Hecke Operators on Vector-Valued Modular Forms
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...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2019 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2019
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210181 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| 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. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862623576009474048 |
|---|---|
| author | Bouchard, V. Creutzig, T. Joshi, A. |
| author_facet | Bouchard, V. Creutzig, T. Joshi, A. |
| citation_txt | 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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].
|
| first_indexed | 2025-12-07T21:24:39Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210181 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:39Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bouchard, V. Creutzig, T. Joshi, A. 2025-12-03T14:28:09Z 2019 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. 1815-0659 2010 Mathematics Subject Classification: 11F25; 11F27; 17B69; 14N35 arXiv: 1807.07703 https://nasplib.isofts.kiev.ua/handle/123456789/210181 https://doi.org/10.3842/SIGMA.2019.041 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]. We would like to thank Duiliu-Emanuel Diaconescu for interesting discussions and collaboration at the initial stages of this project. We would also like to thank Terry Gannon, Jeff Harvey, and Martin Raum for useful discussions. Finally, we would like to thank the anonymous referees for their very valuable comments. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hecke Operators on Vector-Valued Modular Forms Article published earlier |
| spellingShingle | Hecke Operators on Vector-Valued Modular Forms Bouchard, V. Creutzig, T. Joshi, A. |
| title | Hecke Operators on Vector-Valued Modular Forms |
| title_full | Hecke Operators on Vector-Valued Modular Forms |
| title_fullStr | Hecke Operators on Vector-Valued Modular Forms |
| title_full_unstemmed | Hecke Operators on Vector-Valued Modular Forms |
| title_short | Hecke Operators on Vector-Valued Modular Forms |
| title_sort | hecke operators on vector-valued modular forms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210181 |
| work_keys_str_mv | AT bouchardv heckeoperatorsonvectorvaluedmodularforms AT creutzigt heckeoperatorsonvectorvaluedmodularforms AT joshia heckeoperatorsonvectorvaluedmodularforms |