2019/02/18 by Dohoon Choi, Choi, Dohoon, Subong Lim +1
Mathematics · #11F33 #11F37 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1902.06456
openalex publication_date 2019/02/18 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
Let \λ be an integer, and f(z)=\∑n\≫-\∞ a(n)qn be a\nweakly holomorphic modular form of weight \λ+ frac 12 on \Γ0(4)\nwith integral coefficients. Let \ℓ\≥ 5 be a prime. Assume that the\nconstant term a(0) is not zero modulo \ℓ. Further, assume that, for some\npositive integer m, the Fourier expansion of (f|U\ℓm)(z) =\n\∑n=0^\∞ b(n)qn has the form \(f|U
ellm)(z)
equiv b(0) +\n
sumi=1t
sumn=1
infty b(di n2) qdi n2
pmod
ell, where\nd1, \…, dt are square-free positive integers, and the operator U_\ℓ\non formal power series is defined by \
left(
sumn=0^
infty a(n)qn\n
right)
bigg| U_
ell =
sumn=0^
infty a(
ell n)qn. Then, \λ\n\≡ 0 pmod\(\ℓ-1)/(2). Moreover, if \f denotes the\ncoefficient-wise reduction of f modulo \ℓ, then we have n
biggl
limm
rightarrow
infty
tildef|U
ell2m,
limm\n
rightarrow
infty
tildef|U
ell2m+1
biggr
=
biggl
\na(0)
theta(z), a(0)
theta^
ell(z)
in
mathbbF
ell[[q]]
biggr
, where\n\θ(z) is the Jacobi theta function defined by \θ(z) =\n\∑n\∈\ℤ qn2. By using this result, we obtain the\ndistribution of the Fourier coefficients of weakly holomorphic modular forms in\ncongruence classes. This applies to the congruence properties for traces of\nsingular moduli.\n