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Class fields generated by coordinates of elliptic curves

  • Ho Yun Jung EMAIL logo , Ja Kyung Koo and Dong Hwa Shin
Published/Copyright: October 12, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

Let K be an imaginary quadratic field different from Q ( − 1 ) and Q ( − 3 ) . For a nontrivial integral ideal m of K , let K m be the ray class field modulo m . By using some inequalities on special values of modular functions, we show that a single x -coordinate of a certain elliptic curve generates K m over K .

Keywords: class field theory; complex multiplication; modular functions
MSC 2010: 11F03; 11G15; 11R37

1 Introduction

Let K be an imaginary quadratic field with ring of integers O K . Let E be the elliptic curve with complex multiplication by O K given by the Weierstrass equation:

E : y 2 = 4 x 3 − g 2 x − g 3 with g 2 = g 2 ( O K ) and g 3 = g 3 ( O K ) .

For z ∈ C , let [ z ] denote the coset z + O K in C / O K . Then the map

φ K : C / O K → E ( C ) ( ⊂ P 2 ( C ) ) [ z ] ↦ [ ℘ ( z ; O K ) : ℘ ′ ( z ; O K ) : 1 ] ,

where ℘ is the Weierstrass ℘ -function relative to O K , is a complex analytic isomorphism of complex Lie groups ([1, Proposition 3.6 in Chapter VI]). Corresponding to E , we consider the Weber function h K : E ( C ) → P 1 ( C ) given by

h K ( x , y ) = g 2 g 3 Δ x if j ( E ) ≠ 0 , 1,728 , g 2 2 Δ x 2 if j ( E ) = 1,728 , g 3 Δ x 3 if j ( E ) = 0 ,

where Δ = g 2 3 − 27 g 3 2 ( ≠ 0 ) and j ( E ) = j ( O K ) is the j -invariant of E . For a nontrivial ideal m of O K , by K m we mean the ray class field of K modulo m . In particular, K O K is the Hilbert class field H K of K . Then we obtain by the theory of complex multiplication that H K = K ( j ( E ) ) and

K m = H K ( h K ( x , y ) ) for some m -torsion point ( x , y ) on E

if m is proper [2, Chapter 10]. In a letter to Hecke concerning Kronecker’s Jugendtraum (= Hilbert 12th problem), Hasse asked whether every abelian extension of K can be generated only by a single value of the Weber function h K over K [3, p. 91] and Sugawara first gave partial answers to this question [4,5]. Recently, Jung et al. [6] proved that if m = N O K and N ∈ { 2 , 3 , 4 , 6 } , then

(1) K m = K h K φ K 1 N or K m = K h K φ K 2 N .

Koo et al. [7] further showed by utilizing the second Kronecker’s limit formula that (1) holds for N = 5 and N ≥ 7 . Besides, it is worth noting that Ramachandra [8] constructed a complicated primitive generator of K m over K by using special values of the product of high powers of the discriminant Δ function and Siegel functions, which is beautiful in theory.

Now, we assume that K is different from Q ( − 1 ) and Q ( − 3 ) , and so g 2 g 3 ≠ 0 and j ( O K ) ≠ 0 , 1,728 ([9, p. 200]). Let { E K , n } n ∈ Z ≥ 0 be the family of elliptic curves isomorphic to E given by the affine models

(2) E K , n : y 2 = 4 x 3 − J K ( J K − 1 ) 27 ℓ K 2 n x − J K ( J K − 1 ) 2 2 7 2 ℓ K 3 n ,

where

J K = 1 1,728 j ( O K ) and ℓ K = J K 2 ( J K − 1 ) 3 .

Then for each n ∈ Z ≥ 0 we have a parametrization

C / O K → E K , n ( C ) ( ⊂ P 2 ( C ) ) [ z ] ↦ [ x K , n ( z ) : y K , n ( z ) : 1 ] ,

with

x K , n ( z ) = ℓ K n g 2 g 3 Δ ℘ ( z ; O K ) and y K , n ( z ) = ℓ K n g 2 g 3 Δ 3 ℘ ′ ( z ; O K ) .

Here we note that

(3) x K , n ( z ) = ℓ K n h K ( φ K ( [ z ] ) ) ( z ∈ C ) .

Let m be a proper nontrivial ideal of O K in such a way that K m properly contains H K . Let ω be an element of K so that [ ω ] = ω + O K is a generator of the O K -module m − 1 O K / O K . In this article, we shall prove the following three assertions (Theorems 4.4, 5.2, and 6.4):

  1. We have K m = K ( x K , n ( ω ) , y K , n ( ω ) 2 ) for every n ∈ Z ≥ 0 .

  2. We obtain K m = K ( x K , n ( ω ) ) for every n ∈ Z ≥ 0 satisfying

    n ≥ 13 24 π ∣ d K ∣ + 6 ln 229 76 N m 5 2 π ∣ d K ∣ − ln 877,383 − 1 6 ,

    where d K is the discriminant of K and N m is the least positive integer in m .

  3. If m = N O K for an integer N ( ≥ 2 ) whose prime factors are all inert in K , then K m = K ( x K , n ( ω ) ) for every n ∈ Z ≥ 0 .

To this end, we shall make use of some inequalities on special values of the elliptic modular function and Siegel functions (Lemmas 4.1 and 5.1), rather than using L -function arguments adopted in [7,8,10].

Finally, we hope to utilize the aforementioned results (i), (ii), and (iii) to investigate the images of (higher dimensional) Galois representations attached to elliptic curves with complex multiplication.

