomorphism which is induced from Propositions 4.2, 4.6 and Remark 4.5, yields the isomorphism stated in (49), as desired.

Finding representatives of extended form classes
In this last section, by improving the proof of Proposition 2.3 further, we shall explain how to find all quadratic forms which represent distinct classes in Q N (d K )/ ∼ Γ . For a given Q = ax 2 + bxy + cy 2 ∈ Q N (d K ) we define an equivalence relation ≡ Q on M 1, 2 (Z) as follows: Let r s , u v ∈ M 1, 2 (Z). Then, r s ≡ Q u v if and only if Here, a −1 is an integer satisfying aa −1 ≡ 1 (mod N ).
Proposition 6.2. One can explicitly find quadratic forms representing all distinct classes in Q N (d K )/ ∼ Γ .