Abstract
Let λ be an integer, and f(z) = ∑ n ≫ - ∞a(n) qn be a weakly holomorphic modular form of weightλ+12 on Γ (4) with integral coefficients. Let ℓ≥ 5 be a prime. Assume that the constant term a(0) is not zero modulo ℓ. Further, assume that, for some positive integer m, the Fourier expansion of (f|Uℓm)(z)=∑n=0∞b(n)qn has the form (f|Uℓm)(z)≡b(0)+∑i=1t∑n=1∞b(din2)qdin2(modℓ),where d1, … , dt are square-free positive integers, and the operator Uℓ on formal power series is defined by (∑n=0∞a(n)qn)|Uℓ=∑n=0∞a(ℓn)qn.Then, λ≡0(modℓ-12). Moreover, if f~ denotes the coefficient-wise reduction of f modulo ℓ, then we have {limm→∞f~|Uℓ2m,limm→∞f~|Uℓ2m+1}={a(0)θ(z),a(0)θℓ(z)∈Fℓ[[q]]},where θ(z) is the Jacobi theta function defined by θ(z)=∑n∈Zqn2. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.
| Original language | English |
|---|---|
| Journal | Ramanujan Journal |
| DOIs | |
| Publication status | Published - 2019 Jan 1 |
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Keywords
- Congruence
- Modular forms of half-integral weight
- Trace of singular moduli
ASJC Scopus subject areas
- Algebra and Number Theory
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Congruences involving the Uℓ operator for weakly holomorphic modular forms. / Choi, Dohoon; Lim, Subong.
In: Ramanujan Journal, 01.01.2019.Research output: Contribution to journal › Article
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TY - JOUR
T1 - Congruences involving the Uℓ operator for weakly holomorphic modular forms
AU - Choi, Dohoon
AU - Lim, Subong
PY - 2019/1/1
Y1 - 2019/1/1
N2 - Let λ be an integer, and f(z) = ∑ n ≫ - ∞a(n) qn be a weakly holomorphic modular form of weightλ+12 on Γ (4) with integral coefficients. Let ℓ≥ 5 be a prime. Assume that the constant term a(0) is not zero modulo ℓ. Further, assume that, for some positive integer m, the Fourier expansion of (f|Uℓm)(z)=∑n=0∞b(n)qn has the form (f|Uℓm)(z)≡b(0)+∑i=1t∑n=1∞b(din2)qdin2(modℓ),where d1, … , dt are square-free positive integers, and the operator Uℓ on formal power series is defined by (∑n=0∞a(n)qn)|Uℓ=∑n=0∞a(ℓn)qn.Then, λ≡0(modℓ-12). Moreover, if f~ denotes the coefficient-wise reduction of f modulo ℓ, then we have {limm→∞f~|Uℓ2m,limm→∞f~|Uℓ2m+1}={a(0)θ(z),a(0)θℓ(z)∈Fℓ[[q]]},where θ(z) is the Jacobi theta function defined by θ(z)=∑n∈Zqn2. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.
AB - Let λ be an integer, and f(z) = ∑ n ≫ - ∞a(n) qn be a weakly holomorphic modular form of weightλ+12 on Γ (4) with integral coefficients. Let ℓ≥ 5 be a prime. Assume that the constant term a(0) is not zero modulo ℓ. Further, assume that, for some positive integer m, the Fourier expansion of (f|Uℓm)(z)=∑n=0∞b(n)qn has the form (f|Uℓm)(z)≡b(0)+∑i=1t∑n=1∞b(din2)qdin2(modℓ),where d1, … , dt are square-free positive integers, and the operator Uℓ on formal power series is defined by (∑n=0∞a(n)qn)|Uℓ=∑n=0∞a(ℓn)qn.Then, λ≡0(modℓ-12). Moreover, if f~ denotes the coefficient-wise reduction of f modulo ℓ, then we have {limm→∞f~|Uℓ2m,limm→∞f~|Uℓ2m+1}={a(0)θ(z),a(0)θℓ(z)∈Fℓ[[q]]},where θ(z) is the Jacobi theta function defined by θ(z)=∑n∈Zqn2. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.
KW - Congruence
KW - Modular forms of half-integral weight
KW - Trace of singular moduli
UR - http://www.scopus.com/inward/record.url?scp=85069820621&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85069820621&partnerID=8YFLogxK
U2 - 10.1007/s11139-019-00154-z
DO - 10.1007/s11139-019-00154-z
M3 - Article
AN - SCOPUS:85069820621
JO - Ramanujan Journal
JF - Ramanujan Journal
SN - 1382-4090
ER -