2 Fricke and Siegel functions

In this preliminary section, we recall the definitions and basic properties of Fricke and Siegel functions.

Let H be the complex upper half-plane, that is, H = { τ ∈ C ∣ Im ( τ ) > 0 } . Let j be the elliptic modular function on H given by

j ( τ ) = j ( [ τ , 1 ] ) ( τ ∈ H ) ,

where [ τ , 1 ] stands for the lattice Z τ + Z in C and j ( [ τ , 1 ] ) is the j -invariant of an elliptic curve isomorphic to C / [ τ , 1 ] . Define the function J on H by

J ( τ ) = 1 1,728 j ( τ ) ( τ ∈ H ) .

Furthermore, for v = v 1 v 2 ∈ M 1 , 2 ( Q ) ⧹ M 1 , 2 ( Z ) we define the Fricke function f v on H by

f v ( τ ) = − 2 7 3 5 g 2 ( τ ) g 3 ( τ ) Δ ( τ ) ℘ ( v 1 τ + v 2 ; [ τ , 1 ] ) ( τ ∈ H ) ,

where g 2 ( τ ) = g 2 ( [ τ , 1 ] ) , g 3 ( τ ) = g 3 ( [ τ , 1 ] ) , and Δ ( τ ) = Δ ( [ τ , 1 ] ) . Note that for u , v ∈ M 1 , 2 ( Q ) ⧹ M 1 , 2 ( Z )

(4) f u = f v ⇔ u ≡ v or − v ( mod M 1 , 2 ( Z ) )

([9, Lemma 10.4]). For a positive integer N , let ℱ N be the field given by

ℱ N = Q ( j ) if N = 1 , ℱ 1 f v ∣ v ∈ 1 N M 1 , 2 ( Z ) ⧹ M 1 , 2 ( Z ) if N ≥ 2 .

Then, ℱ N is a Galois extension of ℱ 1 whose Galois group is isomorphic to GL 2 ( Z / N Z ) / ⟨ − I 2 ⟩ ([11, Theorem 6.6]). It coincides with the field of meromorphic modular functions of level N whose Fourier coefficients belong to the N th cyclotomic field ([11, Proposition 6.9]).

Proposition 2.1

If N ≥ 2 , v ∈ 1 N M 1 , 2 ( Z ) ⧹ M 1 , 2 ( Z ) , and γ ∈ GL 2 ( Z / N Z ) / ⟨ − I 2 ⟩ , then

f v γ = f v γ .

Moreover, if γ ∈ SL 2 ( Z / N Z ) / ⟨ − I 2 ⟩ , then

f v γ = f v ∘ α ,

where α is any element of SL 2 ( Z ) (acting on H as fractional linear transformation) whose image in SL 2 ( Z / N Z ) / ⟨ − I 2 ⟩ is γ .

Proof

See [2, Theorem 3 in Chapter 6] or [11, Theorem 6.6].□

For v = v 1 v 2 ∈ M 1 , 2 ( Q ) ⧹ M 1 , 2 ( Z ) , the Siegel function g v on H is given by the infinite product expansion

(5) g v ( τ ) = − e π i v 2 ( v 1 − 1 ) q τ 1 2 ( v 1 2 − v 1 + 1 6 ) ( 1 − q z ) ∏ n = 1 ∞ ( 1 − q τ n q z ) ( 1 − q τ n q z − 1 ) ( τ ∈ H ) ,

where q τ = e 2 π i τ and q z = e 2 π i z with z = v 1 τ + v 2 . Observe that g v has neither a zero nor a pole on H .

Proposition 2.2

Let N be an integer such that N ≥ 2 , and let v ∈ 1 N M 1 , 2 ( Z ) ⧹ M 1 , 2 ( Z ) .

  1. If u ∈ 1 N M 1 , 2 ( Z ) ⧹ M 1 , 2 ( Z ) such that u ≡ v or − v ( mod M 1 , 2 ( Z ) ) , then g u 12 N = g v 12 N .

  2. The function g v 12 N belongs to ℱ N and satisfies

    ( g v 12 N ) γ = g v γ 12 N ( γ ∈ GL 2 ( Z / N Z ) / ⟨ − I 2 ⟩ ≃ Gal ( ℱ N / ℱ 1 ) ) .

Proof

  1. See [12, Theorem 1.1 in Chapter 2 and p. 29].

  2. See [12, Theorem 1.2 and Proposition 1.3 in Chapter 2].□

Lemma 2.3

Let u , v ∈ M 1 , 2 ( Q ) ⧹ M 1 , 2 ( Z ) such that u ≢ v and − v ( mod M 1 , 2 ( Z ) ) . Then we have

( f u − f v ) 6 = J 2 ( J − 1 ) 3 3 9 g u + v 6 g u − v 6 g u 12 g v 12 .

Proof

See [12, p. 51].□

3 Extended form class groups

In this section, we review some necessary consequences of the theory of complex multiplication, and introduce extended form class groups which might be an extension of Gauss’ form class group.

Let K be an imaginary quadratic field of discriminant d K . For a positive integer N , let 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 , that is,

Q N ( d K ) = Q x y = a Q x 2 + b Q x y + c Q y 2 ∈ Z [ x , y ] gcd ( a Q , b Q , c Q ) = 1 , gcd ( a Q , N ) = 1 , a Q > 0 , b Q 2 − 4 a Q c Q = d K .

