The transcendence of zeros of canonical basis elements of the space of weakly holomorphic modular forms for Γ0(2)

Journal of Number Theory 204 (2019) 423–434

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General Section

The transcendence of zeros of canonical basis elements of the space of weakly holomorphic modular forms for Γ0 (2) ✩ SoYoung Choi a , Bo-Hae Im b,∗ a

Department of Mathematics Education and RINS, Gyeongsang National University, 501 Jinjudae-ro, Jinju, 52828, South Korea b Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, South Korea

a r t i c l e

i n f o

Article history: Received 10 January 2019 Received in revised form 21 April 2019 Accepted 26 April 2019 Available online 17 May 2019 Communicated by L. Smajlovic MSC: 11F03 11F11

a b s t r a c t ε We consider the canonical basis elements fk,m for the space of weakly holomorphic modular forms of weight k for the Hecke congruence group Γ0 (2) and we prove that for all m ≥ c(k) for some constant c(k), if z0 in a fundamenε , then either z0 is in tal domain for Γ0 (2) is a zero of fk,m  √ √  i 1 i 1 i −1+i 7 1+i 7 √ , − , or z0 is transcen+ , + , , 2 2 2 2 4 4 2 dental. © 2019 Elsevier Inc. All rights reserved.

Keywords: Weakly holomorphic modular form

✩ SoYoung Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Education) (No. 2017R1D1A1A09000691), and Bo-Hae Im who is the corresponding author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2017R1A2B4002619). * Corresponding author. E-mail addresses: [email protected] (S. Choi), [email protected] (B.-H. Im).

https://doi.org/10.1016/j.jnt.2019.04.012 0022-314X/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction and the main theorem The location of zeros of certain modular forms for various discrete subgroups of SL2 (R) has been studied actively. For each even integer k > 2, Rankin and Swinnerton-Dyer ([14]) showed that the zeros in the standard fundamental domain of the Eisenstein series Ek (z) of weight k for the full modular group SL2 (Z) lie on the unit circle. Recall the definition of the Eisenstein series Ek (z) given by Ek (z) = 1 − where q = e2πiz , σk−1 (n) =

∞ ∞ 2k  2k  nk−1 q n σk−1 (n)q n = 1 − , Bk n=1 Bk n=1 1 − q n



dk−1 and Bk is the kth Bernoulli number. For the study

1≤d|n

of the location of the zeros of certain modular forms for SL2 (Z), refer to [6,12,15,8], and in the higher level cases, Shigezumi ([13]) investigated the zeros of the Eisenstein series + + Ek+ (z) for Γ+ group generated by the Hecke congruence 0 (2) and Γ0 (3). Here Γ0 (N ) is the  √  0 −1/ N . group Γ0 (N ) and the Fricke involution wN := √ N 0 Along with the study of zeros of such forms, the property of the transcendence or algebraicity of zeros has been investigated. For instance, Kohnen has showed in [11] that the nontrivial zeros of Ek in the upper half-plane are transcendental. In [6], Duke and Jenkins have constructed a canonical basis {Fk,m }m≥− for the space of weakly holomorphic modular forms of weight k for SL2 (Z). Each basis element Fk,m has the Fourier expansion of the form, Fk,m (z) = q −m +



ak,m (n)q n ,

n≥+1

where k = 12 + k with k ∈ {0, 4, 6, 8, 10, 14} and m ≥ −. Recently Garthwaite and Jenkins [7] have proved that elements in a family of basis elements for the space of weakly holomorphic modular forms for Γ0 (2) with poles only at the cusp at ∞ have many zeros lying on the lower boundary of the fundamental domain for Γ0 (2). Also, in [10] Jennings-Shaffer and Swisher proved that the zeros of Fk,m other than i and e2iπ/3 in the standard fundamental domain are transcendental for each m ≥ || −. After then Gun and Saha [9] studied the nature of zeros of weakly holomorphic modular forms for SL2 (Z) and Γ0 (p) under some assumptions. In particular, Gun and Saha [9] showed that if a nonzero weakly holomorphic modular form f for SL2 (Z) with rational Fourier coefficients has all zeros in the arc {eiθ : π/2 ≤ θ ≤ 2π/3}, then all of its zeros are either CM or transcendental, and proved an analogous result for Γ0 (p) under a certain assumption on the location of zeros. Recall that a CM point is an element in the complex upper plane H belonging to an imaginary quadratic field. Also, Gun and Saha [9, Theorem 5] have shown that all of the zeros of the Eisenstein series Ek,2 for Γ+ 0 (2) are either CM or transcendental. In this paper, we extend this result to the canonical basis elements

