Congruences between cusp forms and Eisenstein series of half-integral weight
Let p be a prime with \(p\equiv 3 (mod 4)\) and h(-p) denote the class number of \({\mathbb{Q}}(\sqrt{-p}).\) We know that \[ G_{2,p}(z)=24^{-1}(p- 1)+\sum_{n\geq 1}\sum_{d| n, p\nmid d}d e^{2\pi inz} \] is the Eisenstein series of weight 2 on \(\Gamma_ 0(p)\), and \[ G_{1,p}=2^{-1} h(-p)+\sum_{n\geq 1}(\sum_{d | n}(d/p))e^{2\pi inz} \] is the Eisenstein series of weight 1 and Nebentypus ( /p) on \(\Gamma_ 0(p)\). Then \(C_ p=12^{-1}(p-1)G^ 2_{1,p}-h(-p)^ 2 G_{2,p}\) is a cusp form of weight 2 on \(\Gamma_ 0(p)\). If \(\ell\) is a prime which divides the exact numerator of (p-1)/12 but does not divide h(-p), then \(C_ p\) has \(\ell\)-integral Fourier coefficients not all zero modulo \(\ell\) and is congruent modulo \(\ell\) to \(-h(-p)^ 2 G_{2,p}.\) The purpose of this note is to show that this congruence between modular forms of integral weight is induced via a Shimura lifting \({\mathcal S}_ p\) from a congruence modulo \(\ell\) between a cusp form \({\mathcal C}_ p\) and the Cohen-Eisenstein series \({\mathcal H}_{3/2,p}\) of weight 3/2 on \(\Gamma_ 0(4p)\), i.e. it is proved that \({\mathcal S}_ p{\mathcal C}_ p=C_ p\) and \({\mathcal C}\equiv -h(-p){\mathcal H}_{3/2,p}(mod \ell),\) where \({\mathcal C}_ p=12^{-1}(p-1)G_{1,p}(4z)\vartheta (pz)-h(-p){\mathcal H}_{3/2,p}(z)\) and \(\vartheta\) (z) is the theta function. Moreover, it is also proved that the Hecke module generated by \({\mathcal C}_ p\) is spanned over \({\mathbb{C}}\) by those eigenforms f for which \(L(F,1)c_ f(p)\neq 0\), where \(F={\mathcal S}_ pf\) and \(c_ f(p)\) is the p-th Fourier coefficient of f. The similar phenomenon where congruences between modular forms of integer weight descend to congruences between modular forms of half-integral weight is also investigated by \textit{Y. Maeda} [Hokkaido Math. J. 12, 64- 73 (1983; Zbl 0521.10024)] and \textit{N. Koblitz} [Math. Ann. 274, 199-220 (1986; Zbl 0571.10030)].
congruence between modular forms of integral weight
scientific article; zbMATH DE number 4035923