The congruence subgroup

Γ 1 ( N ) = γ ∈ SL 2 ( Z ) ∣ γ ≡ 1 ∗ 0 1 ( mod N M 2 ( Z ) )

defines an equivalence relation ∼ N on the set Q N ( d K ) as

Q ∼ N Q ′ ⇔ Q ′ = Q γ x y for some γ ∈ Γ 1 ( N ) .

Let

C N ( d K ) = Q N ( d K ) / ∼ N

be the set of equivalence classes. For each Q = a Q x 2 + b Q x y + c Q y 2 ∈ Q N ( d K ) , let [ Q ] N be its class in C N ( d K ) , and let

τ Q = − b Q + d K 2 a Q ,

which is the zero of the quadratic polynomial Q ( x , 1 ) lying in H . For a nontrivial ideal m of O K , let us denote by Cl ( m ) the ray class group modulo m , namely, Cl ( m ) = I K ( m ) / P K , 1 ( m ) , where I K ( m ) is the group of fractional ideals of K relatively prime to m and P K , 1 ( m ) is the subgroup of P K ( m ) (the subgroup of I K ( m ) consisting of principal fractional ideals) defined by

P K , 1 ( m ) = ⟨ ν O K ∣ ν ∈ O K ⧹ { 0 } such that ν ≡ 1 ( mod m ) ⟩ .

Then, when m = N O K , the map

C N ( d K ) → Cl ( N O K ) [ Q ] N ↦ [ [ τ Q , 1 ] ] = [ Z τ Q + Z ]

is a well-defined bijection, through which we may regard C N ( d K ) as a group isomorphic to Cl ( N O K ) ([13, Theorem 2.9] or [14, Theorem 2.5 and Proposition 5.3]). The identity element of C N ( d K ) is the class [ Q pr ] N of the principal form

Q pr ≔ x 2 + b K x y + c K y 2 = x 2 − d K 4 y 2 if d K ≡ 0 ( mod 4 ) , x 2 + x y + 1 − d K 4 y 2 if d K ≡ 1 ( mod 4 ) .

We call this group C N ( d K ) the extended form class group of discriminant d K and level N .

In particular, C 1 ( d K ) is the classical form class group of discriminant d K , originated and developed by Gauss [15] and Dirichlet [16]. A form Q = a Q x 2 + b Q x y + c Q y 2 in Q 1 ( d K ) is said to be reduced if

− a Q < b Q ≤ a Q < c Q or 0 ≤ b Q ≤ a Q = c Q .

This condition yields

(6) a Q ≤ ∣ d K ∣ 3 .

If we let Q 1 , Q 2 , … , Q h be all the reduced forms of discriminant d K , then we have h = ∣ C 1 ( d K ) ∣ and

(7) C 1 ( d K ) = { [ Q 1 ] 1 , [ Q 2 ] 1 , … , [ Q h ] 1 }

([9, Theorem 2.8]). Set

τ K = d K 2 if d K ≡ 0 ( mod 4 ) , − 1 + d K 2 if d K ≡ 1 ( mod 4 ) ,

and then τ K = τ Q pr and O K = [ τ K , 1 ] . By the theory of complex multiplication, we obtain the following results.

Proposition 3.1

Let K be an imaginary quadratic field and m be a nontrivial ideal of O K .

  1. If m = O K , then we obtain

    K m = H K = K ( j ( τ K ) ) .

    Furthermore, if Q i ( i = 1 , 2 , … , h = ∣ C 1 ( d K ) ∣ ) are reduced forms of discriminant d K , then the singular values j ( τ Q i ) are distinct (Galois) conjugates of j ( τ K ) over K .

  2. If m ≠ O K , then we have

    K m = H K ( h K ( φ K ( [ ω ] ) ) )

    for any element ω of K for which [ ω ] = ω + O K is a generator of the O K -module m − 1 O K / O K . All such h K ( φ K ( [ ω ] ) ) are conjugate over H K . More precisely, if ξ i ( i = 1 , 2 , … , [ K m : H K ] ) are nonzero elements of O K such that

    { ( ξ i ) P K , 1 ( m ) ∣ i = 1 , 2 , … , [ K m : H K ] } = P K ( m ) / P K , 1 ( m ) ( ≃ Gal ( K m / H K ) ) ,

    then h K ( φ K ( [ ξ i ω ] ) ) are all distinct conjugates of h K ( φ K ( [ ω ] ) ) over H K .

Proof

  1. See [2, Theorem 1 in Chapter 10] and [9, Theorem 7.7 (ii)].

  2. See [2, Theorem 7 and its Corollary in Chapter 10].□

By modifying Shimura’s reciprocity law ([11, Theorem 6.31, Propositions 6.33 and 6.34]), Eum et al. established the following proposition.

Proposition 3.2

Let K be an imaginary quadratic field, N be a positive integer, and K ( N ) be the ray class field of K modulo the ideal ( N ) . Then the map

C N ( d K ) → Gal ( K ( N ) / K ) [ Q ] N ↦ f ( τ K ) ↦ f a Q ( b Q − b K ) / 2 0 1 ( τ Q ) ∣ f ∈ ℱ N is f i n i t e a t τ K

is a well-defined isomorphism.

Proof

See [13, Remark 3.3 and Theorem 3.10].□

Remark 3.3

If M and N are positive integers such that M ∣ N , then the natural map

C N ( d K ) → C M ( d K ) [ Q ] N ↦ [ Q ] M

is a surjective homomorphism ([13, Remark 2.10 (i)]).

4 Some applications of inequality on singular values of j

Let K be an imaginary quadratic field of discriminant d K . By using inequality argument on singular values of j developed in [6], we show that coordinates of elliptic curves in the family { E K , n } n ∈ Z ≥ 0 described in (2) can be used in order to generate the ray class fields of K .