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425

of the space of weakly holomorphic modular forms for Γ+ 0 (2) and to the canonical basis elements of the space of weakly holomorphic modular forms for Γ0 (2) which are different from ones in [7] and we give a complete answer to the algebraicity and transcendency for them without any restriction on the location of zeros. Also, we remark that the zeros of these basis elements don’t lie in the region given in the assumption of the result [9, Theorem 4] of Gun and Saha, so our consideration is not covered by [9, Theorem 4]. We let F(2) and F+ (2) be the standard fundamental domains for Γ0 (2) and Γ+ 0 (2) respectively. Then referring to [1], they are given by

1 ≥ 1/2, −1/2 ≤ Re(z) ≤ 0 2

 1 z ∈ C : z − > 1/2, 0 ≤ Re(z) < 1/2 , 2

F(2) = z ∈ C : z + 

and   √ F+ (2) = z ∈ C : |z| ≥ 1/ 2, −1/2 ≤ Re(z) ≤ 0   √ z ∈ C : |z| > 1/ 2, 0 ≤ Re(z) < 1/2 . Following the notations in [2], for a given even integer k ∈ 2Z, let Mk! (Γ0 (2)) be the space of weakly holomorphic modular forms of weight k for Γ0 (2), and let +

Mk! (Γ0 (2)) = {f ∈ Mk! (Γ0 (2)) : f |k W2 = f }, and −

Mk! (Γ0 (2)) = {f ∈ Mk! (Γ0 (2)) : f |k W2 = −f }. Then +

−

Mk! (Γ0 (2)) = Mk! (Γ0 (2)) ⊕ Mk! (Γ0 (2)). +

We also recall that Mk! (Γ0 (2)) is the space of weakly holomorphic modular forms of weight k for Γ+ 0 (2) as in [2]. For convenience, we let ε = + or − . For a given even integer k ∈ 2Z, we can write k = 8εk + rkε ,

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for unique integers εk and rkε ∈ Z, where rk+ ∈ {0, 4, 6, 10} and rk− ∈ {2, 4, 6, 8}. Each ε ε canonical basis element fk,m for the space Mk! (Γ0 (2)) is the unique weakly holomorphic modular form with the Fourier expansion of the form ε fk,m (z) = q −mε + O(q k +1 ). ε ε

(See [3,4].) In particular, we have ε

+ ε k ε fk,m = (Δ+ ε Fk,ε +mε (j ), 2 ) Erk 2 ε k

where Fk,D (x) is a monic polynomial of degree D in x with integer coefficients and ⎧ ∞  ⎪ (1 − q n ), ⎪ η(z) = q 1/24 ⎪ ⎪ n=1 ⎪ ⎪ ⎪ ⎪Δ(z) = η(z)24 , ⎪ ⎪ ⎨ ∞ ∞   8 Δ+ (1 − q n )8 (1 − q 2m )8 , 2 (z) = (η(z)η(2z)) = q ⎪ n=1 m=1 ⎪ ⎪ ⎪ ε ε ⎪ ⎪ E (z) := E2,r (z) = 1+ε21rk /2 (Erk (z) + ε2rk /2 Erk (2z)), ⎪ k ⎪ rk    24 ⎪ 24 ⎪ ⎩j + (z) = η(z) 12 η(2z) + 24 + 2 . 2 η(2z) η(z) Let  V :=

1 √ eiθ 2

:

3π π ≤θ≤ 2 4



be the left arc boundary of F+ (2). It follows from [1] and [2] that if mε ≥ 2|εk | − εk + 8, ε then all of the zeros in F(2) of fk,m lie on V ∪ W2 (V ), since ε F(2) = F+ (2) ∪ W2 (F+ (2)) and by the transformations described in the appendix of [1] and the figure in [2] as Fig. 1. Note that the union of V and W2 (V ) forms the lower arc boundary of F+ (2), and W2 (L−1/2 ) and W2 (L1/2 ) are the right and left arc boundaries of F(2), respectively. ε In this paper we study the transcendence and algebraicity of zeros in F(2) of fk,m . ε In particular, by combining the main idea in [9] and [10], we prove the following main result. Theorem 1.1. Under the notations in the above, if mε ≥ 2|εk | − εk + 8 and z 0 is a zero √ √ −1+i 7 1+i 7 i 1 i 1 i ε √ of fk,mε in F(2), then either z0 is in , − 2 + 2, 2 + 2, , , or z0 is 4 4 2 transcendental.

