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This article is cited in 3 scientific papers (total in 3 papers)
Hecke Operators on Vector-Valued Modular Forms
Vincent Boucharda, Thomas Creutzigba, Aniket Joshia a Department of Mathematical & Statistical Sciences, University of Alberta,
632 Central Academic Building, Edmonton T6G 2G1, Canada
b Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Abstract:
We study Hecke operators on vector-valued modular forms for the Weil representation $\rho_L$ of a lattice $L$. We first construct Hecke operators $\mathcal{T}_r$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of type $\rho_{L(r)}$, where $L(r)$ is the lattice $L$ with rescaled bilinear form $(\cdot, \cdot)_r = r (\cdot, \cdot)$, by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vector-valued Hecke operators $\mathcal{T}_r$ 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 $\mathcal{T}_r$. In the particular case when $r=n^2$ for some positive integer $n$, we compose $\mathcal{T}_{n^2}$ with a projection operator to construct new Hecke operators $\mathcal{H}_{n^2}$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators $\mathcal{H}_{n^2}$, 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].
Keywords:
Hecke operators, vector-valued modular forms, Weil representation.
Received: September 26, 2018; in final form May 13, 2019; Published online May 25, 2019
Citation:
Vincent Bouchard, Thomas Creutzig, Aniket Joshi, “Hecke Operators on Vector-Valued Modular Forms”, SIGMA, 15 (2019), 041, 31 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1477 https://www.mathnet.ru/eng/sigma/v15/p41
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