Let h K denote the class number of K , i.e., h K = ∣ C 1 ( d K ) ∣ = [ H K : K ] . It is well known that

h K = 1 ⇔ d K = − 3 , − 4 , − 7 , − 8 , − 11 , − 19 , − 43 , − 67 , − 163

([9, Theorem 12.34]). So, if h K ≥ 2 , then we have d K ≤ − 15 .

Lemma 4.1

If h K ≥ 2 and d K ≤ − 20 , then we achieve

(8) J ( τ Q ) 2 ( J ( τ Q ) − 1 ) 3 J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 < 877,383 ∣ q τ K ∣ 5 2 ( < 1 )

for all nonprincipal reduced forms Q of discriminant d K .

Proof

See [6, Lemma 6.3 (ii)].□

Remark 4.2

If d K = − 15 , then we obtain C 1 ( d K ) = { [ Q 1 ] 1 , [ Q 2 ] 1 } with

Q 1 = x 2 + x y + 4 y 2 and Q 2 = 2 x 2 + x y + 2 y 2 .

Moreover, we have

j ( τ K ) = j ( τ Q 1 ) = − 52,515 − 85,995 1 + 5 2 and j ( τ Q 2 ) = − 52,515 − 85,995 1 − 5 2

([1, Example 6.2.2]). One can check that inequality (8) also holds true.

Lemma 4.3

Let K be an imaginary quadratic field other than Q ( − 1 ) and Q ( − 3 ) . Then we attain

H K = K ( ℓ K n ) for e v e r y n ∈ Z > 0 .

Proof

If h K = 1 , then the assertion is obvious because H K = K .

Now, assume that h K ≥ 2 . Let σ be an element of Gal ( H K / K ) , which fixes ℓ K n . Then we find by Proposition 3.1 (i) that

1 = ( ℓ K n ) σ ℓ K n = { J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 } σ J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 n = J ( τ Q ) 2 ( J ( τ Q ) − 1 ) 3 J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 n

for some reduced form Q of discriminant d K . Thus, Q must be Q pr by Lemma 4.1 and Remark 4.2, and hence σ is the identity element of Gal ( H K / K ) again by Proposition 3.1 (i). This observation implies by the Galois theory that H K is generated by ℓ K n over K .□

Theorem 4.4

Let K be an imaginary quadratic field other than Q ( − 1 ) and Q ( − 3 ) , and let m be a nontrivial proper integral ideal of O K . Let ω be an element of K such that ω + O K is a generator of the O K -module m − 1 / O K . If K m properly contains H K , then we have

K m = K ( x K , n ( ω ) , y K , n ( ω ) 2 ) for every n ∈ Z ≥ 0 .

Proof

For simplicity, let

X = x K , n ( ω ) = ℓ K n h K ( φ K ( [ ω ] ) ) and Y = y K , n ( ω ) .

Set L = K ( X , Y 2 ) which is a subfield of K m by Proposition 3.1 and the Weierstrass equation for E K , n stated in (2).

Suppose on the contrary that K m ≠ L . Then there is a nonidentity element σ of Gal ( K m / K ) , which leaves both X and Y 2 fixed. Note further that

(9) σ ∉ Gal ( K m / H K )

because K m = H K ( X ) by Proposition 3.1 (ii). By applying σ on both sides of the equality

Y 2 = 4 X 3 − A X − B with A = J K ( J K − 1 ) 27 ℓ K 2 n and B = J K ( J K − 1 ) 2 2 7 2 ℓ K 3 n ,

we obtain

Y 2 = 4 X 3 − A σ X − B σ .

It then follows that

(10) ( A σ − A ) X = B − B σ .

Since

A B = ℓ K 5 n + 1 2 7 3

generates H K over K by Lemma 4.3, we deduce by (9) and (10) that A σ − A ≠ 0 and

X = − B σ − B A σ − A ∈ H K .

Then we obtain

H K = H K ( X ) = K m ,

which contradicts the hypothesis K m ⊋ H K .

Therefore, we conclude that

□ K m = L = K ( x K , n ( ω ) , y K , n ( ω ) 2 ) .

Proposition 4.5

Let K be an imaginary quadratic field other than Q ( − 1 ) and Q ( − 3 ) , and let m be a nontrivial proper integral ideal of O K . Let ω be an element of K such that ω + O K is a generator of the O K -module m − 1 / O K . Then we have

K m = K ( x K , n ( ω ) ) for sufficiently large n ∈ Z ≥ 0 .

Proof

Note that ℓ K = J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 is nonzero because K is different from Q ( − 1 ) and Q ( − 3 ) . There are two possible cases: h K = 1 or h K ≥ 2 .

  1. If h K = 1 (and so H K = K ), then for any n ∈ Z ≥ 0

    K ( x K , n ( ω ) ) = H K ( ℓ K n h K ( φ K ( [ ω ] ) ) ) by (3) = H K ( h K ( φ K ( [ ω ] ) ) ) by Proposition 3.1 (i) = K m by Proposition 3.1 (ii) .

