On a problem of Hasse and Ramachandra

Ja Kyung Koo; Dong Hwa Shin; Dong Sung Yoon

doi:10.1515/math-2019-0013

Artikel Open Access

On a problem of Hasse and Ramachandra

Ja Kyung Koo , Dong Hwa Shin und Dong Sung Yoon

Veröffentlicht/Copyright: 10. März 2019

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Aus der Zeitschrift Open Mathematics Band 17 Heft 1

Abstract

Let K be an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal of K. Hasse and Ramachandra asked whether the ray class field of K modulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = (N) for any positive integer N excluding 2, 3, 4 and 6.

Keywords: class field theory; complex multiplication; Weber function

MSC 2010: Primary 11R37; Secondary 11G15; 11G16

1 Introduction

Let K be an imaginary quadratic field with ring of integers 𝓞_K, and let E be an elliptic curve with complex multiplication by 𝓞_K. When E is given by the affine model

y2=4x3−g2x−g3with g2=g2(OK) and g3=g3(OK),

the Weber function h : ℂ/𝓞_K → ℙ¹(ℂ) is defined by

h(z)=(g22/Δ)℘(z)2if K=Q(−1),(g3/Δ)℘(z)3if K=Q(−3),(g2g3/Δ)℘(z)otherwise, (1)

where Δ=g23−27g32 and ℘(z) = ℘(z; 𝓞_K). This map gives rise to an isomorphism of E/Aut(E) onto ℙ¹(ℂ) ([8, Theorem 7 in Chapter 1]).

Let 𝔣 be a proper nontrivial ideal of 𝓞_K. We denote by H the Hilbert class field of K, and by K_𝔣 the ray class field of K modulo 𝔣. As a consequence of the main theorem of the theory of complex multiplication, Hasse proved in [4] that

H=K(j) with j=1728g23ΔandKf=Hh(z0) for some z0∈f−1. (2)

See also [8, Chapter 10]. In his letter to Hecke, Hasse further asked whether K_𝔣 can be generated by a single value of h without the j-invariant ([3, p. 91]), and Ramachandra also mentioned this problem later in [10]. It was Sugawara who first gave a partial answer to this question ([12] and [13]), however, it still remains an open question.

In this paper, through careful understanding about the characters on class groups and the second Kronecker limit formula, we shall eventually resolve Hasse-Ramachandra’s problem for 𝔣 = (N) with any positive integer N excluding 2, 3, 4 and 6 (Theorem 5.1).

2 The second Kronecker limit formula

For v=r1r2∈(Q∖Z)2, we define the (first) Fricke function f_v(τ) on the upper half-plane ℍ by

fv(τ)=g2(τ) g 3 (τ)Δ(τ)℘ (r1τ+r2), (3)

where g₂(τ) = g₂([τ, 1]), g₃(τ) = g₃([τ, 1]), Δ(τ) = Δ([τ, 1]) and ℘(z) = ℘(z; [τ, 1]). This function depends only on ±v(mod ℤ²), and is holomorphic on ℍ ([8, Chapters 3 and 6]). Furthermore, we define the Siegel function g_v(τ) on ℍ by the following infinite product

gv(τ)=−eπir2(r1−1)q(1/2)(r12−r1+1/6)(1−qr1e2πir2)∏n=1∞(1−qn+r1e2πir2)(1−qn−r1e−2πir2),

where q = e^2πiτ. If N is a positive integer so that Nv ∈ ℤ², then g_v(τ)^12N depends only on ±v(mod ℤ²), and has neither zeros nor poles on ℍ ([6, § 2.1]).

Lemma 2.1

Let u, v ∈ (ℚ ∖ ℤ)² such that u ≢ ±v(mod ℤ²). Then we have the relation

fu(τ)−fv(τ)6=j(τ)2(j(τ)−1728)3230324gu+v(τ)6gu−v(τ)6gu(τ)12gv(τ)12.

