The arithmetic of modular grids

Abstract

A modular grid is a pair of sequences ( f m ) m $ (f_m ) _m$ and ( g n ) n $ (g_n ) _n$ of weakly holomorphic modular forms such that for almost all m and n, the coefficient of q n $q^n$ in f m $f_m$ is the negative of the coefficient of q m $q^m$ in g n $g_n$ . Zagier proved this coefficient duality in weights 1/2 and 3/2 in the Kohnen plus space, and such grids have appeared for Poincaré series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group Γ ⊆ SL 2 ( R ) $\Gamma \subseteq {\text{\rm SL}}_2(\mathbb {R})$ commensurable with SL 2 ( Z ) ${\text{\rm SL}}_2(\mathbb {Z})$ . We construct bivariate generating functions that encode these modular forms, and study linear operations on the resulting modular grids.