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427

F+ (2) L−1/2

L1/2 V

W2 (V ) F(2) = F+ (2) ∪ W2 (F+ (2))

W2 (F+ (2)) W2 (L1/2 )

W2 (L−1/2 )

Fig. 1. The closure of the fundamental domain F(2) for Γ0 (2) as the union of the closures of F+ (2) for Γ+ 0 (2) and W2 (F+ (2)).

We note that 

i 1 1 i √ , − + , + 2 2 2 2  i i 1 = √ , − + , 2 2 2

√ √

i −1 + i 7 1 + i 7 , , 2 4 4 √  √ 

     i 1 −1 + i 7 −1 + i 7 i , W2 − + ∪ W2 √ , W2 . 4 2 2 4 2

Section 2 is devoted to the proof of Theorem 1.1. Acknowledgments. We would also like to thank KIAS (Korea Institute for Advanced Study) for its hospitality while we have worked on this result. 2. The proof of the main theorem In this section, we prove Theorem 1.1. First, recall that the j-invariant j(z) is defined by

j(z) = 1728

E4 (z)3 . E4 (z)3 − E6 (z)2

Proposition 2.1. If z is in the upper half-plane and j2+ (z) is algebraic, then z is either transcendental or imaginary quadratic. (Here, z is said to be imaginary quadratic, if Q(z) is a quadratic extension of Q with z ∈ / R.) Proof. It follows from Schneider’s Theorem in [16] and from the modular relation for Γ+ 0 (2) given by

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  j(z)2 + −t(z)2 − 49t(z) + 6656 j + t(z)3 + 816t(z)2 + 221952t(z) + 20123648 = 0, where t(z) = j2+ (z) + c for some integer c, referring to [5, page 285]. 2 ε

Proposition 2.2. If f ∈ Mk! (Γ0 (2)) has rational Fourier coefficients, then for F (z) :=



f |k γ

γ∈SL2 (Z)/Γ0 (2)

which is a weakly holomorphic modular form of weight 3k for SL2 (Z), F (z) has rational Fourier coefficients. (Here |k is the usual slash operator.) 

     1 0 0 1 1 1 ,S= and T = ∈ SL2 (Z). Since I, T and ST 0 1 −1 0 0 1 are coset representatives of SL2 (Z)/Γ0 (2), we have that

Proof. Let I =

F = f (f |k S) (f |k ST ).

(1)

So 

 F |k T = f (f |k ST ) (f |k ST 2 ) = f (f |k ST ) 

√ 1/ 2 0

f |k

1 0 −2 1

  S = F.

 √0 , we have that (f |k S)(z) = ε2−k/2 f (z/2) and (f |k ST )(z) = 2

Since S = W2  ε2−k/2 f (z + 1)/2 and this implies that F has rational Fourier coefficients. 2

ε Now, we prove Theorem 1.1. By the results in [1] and [2], the zeros of fk,m in F(2) ε lie on V ∪ W2 (V ), and moreover, it is enough to consider the zeros only in V , since if ε ε fk,m (z) = 0 for z ∈ W2 (V ) \ V , then W2 z ∈ V and fk,m (W2 z) = 0. ε ε ε For now, for a given fk,mε (z), we let ε fk,m = fk,m (z), and rk = rkε , k = εk . ε ε

+ k ε ε Then, since fk,m = (Δ+ 2 ) Erk Fk,k +mε (j2 ), the only zeros of fk,m in V are either + i 1 i ε √ , − 2 + 2 which are zeros of Erk or the zeros of Fk,k +m (j2 (z)). So it is enough to 2 consider the zeros of Fk,k +m (j2+ (z)). From now on, we suppose that z0 ∈ V is a zero of Fk,k +m (j2+ (z)) = 0. Then, since Fk,k +m (x) is a polynomial with integer coefficients, j2+ (z0 ) is an algebraic number. By Proposition 2.1, z0 is either transcendental or imaginary quadratic. Suppose that z0 is imaginary quadratic and z0 is a root of a polynomial p(x) = 2 ax + bx + c with discriminant D0 = b2 − 4ac < 0, where a > 0 and gcd(a, b, c) = 1. Consider the point z1 ∈ C defined by

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 z1 =

√ i −D0 2 √, −1+i −D0 , 2

429

if D0 ≡ 0 (mod 4), if D0 ≡ 1 (mod 4).

