Hecke operators on vector-valued modular forms - Bouchard, Vincent; Creutzig, Thomas; Joshi, Aniket (Symmetry Integrability and Geometry-Methods and Applications, 2019)
We study Hecke operators on vector-valued modular forms for the Weil representation ρL of a lattice L. We first construct Hecke operators Tr that map vector-valued modular forms of type ρL into vector-valued modular forms of type ρL(r), where L(r) is the lattice L with rescaled bilinear form (·, ·)r = r(·, ·), by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vectorvalued Hecke operators Tr have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau threefolds. We study algebraic relations satisfied by the Hecke operators Tr. In the particular case when r = n2 for some positive integer n, we compose Tn 2 with a projection operator to construct new Hecke operators Hn 2 that map vector-valued modular forms of type ρL into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators Hn 2, and compare our operators with the alternative construction of BruinierStein [Math. Z. 264 (2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].
CRIS-Team Scopus Importer:2024-06-24
Keywords: Hecke operators; Vector-valued modular forms; Weil representation