Proof

See [8, Theorem 2 in Chapter 18] and [6, p. 29 and p. 51]. □

Let K be an imaginary quadratic field, let 𝔣 be a proper nontrivial ideal of 𝓞_K and let N (> 1) be the smallest positive integer in 𝔣. We denote by Cl(𝔣) the ray class group of K modulo 𝔣. Then Gal(K_𝔣/K) is isomorphic to Cl(𝔣) via the Artin map σ = σ_𝔣 : Cl(𝔣) → Gal(K_𝔣/K). Let C ∈ Cl(𝔣). Take any integral ideal 𝔠 in the class C and express

fc−1=[ω1,ω2]for some ω1,ω2∈C such that ω=ω1ω2∈H,1=r1ω1+r2ω2for some r1,r2∈(1/N)Z.

We define the Fricke invariant f_𝔣(C) and the Siegel-Ramachandra invariant g_𝔣(C) by

ff(C)=fr1r2(ω)andgf(C)=gr1r2(ω)12N, (4)

respectively. These values depend only on the class C, not on the choices of 𝔠, ω₁ and ω₂ ([8, § 6.2 and § 6.3] and [6, § 2.1 and 11.1]).

Proposition 2.2

The invariants f_𝔣(C) and g_𝔣(C) belong to K_𝔣. Furthermore, they satisfy

ff(C)σ(C′)=ff(CC′)andgf(C)σ(C′)=gf(CC′)forall C′∈Cl(f).

Proof

See [6, Theorem 1.1 in Chapter 11]. □

Let χ be a nonprincipal character of Cl(𝔣). We define the Stickelberger element S(χ) = S_𝔣(χ) by

S(χ)=∑C∈Cl(f)χ(C)ln⁡|gf(C)|, (5)

and the L-function L_𝔣(s, χ) by

Lf(s,χ)=∑aχ([a])NK/Q(a)s(s∈C),

where 𝔞 runs over all nontrivial ideals of 𝓞_K prime to 𝔣 and [𝔞] stands for the class in Cl(𝔣) containing the ideal 𝔞. We shall denote by 𝔣_χ the conductor of the character χ.

Proposition 2.3

Let χ₀ be the primitive character of χ on Cl(𝔣_χ). If 𝔣_χ ≠ 𝓞_K, then we obtain the relation

∏p : primeidealsofOK suchthat p|f, p∤fχ(1−χ0¯([p]))Lfχ(1,χ0)=−πχ0([γdKfχ])3N(fχ)|dK|ω(fχ)Tγ(χ0¯)S(χ¯),

where 𝔡_K is the different ideal of the extension K/ℚ, γ is an element of K so that γ𝔡_K𝔣_χ is a nontrivial ideal of 𝓞_K prime to 𝔣_χ, N(𝔣_χ) is the least positive integer in ω(fχ)=|{α∈OK∗ | α≡1(mod fχ)}| and

Tγ(χ0¯)=∑α+fχ∈(OK/fχ)∗χ0¯([αOK])e2πiTrK/Q(αγ).

Proof

See [11, Theorem 9 in Chapter II] or [6, Theorem 2.1 in Chapter 11]. □

Remark 2.4

Since χ₀ is a nonprincipal character of Cl(𝔣_χ) by the assumption 𝔣_χ ≠ 𝓞_K, we have L_{𝔣_χ}(1, χ₀) ≠ 0 ([5, Theorem 10.2 in Chapter V]). Thus, if every prime ideal factor of 𝔣 divides 𝔣_χ, then we derive by Proposition 2.3 that S(χ) ≠ 0.

3 Differences of Weber functions

For an imaginary quadratic field K, fix an element τ_K of ℍ so that 𝓞_K = [τ_K, 1]. From now on, we assume that K is different from Q(−1) and Q(−3), and let N > 1. We then have j(τ_K) ≠ 0, 1728 ([1, p. 261]) and

h(r1τK+r2)=fr1r2(τK)for all r1r2∈(Q∖Z)2

by the definitions (1) and (3).

Let Hf_N be the ring class field of the order of conductor N in K. Then we have a tower of fields

K⊆H⊆HN⊆K(N)

([1, § 7]). For an integer t prime to N, by C_t = C_{N, t} we mean the class in the ray class group Cl(N) of K modulo (N) containing the ideal (t). Note that C₁ is the identity element of Cl(N).

Lemma 3.1

If t is an integer prime to N, then we get

f(N)(Ct)=f0t/N(τK)andg(N)(Ct)=g0t/N(τK)12N.