Following the arguments in [10, pages 4-5], we have that j(z0) and j(z1 ) are conjugate √ and so we take an automorphism σ of Q( D0 )(j(z0 )) such that σ(j(z0 )) = j(z1 ). Now we consider the weakly holomorphic modular form Fm of weight 3k for SL2 (Z) given by 

Fm =

fk,m |k γ.

γ∈SL2 (Z)/Γ0 (2)

Then, since z0 is a zero of fk,m , we see that Fm (z0 ) = 0. Noticing that Fm has rational Fourier coefficients by Proposition 2.2, from the property of the basis elements Fk,m given by the results of Duke and Jenkins in [6], Fm can be written as the product, Fm (z) = Δ(z) Ek (z)Pm (j(z)), where 3k = 12 +k for a unique integer  ∈ Z and a unique integer k ∈ {0, 4, 6, 8, 10, 14}, and Pm (x) is a polynomial with rational coefficients. Since Fm (z0 ) = 0, we have that Ek (z0 ) = 0 or Pm (j(z0 )) = 0. Then together with the following Lemma 2.3, it is enough to consider the case when Pm (j(z0 )) = 0. Lemma 2.3. If Ek (z0 ) = 0 for z0 ∈ V , then z0 = − 12 + 2i .  Proof. If Ek (z0 ) = 0, then for γ =

A C

B D

 ∈ SL2 (Z), 

z0 = γi, or z0 = γ

√  −1 3i + . 2 2

√ Ai+B 1 If z0 = γi = Ci+D ∈ V , since 1/2 ≤ Im(z0 ) = C 2 +D 2, C 2 + D2 = 2, so 2 ≤ 1/ C = ±1 andD = ±1.  So z0 = −1/2 + i/2 ∈ V . √ √ 3/2 3i −1 If z0 = γ 2 + 2 , then similarly, we have that since 1/2 ≤ Im(z0 ) = C 2 −CD+D 2 ≤ √ √ √ 1/ 2, 26 ≤ C 2 − CD + D2 ≤ 3, but there are no such integers C and D. 2 Now suppose that Pm (j(z0 )) = 0. Then we have that 0 = σ(Pm (j(z0 ))) = Pm (σ(j(z0 ))) = Pm (j(z1 )) since σ fixes Pm . So recalling (1), we have that 0 = Fm (z1 ) = fk,m (z1 )(fk,m |k S)(z1 )(fk,m |k ST )(z1 ), which implies that fk,m (z1 ) = 0 or fk,m (Sz1 ) = 0 or fk,m (ST z1 ) = 0. Due to Lemma 2.4 below, it is enough to consider each of the cases when fk,m (Sz1 ) = 0 or fk,m (ST z1 ) = 0. First, we let for a positive integer n ≥ 1,

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D0 =

 −4n, −4n + 1,

if D0 ≡ 0 (mod 4), if D0 ≡ 1 (mod 4),

so z1 =

√ i n,

√ −1+i 4n−1 , 2

if D0 ≡ 0 (mod 4), if D0 ≡ 1 (mod 4).

(2)

Lemma 2.4. If we consider z1 given in (2), then fk,m (z1 ) = 0. Proof. Since n ≥ 1, in both cases of (2) depending on D0 , z1 ∈ F+ (2) but z1 ∈ / V ∪W2 (V ), hence z1 cannot be a zero of fk,m by the results of [1,2]. 2 Now suppose that fk,m (Sz1 ) = 0 or fk,m (ST z1 ) = 0. Note that if fk,m (w) = 0, then γw ∈ V or γW2 w ∈ V for some γ ∈ Γ0 (2). Hence we complete the proof of Theorem 1.1 by considering and proving all possible cases (a)-(h) in Proposition 2.6 below depending on z1 of (2) explicitly. We note that the important part of the proof of Proposition 2.6 is to show that 1 ≤ n ≤ 4 by proving the following lemma. Lemma 2.5. Let n ∈ {1, 2, 3, 4}. Let z0 ∈ V be a zero of the quadratic equation az 2 + bz + c = 0 for integers a, b, c such that a > 0 with discriminant, b2 − 4ac = D0 < 0. Then, √ √ (i) if D0 = −4n (so z1 = i n), then z0 = 22 i with n = 2, or z0 = −1+i with n = 1, 4, 2 and √ √ (ii) if D0 = −4n + 1 (so z1 = −1+i 2 4n−1 ), then z0 = −1+4 7i with n = 2.

