Divisibility properties for weakly holomorphic modular forms with sign vectors (Q2833083)

Let \(N\) be a positive integer and \(\chi\) a primitive quadratic Dirichlet character modulo \(N\). For a sign vector \(\varepsilon=(\varepsilon_p)_{p|N}\) (\(\varepsilon_p=\pm 1\),~\(p:\) prime divisors of \(N\)), let \(A^\varepsilon(N,k,\chi)\) be the space of weakly holomorphic forms \(f\) of weight \(k\) and character \(\chi\) with respect to the modular group \(\Gamma_0(N)\) such that the \(n\)-th Fourier coefficient \(a(n)\) of the expansion \(f\) at the cusp \(\infty\) satisfies the condition \(a(n)=0\) if \(\chi(n)=-\varepsilon_p\) for any \(n\in\mathbb Z\). The space \(A^\varepsilon(N,k,\chi)\) has a basis consisted of reduced modular forms \(f_m=\sum_n a_m(n)q^n,m\in\mathbb Z\). In this article, the author studies the divisibility of Fourier coefficients \(a_m(n)\) of reduced modular forms \(f_m\). For \(k\geq 2\) and \(m=0\), the author shows that if \(2^{\sigma_0(N)}f_0\) has integral Fourier coefficients, then \(r|2^{\sigma_0(N)}a_0(n)\) for \(n>0\), where \(\sigma_0(N)\) is the number of prime divisors of \(N\) and \(r\) is determined by \(L(1-k,\chi)=s/r,~r,s\in\mathbb Z,(r,s)=1\). This result generalizes a result of Siegel to higher level reduced forms. In the case that \(N\) is a prime power, the integer \(r\) is explicitly given. For \(k\leq 0\), by applying Hecke operators and the differential operator\((q\frac{d}{dq})^{1-k}\) the author gives various divisibility properties and examples. One of them is as follows:~for a prime number \(p\) and a reduced form \(f_{mp}\in A^\varepsilon(N,k,\chi)\), if \(2^{\sigma_0((mp,N))}f_{mp}\) has integral Fourier coefficients and \(S^\varepsilon(N,2-k,\chi)=S^{\varepsilon^*}(N,2-k,\chi)=\{0\}\), then \(p^{1-k}|2^{\sigma_0((mp,N))}a_{mp}(n)\) whenever \(p\nmid n\). Here \(S^\varepsilon(N,2-k,\chi)\) is a subspace of cusp forms in \(A^\varepsilon(N,2-k,\chi)\) and \(\varepsilon^*\) is the dual sign vector of \(\varepsilon\). By Borcherds's theory a reduced form \(f_m\in A^\varepsilon(N,k,\chi)\) is lifted to a Hilbert form \(\Psi_{f_m}\) for \(\mathbb Q(\sqrt N)\). The author shows that the weight of \(\Psi_{f_m}\) is divisible by \(5\) if \(N=5,k=2,\varepsilon=1\). Further by a paring between the spaces of modular forms with respect to Weil representation of \(\text{SL}_2(\mathbb Z)\) determined by the pair \((\chi,\varepsilon)\), the author gives a different proof of \textit{D. Zagier} duality for reduced modular forms [Proc. Indian Acad. Sci., Math. Sci. 104, No. 1, 57--75 (1994; Zbl 0806.11022)] to avoid some restrictions in his previous proof. This duality is essentially used in this article.