Proof

Since

(NOK)(tOK)−1=(N/t)OK=[NτK/t,N/t]and1=0(NτK/t)+(t/N)(N/t),

we deduce the lemma by the definition (4). □

For an intermediate field F of the extension K_(N)/K, we shall denote by Cl(K_(N)/F) the subgroup of Cl(N) corresponding to Gal(K_(N)/F).

Lemma 3.2

We have

Cl(K(N)/HN)={Ct | t∈(Z/NZ)∗/{±1}}≃(Z/NZ)∗/{±1}.

Proof

See [2, Proposition 3.8]. □

Let t be an integer such that

gcd(N,t)=1andt≢±1(mod N).

Note that such an integer t always exists except for the four cases N = 2, 3, 4, 6. Express (t+1)/N and (t − 1)/N as

t+1N=n+N+andt−1N=n−N−,

where n₊, N₊, n₋, N₋ are integers such that N₊, N₋ > 0 and gcd(n₊,N₊) = gcd(n₋,N₋) = 1. Observe that the condition t ≢ ±1 (mod N) is equivalent to saying that neither N₊ nor N₋ is equal to 1.

Now, we define

ξt=h(t/N)−h(1/N)12N=f0t/N(τK)−f01/N(τK)12N. (6)

Furthermore, for a character χ of Cl(N) we denote by

S(χ,ξt)=∑C∈Cl(N)χ(C)lnξtσ(C).

Lemma 3.3

If χ is nontrivial on Cl(K_(N)/H), then we obtain

Proof

We derive that

S(χ¯,ξt)=∑C∈Cl(N)χ¯(C)lnj(τK)4N(j(τK)−1728)6N260N348Nσ(C)+∑C∈Cl(N)χ¯(C)lng0n+/N+(τK)12Nσ(C)+∑C∈Cl(N)χ¯(C)lng0n−/N−(τK)12Nσ(C)−∑C∈Cl(N)χ¯(C)lng0t/N(τK)24Nσ(C)−∑C∈Cl(N)χ¯(C)lng01/N(τK)24Nσ(C)by the definition (6) and Lemma 2.1=∑B∈Cl(N)(mod Cl(K(N)/H))∑A∈Cl(K(N)/H)χ¯(AB)lnj(τK)4N(j(τK)−1728)6N260N348Nσ(AB)+(N/N+)∑B+∈Cl(N)(mod Cl(K(N)/K(N+)))∑A+∈Cl(K(N)/K(N+))χ¯(A+B+)lng(N+)(CN+,n+)σ(A+B+)+(N/N−)∑B−∈Cl(N)(mod Cl(K(N)/K(N−)))∑A−∈Cl(K(N)/K(N−))χ¯(A−B−)lng(N−)(CN−,n−)σ(A−B−)−2∑C∈Cl(N)χ¯(C)lng(N)(Ct)σ(C)−2∑C∈Cl(N)χ¯(C)lng(N)(C1)σ(C)by Lemma 3.1=∑Bχ¯(B)lnj(τK)4N(j(τK)−1728)6N260N348Nσ(B)∑Aχ¯(A)+(N/N+)∑B+χ¯(B+)lng(N+)(CN+,n+)σ(B+)∑A+χ¯(A+)+(N/N−)∑B−χ¯(B−)lng(N−)(CN−,n−)σ(B−)∑A−χ¯(A−)−2χ(Ct)∑Cχ¯(CtC)lng(N)(CtC)−2∑Cχ¯(C)lng(N)(C)by (2) and Proposition 2.2=(N/N+)∑B+χ¯(B+)lng(N+)(CN+,n+)σ(B+)∑A+χ¯(A+)+(N/N−)∑B−χ¯(B−)lng(N−)(CN−,n−)σ(B−)∑A−χ¯(A−)−2(χ(Ct)+1)S(χ¯)bytheassumptionthatχisnontrivialonCl(K(N)/H) and the definition (5).

□

4 Lemmas on characters of class groups

If we set

F=Kh(1/N)=Kf01/N(τK),

then we obtain by (2) that

Cl(K(N)/H)∩Cl(K(N)/F)=Cl(K(N)/HF)=Cl(K(N)/K(N))={C1}. (7)

In this section, we shall prove the existence of certain characters of class groups under the assumption that F is properly contained in K_(N).

