Some constructions of modular forms for the Weil representation of \(\mathrm{SL}_2(\mathbb Z)\) (Q2788612)

Vector valued modular for the Weil representation \(\rho_D\) of \(\mathrm{SL}_2(\mathbb Z)\) associated to a discriminant form \(D\) are subject of many recent publications since the singular theta lift (see e. g. [\textit{R. E. Borcherds}, Invent. Math. 132, No. 3, 491--562 (1998; Zbl 0919.11036); \textit{J. H. Bruinier}, Borcherds products on \(\mathrm{O}(2,l)\) and Chern classes of Heegner divisors. Berlin: Springer (2002; Zbl 1004.11021)]) maps them to automorphic forms on orthogonal groups with a product expansion. An important way to construct such vector valued modular forms is to obtain them as a lifting from scalar valued modular forms on some congruence subgroup. The author of the paper under review investigated such a lifting for the group \(\Gamma_0(N)\), see [\textit{N. R. Scheithauer}, Int. Math. Res. Not. 2009, No. 8, 1488--1545 (2009; Zbl 1244.11043)]:NEWLINENEWLINELet \(D\) be a discriminant form of even signature and level dividing \(N\in \mathbb N\). Let \(f\) be a scalar valued modular form on \(\Gamma_0(N)\) of weight \(k\) and character \(\chi_D\) (see [loc. cit.] for the definition of \(\chi_D\)) and \(H\) an isotropic subset of \(D\) which is invariant under \((\mathbb Z/N\mathbb Z)^*\) as a set. Then NEWLINE\[NEWLINEF_{\Gamma_0(N), f, H}=\sum_{M\in \Gamma_0(N)\backslash \mathrm{SL}_2(\mathbb Z)}\sum_{\gamma\in H} f|_k M\rho_D(M^{-1})\chi_\gammaNEWLINE\]NEWLINE is a modular form for \(\rho_D\) of weight \(k\) which is invariant under the automorphisms of the discriminant form that stabilize \(H\) as a set.NEWLINENEWLINEHere \(|_k\) is the usual slash operator and \(\chi_\gamma\) an element of the standard basis of the group ring \(\mathbb C[D]\).NEWLINENEWLINEThe paper under review can be seen as a continuation of [loc. cit.] since it introduces the following two additional liftings which are related but not equal to the one in TheoremNEWLINENEWLINELet \(D\) and \(N\) be as in Theorem and \(\gamma\in D\). {\parindent=0.6cm\begin{itemize}\item[--] Let \(f\) be a scalar valued modular form for \(\Gamma_1(N)\) of weight \(k\) and character \(\chi_\gamma\) (which is defined in the paper). Then NEWLINE\[NEWLINEF_{\Gamma_1(N),f,\gamma}=\sum_{M\in \Gamma_1(N)\backslash \mathrm{SL}_2(\mathbb Z)} f|_kM\rho_D(M^{-1})\chi_\gammaNEWLINE\]NEWLINE is a modular form for \(\rho_D\) of weight \(k\) which is invariant under the stabilizer of \(\gamma\) in \(\operatorname{O}(D)\). \item[--] Let \(f\) be scalar valued modular form on \(\Gamma(N)\) of weight \(k\). Then NEWLINE\[NEWLINEF_{\Gamma(N),f,\gamma}=\sum_{M\in \Gamma(N)\backslash \mathrm{SL}_2(\mathbb Z)} f|_kM\rho_D(M^{-1})\chi_\gammaNEWLINE\]NEWLINE is a modular form for \(\rho_D\) of weight \(k\) which is invariant under the stabilizer of \(\gamma\) in \(\operatorname{O}(D)\). NEWLINENEWLINE\end{itemize}} It is important to note that both of the liftings lead to modular forms which are not symmetric under the action of \(\operatorname{O}(D)\) in contrast to the lifting. Also, the author proves that any vector valued modular form can be written as a linear combination of liftings from \(\Gamma_1(N)\) which is not true for \(\Gamma_0(N)\)-liftings.NEWLINENEWLINEBased on these two liftings, the singular theta lift yields automorphic forms of singular weight, which in turn give rise to denominator identities of some new Kac-Moody algebras.NEWLINENEWLINEIn a third construction, the author describes how vector valued modular forms can obtained from modular forms of the same type associated to an isotropic subgroup of a discriminant form \(D\).NEWLINENEWLINELet \(D\) be a discriminant form of even signature and \(H\) an isotropic subgroup of \(D\). Let \(F_{D_H}=\sum_{\gamma\in D_H} F_{D_H,\gamma}\chi_\gamma\) be a modular form for the Weil representation of the discriminant form \(D_{H}=H^{\perp}\slash H\). Then NEWLINE\[NEWLINEF=\sum_{\gamma\in H^{\perp}} F_{D_H,\gamma +H}\chi_\gammaNEWLINE\]NEWLINE is a modular form for \(\rho_D\).NEWLINENEWLINEThe author notes that modular forms constructed this way can be interpreted as oldforms within the space of modular forms for \(\rho_D\).NEWLINENEWLINEMoreover, the paper gives a criterion when a vector valued modular form for \(\rho_D\) with certain properties can be obtained from a scalar valued modular form as described in the theorem above.NEWLINENEWLINELet \(D\) be a discriminant form of squarefree level \(N\) and \(F=\sum_{\gamma\in D}F_\gamma\chi_\gamma\) a modular form for \(\rho_D\) which is invariant under \(O(D)\). Then the complex vector space \(W\) spanned by the components \(F_\gamma,\; \gamma\in D,\) is generated by the functions \(F_0|_k M,\; M\in \operatorname{SL}_2(\mathbb Z)\). Let \(W_0\) be the subspace of \(W\) with \(T-\) eigenvalue \(e(0)\). Then the map NEWLINE\[NEWLINE\begin{aligned} \Phi: & W_0\longrightarrow W_0,\\ & f\mapsto 0\text{-component of } F_{\Gamma_0(N),f,0}\end{aligned}NEWLINE\]NEWLINE as a bijection. In particular \(F=F_{\Gamma_0(N),f,0}\) for a suitable function \(f\) in \(W_0\).