Proof. In order to have z0 ∈ V with a > 0, the following conditions are satisfied: √ −b + i −D0 ∈V z0 = 2a and √ √ b2 − D0 1 −D0 ≤ 1/ 2, and b ≥ 0, 1/2 ≤ = . 2 2a 4a 2 If D0 = −4n, then the conditions are b ≥ 0,

√

√ 2n ≤ a ≤ 2 n, and b2 + 4n = 2a2 .

We note that if a = 3, then n = 3, 4 and so b2 + 12 = 18 and b2 + 16 = 18 have no integral solutions. Since n ∈ {1, 2, 3, 4}, we can easily check that the possible values for n, a and b are

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431

⎧ ⎪ ⎪ ⎨n = 1, a = 2, b = 2, n = 2, a = 2, b = 0, ⎪ ⎪ ⎩n = 4, a = 4, b = 4. So in each case, we get the corresponding z0 in V . If D0 = −4n + 1, then the conditions are  b ≥ 0,

√ 2(4n − 1) 2 4n − 1 ≤a≤ , and b2 + 4n − 1 = 2a2 . 2 2

If n = 1, then there is no such integer a. If n = 3, 4, then a = 3 but b2 + 11 = 18 and b2 +√15 = 18 have no integral solutions. If n = 2, then a = 3 and b = 1. So z0 = −1+4 7i . 2 Proposition 2.6. If fk,m (Sz1 ) = 0, then we get the following (a)–(d), and if fk,m (ST z1 ) = 0, then we get the following (e)–(h). √ √ If z1 = i n and γSz1 ∈ V for some γ ∈ Γ0 (2), then z0 = 22 i with n = 2. √ √ If z1 = i n and γW2 Sz1 ∈ V for some γ ∈ Γ0 (2), then z0 = 22 i with n = 2. √ √ If z1 = −1+i 2 4n−1 and γSz1 ∈ V for some γ ∈ Γ0 (2), then z0 = −1+4 7i with n = 2. √ √ If z1 = −1+i 2 4n−1 and γW2 Sz1 ∈ V for some γ ∈ Γ0 (2), then z0 = −1+4 7i with n = 2. √ (e) If z1 = i n and γST z1 ∈ V for some γ ∈ Γ0 (2), then z0 = −1+i with n = 1. 2 √ (f ) If z1 = i n and γW2 ST z1 ∈ V for some γ ∈ Γ0 (2), then z0 = −1+i with n = 1. 2 √ √ −1+i 4n−1 −1+ 7i (g) If z1 = and γST z ∈ V for some γ ∈ Γ (2), then z = with 1 0 0 2 4 n = 2. √ √ (h) If z1 = −1+i 2 4n−1 and γW2 ST z1 ∈ V for some γ ∈ Γ0 (2), then z0 = −1+4 7i with n = 2.   A B Proof. Let γ = ∈ Γ0 (2). Then for w in the upper half-plane, the following are C D some of necessary conditions for γw to be in V :

(a) (b) (c) (d)

(i) AD − BC = 1 and C ≡ 0 (mod 2), so D = 0 and A = 0, since if D = 0 or A = 0, then −BC = 1, which is impossible for an even integer C. (ii) −1 2 ≤ Re(γw) ≤ 0. (iii) 12 ≤ Im(γw) ≤ √12 . (iv) |w|2 = 12 . For convenience, we list the following:  γS =

−B −D

A C



 , γST =

−B −D

−B + A −D + C

 ,

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 W2 S =

For (a), if z1 = equivalent to

√

√ 1/ 2 0

√0 2

ni and γSz1 =



 , W2 ST =

√ −B √ni+A −D ni+C

√  √ 1/ 2 1/ √ 2 0 2

(3)

∈ V referring to (3), the condition (iii) is

√ 1 n 1 ≤ 2 ≤√ , 2 C + nD2 2 √ 2 2 √ which implies that nD2 ≤ C +nD ≤ 2. Hence since D = 0 by the condition (i), we have n that D = ±1 and 1 ≤ n ≤ 4. If n = 1, then C = ±1 which contradicts the condition (i). If n = 2, 3, 4, then C = 0, so by (i), AD = 1 so A = ±1 = D. So γSz1 = ±B + √1n i ∈ V . √