Lemma 4.1

Assume that

gcd(72,N)∈{1,8,9,72}.

Then, there is a character χ of Cl(N) satisfying the following properties:

It is trivial on Cl(K_(N)/H_N).
χ(C) ≠ 1 for any chosen C ∈ Cl(K_(N)/H) ∖ Cl(K_(N)/H_N).
Every prime ideal factor of (N) divides the conductor (N)_χ.

Proof

See [7, Lemma 3.4 and Remark 4.5]. □

Lemma 4.2

Suppose that F is properly contained in K_(N). Then, there is a character ρ of Cl(N) satisfying the following properties:

It is trivial on Cl(K_(N)/H), and so (N)_ρ = 𝓞_K.
It is nontrivial on Cl(K_(N)/F).

Here, (N)_ρ stands for the conductor of the character ρ.

Proof

Since |Cl(K_(N)/F)| ≥ 2 and Cl(K_(N)/H) ∩ Cl(K_(N)/F) = {C₁} by (7), one can take a class C ∈ Cl(K_(N)/F) ∖ Cl(K_(N)/H). Thus, if we let μ : Cl(N) → Cl(N)/Cl(K_(N)/H) be the canonical homomorphism, then there is a character ψ of Cl(N)/Cl(K_(N)/H) such that ψ(μ(C)) ≠ 1.

Now, defining a character ρ of Cl(N) by ρ = ψ ∘ μ, we see that it is trivial on Cl(K_(N)/H). Since

Cl(N)/Cl(K(N)/H)≃Cl(H/K)=Cl(OK),

we get (N)_ρ = 𝓞_K. Moreover, ρ(C) = ψ(μ(C)) ≠ 1 implies that ρ is nontrivial on Cl(K_(N)/F). □

Proposition 4.3

Assume that

gcd(72,N)∈{1,8,9,72} andFisproperlycontainedinK(N). (8)

Then, there is a character χ of Cl(N) and an integer t which satisfy the following properties:

χ is nontrivial on Cl(K_(N)/F).
gcd(N, t) = 1 and t ≢ ± 1 (mod N).
S(χ, ξ_t) ≠ 0.

Proof

We divide the proof into three cases in accordance with gcd(72, N).

First, consider the case where gcd(72, N) ∈ {8, 72}. Let C be the class in Cl(N) containing the ideal ((N/2)τ_K+1). We observe by Lemma 3.2 that

C∈Gal(K(N)/K(N/2))∖Gal(K(N)/HN). (9)

Then, by Lemma 4.1 there is a character χ of Cl(N) satisfying (A1)–(A3). If χ is trivial on Cl(K_(N)/F), then we replace χ by χρ, where ρ is a character of Cl(N) given in Lemma 4.2. The new character χ is nontrivial on Cl(K_(N)/F) and preserves the properties (A1)–(A3). Take any integer t such that gcd(N, t) = 1 and t ≢ ± 1 (mod N). Since N, t+1 and t − 1 are all even, we see that N₊ and N₋ divide N/2, from which it follows that

Cl(K(N)/K(N/2))⊆Cl(K(N)/K(N+))∩Cl(K(N)/K(N−)). (10)

We then achieve that

S(χ¯,ξt)=(N/N+)∑B+∈Cl(N)(mod Cl(K(N)/K(N+)))χ¯(B+)lng(N+)(CN+,n+)σ(B+)∑A+∈Cl(K(N)/K(N+))χ¯(A+)+(N/N−)∑B−∈Cl(N)(mod Cl(K(N)/K(N−)))χ¯(B−)lng(N−)(CN−,n−)σ(B−)∑A−∈Cl(K(N)/K(N−))χ¯(A−)−2(χ(Ct)+1)S(χ¯)by Lemma 3.3=−2(χ(Ct)+1)S(χ¯)since χ is nontrivial on Cl(K(N)/K(N+)) and Cl(K(N)/K(N−))by (9), (10) and (A2)=−4S(χ¯)by (A1) and Lemma 3.2≠0by Proposition 2.3 and Remark 2.4.
Second, consider the case where gcd(72, N) = 9. If we let C be the class in Cl(N) containing the ideal ((N/3)τ_K+1), then we see that

C∈Gal(K(N)/K(N/3))∖Gal(K(N)/HN) (11)

by Lemma 3.2. By Lemma 4.1, there exists a character χ of Cl(N) satisfying (A1)–(A3). In a similar way to the above Case 1, we may assume that χ is nontrivial on Cl(K_(N)/F). Take t = 2, and then we get

n+=1, N+=N3andn−=1, N−=N.