  2. Consider the case where h K ≥ 2 . Let Gal ( H K / K ) = { σ 1 = id , σ 2 , … , σ h K } and d = [ K m : H K ] . Observe by Proposition 3.1 (i) that for each i = 1 , 2 , … , h K there is a unique reduced form Q i of discriminant d K , and so J ( τ K ) σ i = J ( τ Q i ) . By Lemma 4.1 and Remark 4.2 we can take a sufficiently large positive integer m so that

    ℓ K σ i ℓ K m d = J ( τ Q i ) 2 ( J ( τ Q i ) − 1 ) 3 J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 m d < N K m / H K ( h K ( φ K ( [ ω ] ) ) ) N K m / H K ( h K ( φ K ( [ ω ] ) ) ) σ i for all i = 2 , 3 , … , h K .

    We then see by (3) and Proposition 3.1 (i) that if n ∈ Z ≥ 0 satisfies n ≥ m and 2 ≤ i ≤ h K , then

    N K m / H K ( x K , n ( ω ) ) σ i N K m / H K ( x K , n ( ω ) ) = ℓ K σ i ℓ K n d N K m / H K ( h K ( φ K ( [ ω ] ) ) ) σ i N K m / H K ( h K ( φ K ( [ ω ] ) ) ) < 1 .

    This observation implies that

    (11) K ( N K m / H K ( x K , n ( ω ) ) ) = H K .

    Hence we derive that if n ≥ m , then

    □ K ( x K , n ( ω ) ) = K ( x K , n ( ω ) , N K m / H K ( x K , n ( ω ) ) ) since K ( x K , n ( ω ) ) ( ⊆ K m ) is an abelian extension of K = H K ( x K , n ( ω ) ) by (11) = K m by (3) and Proposition 3.1 .

5 Generation of ray class fields by x -coordinates of elliptic curves

By using some interesting inequalities on special values of Siegel functions, we shall find a concrete bound of n in Proposition 4.5 for which if n is greater than or equal to this, then x K , n ( ω ) generates K m over K .

Lemma 5.1

Let v ∈ M 1 , 2 ( Q ) ⧹ M 1 , 2 ( Z ) , and let τ ∈ H such that ∣ q τ ∣ = ∣ e 2 π i τ ∣ ≤ e − π 3 .

  1. We have

    ∣ g v ( τ ) ∣ < 2.29 ∣ q τ ∣ − 1 24 .

  2. If v ∈ 1 N M 1 , 2 ( Z ) ⧹ M 1 , 2 ( Z ) for an integer N ≥ 2 , then we obtain

    ∣ g v ( τ ) ∣ > 0.76 ∣ q τ ∣ 1 12 N .

Proof

Let v = v 1 v 2 and z = v 1 τ + v 2 . By Proposition 2.2 (i) we may assume that 0 ≤ v 1 ≤ 1 2 . Set s = ∣ q τ ∣ .

  1. We then derive that

    ∣ g v ( τ ) ∣ ≤ ∣ q τ ∣ 1 2 ( v 1 2 − v 1 + 1 6 ) ( 1 + ∣ q z ∣ ) ∏ n = 1 ∞ ( 1 + ∣ q τ ∣ n ∣ q z ∣ ) ( 1 + ∣ q τ ∣ n ∣ q z ∣ − 1 ) by (5) = s 1 2 ( v 1 2 − v 1 + 1 6 ) ( 1 + s v 1 ) ∏ n = 1 ∞ ( 1 + s n + v 1 ) ( 1 + s n − v 1 ) ≤ s − 1 24 ( 1 + 1 ) ∏ n = 1 ∞ ( 1 + s n − 1 2 ) 2 since 0 ≤ v 1 ≤ 1 2 and v 1 2 − v 1 + 1 6 ≥ − 1 12 ≤ 2 s − 1 24 ∏ n = 1 ∞ e 2 ( e − π 3 ) n − 1 2 because 1 + x < e x for x > 0 = 2 s − 1 24 e 2 ∑ n = 1 ∞ ( e − π 3 ) n − 1 2 = 2 s − 1 24 e 2 e − π 3 2 1 − e − π 3 < 2.29 s − 1 24 .

  2. Furthermore, we see that

    □ ∣ g v ( τ ) ∣ ≥ s 1 2 ( v 1 2 − v 1 + 1 6 ) ∣ 1 − q z ∣ ∏ n = 1 ∞ ( 1 − s n + v 1 ) ( 1 − s n − v 1 ) by (5) ≥ s 1 12 min ∣ 1 − e 2 π i N ∣ , 1 − s 1 N ∏ n = 1 ∞ ( 1 − s n − 1 2 ) 2 because v 1 2 − v 1 + 1 6 ≤ 1 6 ≥ s 1 12 min sin π N , 1 − ( e − π 3 ) 1 N ∏ n = 1 ∞ e − 4 ( e − π 3 ) n − 1 2 since 1 − x > e − 2 x for 0 < x < 1 2 > s 1 12 1 N e − 4 ∑ n = 1 ∞ ( e − π 3 ) n − 1 2 because both sin ( π x ) and 1 − e − π 3 x are > x for 0 < x ≤ 1 2 = s 1 12 1 N e − 4 e − π 3 2 1 − e − π 3 > 0.76 s 1 12 N .

Theorem 5.2

Let K be an imaginary quadratic field other than Q ( − 1 ) and Q ( − 3 ) . Let m be a proper nontrivial ideal of O K in which N m ( ≥ 2 ) is the least positive integer. Let ω be an element of K such that ω + O K is a generator of the O K -module m − 1 / O K . If K m properly contains H K , then

K m = K ( x K , n ( ω ) )

for every nonnegative integer n satisfying

(12) n ≥ 13 24 π ∣ d K ∣ + 6 ln 229 76 N m 5 2 π ∣ d K ∣ − ln 877,383 − 1 6 .