2 By the condition (ii), B = 0 and so n = 2.  by Lemma 2.5(i), z0 = 2 i.  √Then ni √ √ For (b), if z1 = i n and γW2 Sz1 = γ √22 ∈ V referring to (3), the condition (iii)

is equivalent to √ 2 n 1 1 ≤ ≤√ , 2 nC 2 + 4D2 2 √

2

2

2

+4D which implies that nC ≤ nC2√ ≤ 2. Hence together with the condition (i), C = 0 2 n 2 √1 , which implies or C = ±2 and n = 1. If C = ±2 and n = 1, then 12 ≤ 4+4D 2 ≤ 2 that D = 0 which contradicts√the condition (i). Hence C = 0. So by the condition (i), √ √ 2 n 2 A = ±1 = D. Hence 1/2 ≤ 4 ≤ 1/ 2, so n = 1, 2. Then γW2 Sz1 = ±B + 2 i ∈ V √ and by the condition (ii), B = 0 and n = 2. Hence by Lemma 2.5(i), z0 = 22 i. √ −Bz1 +A For (c), if z1 = −1+i 2 4n−1 and γSz1 = −Dz ∈ V , the condition (iii) is equivalent 1 +C to

√ 2 4n − 1 1 1 ≤ ≤√ , 2 2 2 (2C + D) + (4n − 1)D 2 √

2

2

2

+(4n−1)D √ which implies that 4n−1D ≤ (2C+D) ≤ 2. Then since D = 0 by the 2 2 4n−1 condition (i), we have that D = ±1 and 1 ≤ n ≤ 4. Then from the inequalities, √ √ √ 2 2 4n − 1 ≤ (2C ± 1)2 + (4n − 1) ≤ 4 4n − 1 and C ≡ 0 (mod 2), we get that if n = 1 or 4, then there is no such even integer C, and so√ we have that n = 2 and −1+ 7i C = 0, ±2, or n = 3 and C = 0. By Lemma 2.5(ii), with n = 2. 4  z0 =

For (d), if z1 =

√ −1+i 4n−1 2

and γW2 Sz1 = γ

√ −1+ √ 4n−1i 2 √2

2

condition (iii) is equivalent to √ 4 4n − 1 1 1 ≤ ≤√ , 2 (4D − C)2 + (4n − 1)C 2 2

∈ V referring to (3), the

S. Choi, B.-H. Im / Journal of Number Theory 204 (2019) 423–434 √

2

2

433

2

+(4n−1)C √ which implies that 4n−1C ≤ (4D−C) ≤ 2. Then by the condition (i), C = ±2 4 4 4n−1 and n = 1 whose case is not suitable by Lemma 2.5(ii), so C = 0 and then A = ±1 = D √ by (i) whose case implies that γW2 Sz1 = ±B − 14 + 4n−1 i ∈ V . By the condition (ii), √4 4n−1 1 B = 0, so in order to satisfy (iii), we have that 2 ≤ ≤ √12 , so n = 2. Hence by 4 √

Lemma 2.5(ii), z0 = −1+4 7i with n = 2.  1  √ (z1 +1) √ 2 √ ∈ V referring For (e), if z1 = i n and γST z1 = γW2 (W2 ST )z1 = γW2 2 to (3), let z2 =

1 √ (z +1) 2 √1

2

= 12 +

√ n 2 i

which lies on the right vertical boundary {z = x +iy :

x = 1/2, y ≥ 1/2} of F+ (2). Since γW2 ∈ Γ+ 0 (2), the transformation correspondence given in Fig. 1 (referring to the appendix of [1]) implies that γST z1 = γW2 z2 ∈ V if and only if γW2 =

 T −1 , T

−1

if n = 1, or W2 ,

if n = 1.