So, we derive that

S(χ¯,ξt)=3∑B+∈Cl(N)(mod Cl(K(N)/K(N/3)))χ¯(B+)lng(N/3)(C(N/3),1)σ(B+)∑A+∈Cl(K(N)/K(N/3))χ¯(A+)+S(χ¯)−2(χ(Ct)+1)S(χ¯)by Lemma 3.3=−(2χ(Ct)+1)S(χ¯)since χ is nontrivial on Cl(K(N)/K(N/3)) by (11) and (A2)=−3S(χ¯)by (A1) and Lemma 3.2≠0by Proposition 2.3 and Remark 2.4.
Lastly, consider the case where gcd(72, N) = 1. By Lemma 4.1, there is a character χ of Cl(N) satisfying (A1)–(A3) for any chosen C ∈ Cl(K_(N)/H) ∖ Cl(K_(N)/H_N). In like manner as above, we may assume that χ is nontrivial on Cl(K_(N)/F). Take t = 2, then it follows that

n+=3, N+=Nandn−=1, N−=N.

Therefore, we obtain

S(χ¯,ξt)=χ(Cn+)S(χ¯)+S(χ¯)−2(χ(Ct)+1)S(χ¯)by Lemma 3.3=−2S(χ¯)by (A1) and Lemma 3.2≠0by Proposition 2.3 and Remark 2.4.

This proves the lemma. □

Lemma 4.4

Assume that

gcd(72,N)∈{2,3,4,6,12,18,24,36}andN≠2,3,4,6. (12)

Then, there exists an integer t satisfying the following properties:

gcd(N, t) = 1 and t ≢ ±1 (mod N).
There are prime factors p₊, p₋ of N (not necessarily distinct) such that gcd(p_±, N_±) = 1 (Note that N_± depends on the choice of t).

Proof

Let ℓ be an integer such that ℓ > 1 and gcd(6, ℓ) = 1. One can take t as listed in Table 1.

Table 1

An integer t satisfying (C1) and (C2)

N	t	N₊	N₋	p₊	p₋
12	5	2	3	3	2
18	5	3	9	2	2
24	7	3	4	2	3
36	17	2	9	3	2
2ℓ	ℓ+2	ℓ	ℓ	2	2
4ℓ	2ℓ+1	ℓ	2	2	a prime factor of ℓ
2a3bℓ witha≥0, b≥1	a solution ofx≡1(mod 2aℓ),x≡−1(mod 3b)	a divisor of 2^aℓ	a divisor of 3^b	3	a prime factor of ℓ

Let (N) = ∏_𝔭𝔭^n_𝔭 be the prime ideal factorization of (N). Then we get

[K(N):H]=ω(N)2∏p|(N)NK/Q(p)−1NK/Q(p)np−1,

where ω(N) is the number of roots of unity in K which are congruent to 1 modulo (N) ([8, Theorem 1 in Chapter VI]). One can then readily deduce that

K(N)=K(M) for a proper divisor M of N⟺ 2∥N and 2 splits in K.

In this case, we have

K(N)=K(N/2). (13)

Furthermore, it is well known that

[HN:H]=N∏p|N1−dKp1p, (14)

where (d_K/p) is the Legendre symbol for an odd prime p, and (d_K/2) is the Kronecker symbol ([1, Theorem 7.24]).

Lemma 4.5

Assume that if 2 ∥N, then 2 does not split in K. Let p be a prime factor of N with p^e ∥ N. Then, there is a nontrivial character χ_p of Cl(N) satisfying the following properties:

It is trivial on Cl(K_(N)/H_p^e), and so (N)_{χ_p} divides (p^e).
(N)_{χ_p} is divisible by every prime ideal factor of (p).