Proof

Since N m O K ⊆ m and ω ∈ m − 1 ⧹ O K , we have

(13) N m ω = a τ K + b for some a , b ∈ Z such that a b ∉ N m M 1 , 2 ( Z ) .

Let n be a nonnegative integer satisfying (12). If h K = 1 , then the assertion holds by the proof (Case 1) of Proposition 4.5.

Now, we assume h K ≥ 2 . Since K m properly contains H K , one can take a nonidentity element ρ of Gal ( K m / H K ) . Note that ρ does not fix x K , n ( ω ) due to the fact K m = H K ( x K , n ( ω ) ) by (3) and Proposition 3.1 (ii). Suppose on the contrary that x K , n ( ω ) does not generate K m over K . Then there exists at least one nonidentity element σ in Gal ( K m / K ( x K , n ( ω ) ) ) = Gal ( H K ( x K , n ( ω ) ) / K ( x K , n ( ω ) ) ) . Let P be a quadratic form in Q N m ( d K ) such that [ P ] N m maps to σ through the surjection

C N m ( d K ) ⟶ ∼ Gal ( K ( N m ) / K ) ⟶ restriction Gal ( K m / K ) μ ↦ μ ∣ K m

whose first map is the isomorphism given in Proposition 3.2. It follows from (7), Proposition 3.2, and Remark 3.3 that

P = Q γ for some nonpricipal reduced form  Q   and γ ∈ SL 2 ( Z ) .

Here we observe that

(14) τ P = γ − 1 ( τ Q ) and a Q ≥ 2 .

Then we deduce that

1 = ( x K , n ( ω ) − x K , n ( ω ) ρ ) σ x K , n ( ω ) − x K , n ( ω ) ρ because  σ  is the identity on  K ( x K , n ( ω ) )  which contains x K , n ( ω ) ρ = ( ℓ K n f u ( τ K ) − ( ℓ K n f u ( τ K ) ) ρ ) σ ℓ K n f u ( τ K ) − ( ℓ K n f u ( τ K ) ) ρ with u = a N m b N m by (3), (13), and the definitions of  h K , φ K  and a Fricke function = { J ( τ K ) 2 n ( J ( τ K ) − 1 ) 3 n ( f u ( τ K ) − f v ( τ K ) ) } σ J ( τ K ) 2 n ( J ( τ K ) − 1 ) 3 n ( f u ( τ K ) − f v ( τ K ) ) for some v ∈ 1 N m M 1 , 2 ( Z ) ⧹ M 1 , 2 ( Z ) such that u ≢ v and − v ( mod M 1 , 2 ( Z ) ) by Proposition 3.1(ii) and (4) since ρ ∈ Gal ( K m / H K ) ⧹ { id K m } = J ( τ P ) 2 ( J ( τ P ) − 1 ) 3 J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 n f u ′ ( τ P ) − f v ′ ( τ P ) f u ( τ K ) − f v ( τ K ) for some u ′ , v ′ ∈ 1 N m M 1 , 2 ( Z ) ⧹ M 1 , 2 ( Z ) such that u ′ ≢ v ′ and − v ′ ( mod M 1 , 2 ( Z ) ) by Proposition 3.2 = J ( γ − 1 ( τ Q ) ) 2 ( J ( γ − 1 ( τ Q ) ) − 1 ) 3 J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 n f u ′ ( γ − 1 ( τ Q ) ) − f v ′ ( γ − 1 ( τ Q ) ) f u ( τ K ) − f v ( τ K ) by (14) = J ( τ Q ) 2 ( J ( τ Q ) − 1 ) 3 J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 n f u ″ ( τ Q ) − f v ″ ( τ Q ) f u ( τ K ) − f v ( τ K ) with u ″ = u ′ γ − 1 and v ″ = v ′ γ − 1 by the fact  J ∈ ℱ 1  and Proposition 2.1 = J ( τ Q ) 2 ( J ( τ Q ) − 1 ) 3 J ( τ K ) 2 ( J ( τ K ) − 1 ) 3 n + 1 6 g u ″ + v ″ ( τ Q ) g u ″ − v ″ ( τ Q ) g u ( τ K ) 2 g v ( τ K ) 2 g u + v ( τ K ) g u − v ( τ K ) g u ″ ( τ Q ) 2 g v ″ ( τ Q ) 2 by Lemma 2.3 < 877,383 ∣ q τ K ∣ 5 2 n + 1 6 2.2 9 6 ∣ q τ Q ∣ − 1 12 ∣ q τ K ∣ − 1 6 0.76 N m 6 ∣ q τ K ∣ 1 6 ∣ q τ Q ∣ 1 3 by Lemmas 4.1, 5.1, and Remark 4.2 because ∣ q τ K ∣ = e − π ∣ d K ∣ ≤ e − π 15 and ∣ q τ Q ∣ = e − π ∣ d K ∣ a Q ≤ e − π 3 by (6) = 877,383 ∣ q τ K ∣ 5 2 n + 1 6 229 76 N m 6 ∣ q τ Q ∣ − 5 12 ∣ q τ K ∣ − 1 3 ≤ 877,383 ∣ q τ K ∣ 5 2 n + 1 6 229 76 N m 6 ∣ q τ K ∣ − 5 24 ∣ q τ K ∣ − 1 3 since ∣ q τ K ∣ − 1 > 1 and ∣ q τ Q ∣ − 1 = ( ∣ q τ K ∣ − 1 ) 1 a Q ≤ ( ∣ q τ K ∣ − 1 ) 1 2 by (14) = 877,383 e − 5 2 π ∣ d K ∣ n + 1 6 229 76 N m 6 e 13 24 π ∣ d K ∣ .