 √  √ Hence, if n = 1, then γST z1 = T −1 12 + 2n i = −1+2 ni , which is not in V . If n = 1,     then γST z1 = T −1 12 + 12 i = −1+i = W2 12 + 12 i , which is in V . So by Lemma 2.5(i), 2 z0 = −1+i with n = 1. 2   1 √ (z1 +1) √ 2 √ ∈ V referring to (3), then by the For (f), if z1 = i n and γW2 ST z1 = γ 2 similar argument with γ instead of γW2 as in (e), we conclude that n = 1 and z0 = −1+i 2 by Lemma 2.5(i).   √ √ For (g), if z1 = −1+i 2 4n−1 and γST z1 = γW2 (W2 ST )z1 = γW2 12 −1+i 2 4n−1 ∈ V , √

√

let z2 = 12 −1+i 2 4n−1 = −1+i 4 4n−1 . If n > 2, then z2 lies in the interior of F+ (2), hence since γW2 ∈ Γ+ / V ∪W2 (V ) by the transformation correspondence 0 (2), γST z1 = γW2 z2 ∈ in [2, Appendix], which is a contradiction. Hence n = 1 or 2. By Lemma 2.5(ii), n = 2 √ −1+ 7i and z0 = . 4   √ −1+i 4n−1 2 √ 1+i 4n−1 . As 4

For (h), if z1 = 1 √

(z1 +1)

and γW2 ST z1 = γ

1 √ (z +1) 2 √1

2

∈ V referring to (3), let

z2 = 2 √2 = in (g), if n > 2, then z2 lies in the interior of F+ (2), hence since γ ∈ Γ+ / V ∪ W2 (V ) by the transformation correspondence 0 (2), γW2 ST z1 = γz2 ∈ in [2, Appendix] or Fig. 1 in Section 1, which is a contradiction. Hence n = 1 or 2. By √ −1+ 7i Lemma 2.5(ii), n = 2 and z0 = . 2 4 References [1] S. Choi, B.H. Im, On the zeros of certain weakly holomorphic modular forms for Γ+ 0 (2), J. Number Theory 166 (2016) 298–323. [2] S. Choi, B.H. Im, A family of weakly holomorphic modular forms for Γ0 (2) with all zeros on a certain geodesic, Acta Arith. (2019), in press. [3] S. Choi, C.H. Kim, Basis for the space of weakly holomorphic modular forms in higher level cases, J. Number Theory 133 (4) (2013) 1300–1311. [4] S. Choi, C.H. Kim, K.S. Lee, Arithmetic properties for the minus space of weakly holomorphic modular forms, J. Number Theory 196 (3) (2019) 309–336.

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[5] I. Chen, N. Yui, Singular values of Thompson series, in: Groups, Difference Sets, and the Monster, Columbus, OH, 1993, in: Ohio State Univ. Math. Res. Inst. Publ., vol. 4, de Gruyter, Berlin, 1996, pp. 255–326 (English summary). [6] W. Duke, P. Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl. Math. Q. 4 (4) (2008), Special Issue: In honor of Jean-Pierre Serre. Part 1, 1327–1340. [7] S.A. Garthwaite, P. Jenkins, Zeros of weakly holomorphic modular forms of levels 2 and 3, Math. Res. Lett. 20 (4) (2013) 657–674. [8] S. Gun, On the zeros of certain cusp forms, Math. Proc. Cambridge Philos. Soc. 141 (2) (2006) 191–195. [9] S. Gun, B. Saha, On the zeros of weakly holomorphic modular forms, Arch. Math. (Basel) 102 (6) (2014) 531–543 (English summary) 11F03 (11J81 11J91). [10] C. Jennings-Shaffer, H. Swisher, A note on the transcendence of zeros of a certain family of weakly holomorphic forms, Int. J. Number Theory 10 (2) (2014) 309–317 (English summary). [11] W. Kohnen, Transcendence of zero of Eisenstein series and other modular functions, Comment. Math. Univ. St. Pauli 52 (1) (2003) 55–57. [12] W. Kohnen, Zeros of Eisenstein series, Kyushu J. Math. 58 (2) (2004) 251–256. [13] T. Miezaki, H. Nozaki, J. Shigezumi, On the zeros of Eisenstein series for Γ∗0 (2) and Γ∗0 (3), J. Math. Soc. Japan 59 (3) (2007) 693–706. [14] F.K.C. Rankin, H.P.F. Swinnerton-Dyer, On the zeros of Eisenstein series, Bull. Lond. Math. Soc. 2 (1970) 169–170. [15] Z. Rudnick, On the asymptotic distribution of zeros of modular forms, Int. Math. Res. Not. (34) (2005) 2059–2074. [16] T. Schneider, Arithmetische Untersuchungen elliptischer Integral, (German) Math. Ann. 113 (1) (1937) 1–13.