Proof

Note that the assumption implies [H_p^e : H] ≥ 2 by (14). Therefore, the lemma is an immediate consequence of [7, Lemma 3.3]. □

Proposition 4.6

Assume that

Nsatisfies(12) andFisproperlycontainedinK(N).

Under this assumption instead of (8), Proposition 4.3 also holds.

Proof

Let

χ=∏p|Nχp,

where χ_p is a character of Cl(N) given in Lemma 4.5 for each prime factor p of N. If χ is trivial on Cl(K_(N)/F), then we replace χ by χρ where ρ is a character of Cl(N) given in Lemma 4.2. Then, χ satisfies the following properties:

It is trivial on Cl(K_(N)/H_N).
It is nontrivial on Cl(K_(N)/F).
(N)_χ is divisible by every prime ideal factor of (N).

Now, take an integer t satisfying (C1) and (C2) in Lemma 4.4. We then derive that

S(χ¯,ξt)=(N/N+)∑B+∈Cl(N)(mod Cl(K(N)/K(N+)))χ¯(B+)lng(N+)(CN+,n+)σ(B+)∑A+∈Cl(K(N)/K(N+))χ¯(A+)+(N/N−)∑B−∈Cl(N)(mod Cl(K(N)/K(N−)))χ¯(B−)lng(N−)(CN−,n−)σ(B−)∑A−∈Cl(K(N)/K(N−))χ¯(A−)−2(χ(Ct)+1)S(χ¯)by Lemma 3.3=−4S(χ¯)because χ is nontrivial on Cl(K(N)/K(N+)) and Cl(K(N)/K(N−))by (iii) and (C2) and χ(Ct)=1 by (i) and Lemma 3.2≠0by Proposition 2.3, Remark 2.4 and (iii).

5 Main theorem

Now, we are ready to prove our main theorem. Note by (2) that the problem of Hasse and Ramachandra is trivial if the class number of K is one.

Theorem 5.1

Let K be an imaginary quadratic field other than Q(−1)andQ(−3), and let N > 1 be an integer such that N ≠ 2, 3, 4, 6. Then we have

K(N)=K(h(1/N))if2∦Nor2doesnotsplitinK,K(h(2/N))otherwise.

Proof

First, consider the case where 2 ∦ N or 2 does not split in K. Suppose on the contrary that F = K (h(1/N)) is properly contained in K_(N). Then, by Propositions 4.3 and 4.6, there exist a character χ of Cl(N) and an integer t such that

χ is nontrivial on Cl(K_(N)/F),
gcd(N, t) = 1 and t ≢ ±1 (mod N),
S(χ, ξ_t) ≠ 0.

On the other hand, since F is a Galois extension of K, it contains the Galois conjugate h(1/N)^σ(C_t) of h(1/N). We then see by Proposition 2.2 and Lemma 3.1 that

h(1/N)σ(Ct)=f01/N(τK)σ(Ct)=f(N)(C1)σ(Ct)=f(N)(Ct)=f0t/N(τK)=h(t/N).

Thus F contains the element ξ_t = (h(t/N) − h(1/N))^12N. Now, we derive that

S(χ¯,ξt)=∑C∈Cl(N)χ¯(C)lnξtσ(C)=∑B∈Cl(N)(mod Cl(K(N)/F))∑A∈Cl(K(N)/F)χ¯(AB)lnξtσ(AB)=∑B∈Cl(N)(mod Cl(K(N)/F))χ¯(B)lnξtσ(B)∑A∈Cl(K(N)/F)χ¯(A)because ξt∈F=0by (B1),

which contradicts (B3). Hence, we have K_(N) = K(h(1/N)) as desired.

Second, consider the case where 2 ∥ N and 2 splits in K. Then we have

K(N)=K(N/2)as mentioned in (13)=K(h(2/N))by the first case of the theorem.

This completes the proof. □

Acknowledgement

The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (2017R1A2B1006578), and by Hankuk University of Foreign Studies Research Fund of 2018. The third (corresponding) author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2017R1D1A1B03030015), and Pusan National University Research Grant, 2018.

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Received: 2018-10-04

Accepted: 2019-01-10

Published Online: 2019-03-10

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0013

Schlagwörter für diesen Artikel

class field theory; complex multiplication; Weber function

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