Now, by taking logarithm we obtain the inequality

0 < n + 1 6 ln 877,383 − 5 2 π ∣ d K ∣ + 6 ln 229 76 N m + 13 24 π ∣ d K ∣

with

ln 877,383 − 5 2 π ∣ d K ∣ ≤ ln 877,383 − 5 2 π 15 < 0 .

But this contradicts (12). Therefore, we conclude that K m = K ( x K , n ( ω ) ) .□

Example 5.3

Let K = Q ( − 5 ) , and so d K = − 20 . Note that

2 is ramified in  K since 2 ∣ d K , 13 is inert in  K due to d K 13 = − 1 , 23 splits completely in  K because d K 23 = 1

([9, Proposition 5.16]). Let m = p 1 p 2 p 3 be the product of three prime ideals

p 1 = [ 1 + − 5 , 2 ] , p 2 = [ 13 − 5 , 13 ] , and p 3 = [ 15 + − 5 , 23 ]

of O K satisfying 2 O K = p 1 2 , 13 O K = p 2 , and 23 O K = p 3 p ¯ 3 . In this case, by checking the degree formula for [ K m : H K ] we see that K m properly contains H K . Since

m = p 1 p 2 p 3 ⊃ p 1 2 p 2 p 3 p ¯ 3 = ( 2 ⋅ 13 ⋅ 23 ) O K ,

we obtain N m = 2 ⋅ 13 ⋅ 23 = 598 , and hence one can estimate

13 24 π ∣ d K ∣ + 6 ln 229 76 N m 5 2 π ∣ d K ∣ − ln 877,383 − 1 6 = 13 24 π 20 + 6 ln 229 76 ⋅ 598 5 2 π 20 − ln 877,383 − 1 6 ≈ 2.286282 .

If ω is an element of K such that ω + O K is a generator of the O K -module m − 1 / O K , then we obtain by Theorem 5.2 that

K m = K ( x K , n ( ω ) ) for all n ≥ 3 .

Remark 5.4

At this stage, we conjecture that Theorem 5.2 may hold for every n ∈ Z ≥ 0 .

6 Ray class fields of special moduli

Let K be an imaginary quadratic field other than Q ( − 1 ) and Q ( − 3 ) , and let N ( ≥ 2 ) be an integer whose prime factors are all inert in K . In this last section, we consider the special case where m = N O K and show that Theorem 5.2 is also true for every n ∈ Z ≥ 0 .

Lemma 6.1

Let f ∈ ℱ 1 . If f has neither a zero nor a pole on H , then it is a nonzero rational number.

Proof

See [2, Theorem 2 in Chapter 5] and [17, Lemma 2.1].□

For an integer N ≥ 2 , let

S N = s t ∈ M 1 , 2 ( Z ) ∣ 0 ≤ s , t < N and gcd ( N , s , t ) = 1 .

We define an equivalence relation ≡ N on the set S N as follows: for u , v ∈ S N

u ≡ N v ⇔ u ≡ v or − v ( mod N M 1 , 2 ( Z ) ) .

Let

P N = { ( u , v ) ∣ u , v ∈ S N such that u ≢ N v } and m N = ∣ P N ∣ .

Since 1 0 , 0 1 represent distinct classes in S N / ≡ N , we claim m N ≥ 2 .

Lemma 6.2

If N is an integer such that N ≥ 2 , then we have

∏ ( u , v ) ∈ P N f 1 N u − f 1 N v 6 = k { J 2 ( J − 1 ) 3 } m N for s o m e k ∈ Q ⧹ { 0 } .

Proof

For a b ∈ M 1 , 2 ( Z ) with gcd ( N , a , b ) = 1 , let π N a b denote the unique element of S N satisfying

π N a b ≡ a b ( mod N M 1 , 2 ( Z ) ) .

Let α ∈ M 2 ( Z ) with gcd ( N , det ( α ) ) = 1 , and let α ˜ be its image in GL 2 ( Z / N Z ) / ⟨ − I 2 ⟩ ( ≃ Gal ( ℱ N / ℱ 1 ) ). Setting

f = ∏ ( u , v ) ∈ P N f 1 N u − f 1 N v 6 ,

we find that

f α ˜ = ∏ ( u , v ) ∈ P N f 1 N u α ˜ − f 1 N v α ˜ 6 by Proposition 2.1 = ∏ ( u , v ) ∈ P N f 1 N π N ( u α ) − f 1 N π N ( v α ) 6 by (4) = f

because the mapping S N → S N , u ↦ π N ( u α ) , gives rise to an injection (and so, a bijection) of P N into itself. This observation implies by the Galois theory that f lies in ℱ 1 .

On the other hand, we attain

f = ∏ ( u , v ) ∈ P N J 2 ( J − 1 ) 3 3 9 g 1 N ( u + v ) 6 g 1 N ( u − v ) 6 g 1 N u 12 g 1 N v 12 by Lemma 2.3 = g J 2 ( J − 1 ) 3 3 9 m N with g = ∏ ( u , v ) ∈ P N g 1 N ( u + v ) 6 g 1 N ( u − v ) 6 g 1 N u 12 g 1 N v 12 .

Since f and J belong to ℱ 1 , so does g . Moreover, since g has neither a zero nor a pole on H , it is a nonzero rational number by Lemma 6.1. Therefore, we obtain

□ f = k { J 2 ( J − 1 ) 3 } m N for some k ∈ Q ⧹ { 0 } .

