A New Look at Hecke’s Indefinite Theta Series

where Q is an indefinite quadratic form on Z, f(m,n) is a doubly periodic function on Z such that the sums of f(m,n)q over all vertical and all horizontal lines in Z vanish. Some of these series appeared as coefficients in univalued triple Massey products on elliptic curves computed via homological mirror symmetry in [3]. In particular, in this context the condition of vanishing of sums over vertical and horizontal lines appears to be related to the standard necessary condition of the existence of triple Massey products (the vanishing of two double products). In the present paper we generalize Theorem 3 of [3] which relates such series to the indefinite theta series considered by Hecke in [1], [2] (our approach is completely elementary and doesn’t use the connection with triple products on elliptic curves). The main consequence of this relation is the modularity of our q-series. We also show that the problem of finding all linear relations between our series is related to the study of orbits of actions of dihedral groups on (Z/NZ).