Hecke operators on vector-valued modular forms of the Weil representation

• Author / Creator

  Joshi, Aniket

• Vector-valued modular forms of the Weil representation are an indispensable tool in diverse areas of mathematics such as enumerative geometry of Calabi-Yau manifolds and rational conformal field theory. In this thesis, we study Hecke operators on vector-valued modular forms of the Weil representation of a lattice L. We first construct Hecke operators Tr that map vector-valued
  modular forms of a certain type into vector-valued modular forms of rescaled lattices by lifting standard Hecke operators for scalar-valued modular forms through Siegel theta functions. We also get a set of algebraic relations satisfied by the Hecke operators Tr similar to the scalar-valued case. In the particular case when r is a square number, the Weil representation of the rescaled lattice carries a sub-representation of the Weil representation of the original lattice and we can compose Tr with a projection operator to construct new Hecke operators Hr that map vector-valued modular forms of a certain type into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators H_r , and compare our operators with the Hecke operators of Bruinier and Stein obtained by a different construction.

• Subjects / Keywords

  • vector-valued modular forms
  • Weil representation
  • Hecke operators

• Graduation date

  Fall 2018

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/R3707X471

• License

  Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.