Hecke Operators on Vector-Valued Modular Forms

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2019
• Main Authors: Bouchard, V., Creutzig, T., Joshi, A.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210181
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Hecke Operators on Vector-Valued Modular Forms / V. Bouchard, T. Creutzig, A. Joshi // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 35 назв. — англ.

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 vector-valued 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 Tn² with a projection operator to construct new Hecke operators Hn² 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 Hₙ², and compare our operators with the alternative construction of Bruinier-Stein [Math. Z. 264 (2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].