\(p\)-adic properties of coefficients of basis for the space of weakly holomorphic modular forms of weight 2
In [\textit{W. Duke} and \textit{P. Jenkins}, Pure Appl. Math. Q. 4, No. 4, 1327--1340 (2008; Zbl 1200.11027)], a canonical basis \(\{f_{k,m}\}\) of the space of weakly holomorphic modular forms of weight \(k\) for \(\text{SL}_2(\mathbb Z)\) was constructed. The functions \(f_{k,m}\) are characterized by \(q\)-expansions:\(f_{k,m}=q^{-m}+\sum_{n\geq 0}^\infty a_k(m,n)q^n\), \(q=e^{2\pi i z},a_k(m,n)\in\mathbb Z\). Further in [\textit{D. Doud} and \textit{P. Jenkins}, Int. Math. Res. Not. 2010, No. 16, 3184--3206 (2010; Zbl 1201.11047)] it was showed that Fourier coefficients \(a_k(m,n)\) are often highly divisible by small primes \(2,3,5\) when \(k=4,6,8,10\) and \(14\). The author proves in the case \(k=2\) similar divisibility properties by the primes \(p=2,3,5,7,11\) of the Fourier coefficients of the functions \(f_{2,m}\). Let \(m,n\) be any positive integers. Assume \(p^r||m,p^s||n\) and \(r\neq s\). Then main results are as follows: If \(r>s\) (resp. \(r<s\)), then \(a_2(m,n)\) is divisible by \(p^{i(r-s)+j}\) (resp. \(p^{(i+1)(s-r)+j}\)), where \((i,j)=(3,8),(2,3),(1,1),(1,0),(0,1)\) for \(p=2,3,5,7,11\), respectively.
scientific article; zbMATH DE number 6031431