Lemma 6.3

Let K be an imaginary quadratic field and N be an integer with N ≥ 2 . If every prime factor of N is inert in K , then the principal ideal ( s τ K + t ) O K is relatively prime to N O K for all s , t ∈ Z such that gcd ( N , s , t ) = 1 .

Proof

We see that

N K / Q ( s τ K + t ) = ( s τ K + t ) ( s τ ¯ K + t ) = τ K τ ¯ K s 2 + ( τ K + τ ¯ K ) s t + t 2 = c K s 2 − b K s t + t 2 .

Now, we claim that N K / Q ( s τ K + t ) is relatively prime to N . Indeed, we have two cases: d K ≡ 0 ( mod 4 ) or d K ≡ 1 ( mod 4 ) .

Case 1. Consider the case where d K ≡ 0 ( mod 4 ) , and then b K = 0 and c K = − d K 4 . Let p be a prime factor of N . Since p is inert in K , it must be odd and satisfy d K p = − 1 . If

N K / Q ( s τ K + t ) = − d K 4 s 2 + t 2 ≡ 0 ( mod p ) ,

then the fact d K p = − 1 forces us to obtain s ≡ 0 ( mod p ) and so t ≡ 0 ( mod p ) . But this contradicts the fact gcd ( N , s , t ) = 1 . Therefore, N K / Q ( s τ K + t ) is relatively prime to p , and hence to N .

Case 2. Let d K ≡ 1 ( mod 4 ) , and so b K = 1 and c K = 1 − d K 4 . Let p be a prime factor of N . Since p is inert in K , we derive

d K ≡ 5 ( mod 8 ) if p = 2 , d K p = − 1 if p > 2 .

Then we find that

N K / Q ( s τ K + t ) = 1 − d K 4 s 2 − s t + t 2 ≡ s 2 + s t + t 2 ( mod p ) if p = 2 , 4 ′ { ( s − 2 t ) 2 − d K s 2 } ( mod p ) if p > 2 ,

where 4 ′ is an integer such that 4 ⋅ 4 ′ ≡ 1 ( mod p ) . When p = 2 , we see that s 2 + s t + t 2 ≡ 1 ( mod p ) because s and t are not both even. When p > 2 , if N K / Q ( s τ K + t ) ≡ 0 ( mod p ) , then the fact d K p = − 1 yields that s ≡ 0 ( mod p ) and so t ≡ 0 ( mod p ) . But this again contradicts gcd ( N , s , t ) = 1 . Hence N K / Q ( s τ K + t ) is relatively prime to p , and so to N . Therefore, the principal ideal ( s τ K + t ) O K is relatively prime to N O K .□

Theorem 6.4

Let K be an imaginary quadratic field other than Q ( − 1 ) and Q ( − 3 ) , and let N be an integer such that N ≥ 2 . Let ω be an element of K so that ω + O K is a generator of the O K -module N − 1 O K / O K . If every prime factor of N is inert in K , then we attain

K ( N ) = K ( x K , n ( ω ) ) for e v e r y n ∈ Z ≥ 0 .

Proof

Since K ( N ) is an abelian extension of K , K ( x K , n ( ω ) ) is also an abelian extension of K containing x K , n 1 N by Proposition 3.1 (ii). Since

( s τ K + t ) O K ∈ P K ( N O K ) for all s t ∈ S N

by Lemma 6.3, we obtain by Proposition 3.1 (ii) that

x K , n s τ K + t N ∈ K ( x K , n ( ω ) ) for all s t ∈ S N .

We then deduce that

K ( x K , n ( ω ) ) ∋ ∏ ( u , v ) ∈ P N x K , n u 1 τ K + u 2 N − x K , n v 1 τ K + v 2 N 6 with u = u 1 u 2 and v = v 1 v 2 = ∏ ( u , v ) ∈ P N ℓ K n h K φ K u 1 τ K + u 2 N − ℓ K n h K φ K v 1 τ K + v 2 N 6 by (3) = ∏ ( u , v ) ∈ P N ℓ K n 2 7 3 5 6 f 1 N u ( τ K ) − f 1 N v ( τ K ) 6 by the definitions of  φ K , h K  and Fricke functions = ℓ K n 2 7 3 5 6 m N ∏ ( u , v ) ∈ P N f 1 N u − f 1 N v 6 ( τ K ) = ℓ K n 2 7 3 5 6 m N [ k { J 2 ( J − 1 ) 3 } m N ] ( τ K ) for some  k ∈ Q ⧹ { 0 }  by Lemma 6.2 = k ℓ K 6 n + 1 2 42 3 30 m N .

Therefore, we achieve that

□ K ( x K , n ( ω ) ) = K x K , n ( ω ) , k ℓ K 6 n + 1 2 42 3 30 m N = H K ( x K , n ( ω ) ) by Lemma 4.3 = K ( N ) by (3) and Proposition 3.1 .

Acknowledgments

Ho Yun Jung was supported by the research fund of Dankook University in 2020 and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01073055). Dong Hwa Shin was supported by Hankuk University of Foreign Studies Research Fund of 2022 and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01048633).

  1. Conflict of interest: All authors state no conflict of interest.

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Received: 2021-11-07
Revised: 2022-05-04
Accepted: 2022-09-09
Published Online: 2022-10-12

© 2022 Ho Yun Jung et al., published by De Gruyter

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

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https://doi.org/10.1515/math-2022-0502
Keywords for this article
class field theory; complex multiplication; modular functions
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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