Geodesic
Hecke Operators on Vector-Valued Modular Forms
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SIGMA_2019_15_a40,
     author = {Vincent Bouchard and Thomas Creutzig and Aniket Joshi},
     title = {Hecke {Operators} on {Vector-Valued} {Modular} {Forms}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a40/}
}
TY  - JOUR
AU  - Vincent Bouchard
AU  - Thomas Creutzig
AU  - Aniket Joshi
TI  - Hecke Operators on Vector-Valued Modular Forms
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2019
VL  - 15
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a40/
LA  - en
ID  - SIGMA_2019_15_a40
ER  - 
%0 Journal Article
%A Vincent Bouchard
%A Thomas Creutzig
%A Aniket Joshi
%T Hecke Operators on Vector-Valued Modular Forms
%J Symmetry, integrability and geometry: methods and applications
%D 2019
%V 15
%U http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a40/
%G en
%F SIGMA_2019_15_a40
Vincent Bouchard; Thomas Creutzig; Aniket Joshi. Hecke Operators on Vector-Valued Modular Forms. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a40/

[1] Ajouz A., Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular form, Ph.D. Thesis, University of Siegen, 2015

[2] Booker T., Davydov A., “Commutative algebras in Fibonacci categories”, J. Algebra, 355 (2012), 176–204, arXiv: 1103.3537 | DOI | MR | Zbl

[3] Borcherds R.E., “Automorphic forms with singularities on Grassmannians”, Invent. Math., 132 (1998), 491–562, arXiv: alg-geom/9609022 | DOI | MR | Zbl

[4] Borcherds R.E., “Reflection groups of Lorentzian lattices”, Duke Math. J., 104 (2000), 319–366, arXiv: math.GR/9909123 | DOI | MR | Zbl

[5] Bouchard V., Creutzig T., Diaconescu D.-E., Doran C., Quigley C., Sheshmani A., “Vertical D4-D2-D0 bound states on K3 fibrations and modularity”, Comm. Math. Phys., 350 (2017), 1069–1121, arXiv: 1601.04030 | DOI | MR | Zbl

[6] Boylan H., Jacobi forms, finite quadratic modules and Weil representations over number fields, Lecture Notes in Math., 2130, Springer, Cham, 2015 | DOI | MR | Zbl

[7] Bruinier J.H., “On the converse theorem for Borcherds products”, J. Algebra, 397 (2014), 315–342, arXiv: 1210.4821 | DOI | MR | Zbl

[8] Bruinier J.H., Stein O., “The Weil representation and Hecke operators for vector valued modular forms”, Math. Z., 264 (2010), 249–270, arXiv: 0704.1868 | DOI | MR | Zbl

[9] Carnahan S., “Generalized moonshine, II: Borcherds products”, Duke Math. J., 161 (2012), 893–950, arXiv: 0908.4223 | DOI | MR | Zbl

[10] Creutzig T., McRae R., Kanade S., Tensor categories for vertex operator superalgebra extensions, arXiv: 1705.05017

[11] Diamond F., Shurman J., A first course in modular forms, Graduate Texts in Mathematics, 228, Springer-Verlag, New York, 2005 | DOI | MR | Zbl

[12] Eholzer W., Skoruppa N.-P., “Conformal characters and theta series”, Lett. Math. Phys., 35 (1995), 197–211, arXiv: hep-th/9410077 | DOI | MR | Zbl

[13] Eichler M., Zagier D., The theory of Jacobi forms, Progress in Mathematics, 55, Birkhäuser Boston, Inc., Boston, MA, 1985 | DOI | MR | Zbl

[14] Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Mathematical Surveys and Monographs, 205, Amer. Math. Soc., Providence, RI, 2015 | DOI | MR | Zbl

[15] Gholampour A., Sheshmani A., “Donaldson–Thomas invariants of 2-dimensional sheaves inside threefolds and modular forms”, Adv. Math., 326 (2018), 79–107, arXiv: 1309.0050 | DOI | MR | Zbl

[16] Harvey J.A., Moore G., “Algebras, BPS states, and strings”, Nuclear Phys. B, 463 (1996), 315–368, arXiv: hep-th/9510182 | DOI | MR | Zbl

[17] Harvey J.A., Moore G., “On the algebras of BPS states”, Comm. Math. Phys., 197 (1998), 489–519, arXiv: hep-th/9609017 | DOI | MR | Zbl

[18] Harvey J.A., Wu Y., “Hecke relations in rational conformal field theory”, J. High Energy Phys., 2018:9 (2018), 032, 37 pp., arXiv: 1804.06860 | DOI | MR

[19] Höhn G., Scheithauer N.R., “A natural construction of Borcherds' fake Baby Monster Lie algebra”, Amer. J. Math., 125 (2003), 655–667, arXiv: math.QA/0312106 | DOI | MR | Zbl

[20] Huang Y.-Z., “Rigidity and modularity of vertex tensor categories”, Commun. Contemp. Math., 10 (2008), 871–911, arXiv: math.QA/0502533 | DOI | MR | Zbl

[21] Huang Y.-Z., Kirillov Jr. A., Lepowsky J., “Braided tensor categories and extensions of vertex operator algebras”, Comm. Math. Phys., 337 (2015), 1143–1159, arXiv: 1406.3420 | DOI | MR | Zbl

[22] Joshi A., Hecke operators on vector-valued modular forms of the Weil representation, MSc. Thesis, University of Alberta, 2018 | Zbl

[23] Maldacena J., Strominger A., Witten E., “Black hole entropy in M-theory”, J. High Energy Phys., 1997:12 (1997), 002, 16 pp., arXiv: hep-th/9711053 | DOI

[24] Raum M., “Computing genus 1 Jacobi forms”, Math. Comp., 85 (2016), 931–960, arXiv: 1212.1834 | DOI | MR | Zbl

[25] Rössler M., Hecke operators and vector valued modular forms, MSc. Thesis, Technische Universität Darmstadt, 2015

[26] Scheithauer N.R., “Generalized Kac–Moody algebras, automorphic forms and Conway's group. I”, Adv. Math., 183 (2004), 240–270 | DOI | MR | Zbl

[27] Scheithauer N.R., “Generalized Kac–Moody algebras, automorphic forms and Conway's group. II”, J. Reine Angew. Math., 625 (2008), 125–154 | DOI | MR | Zbl

[28] Scheithauer N.R., “The Weil representation of ${\rm SL}_2({\mathbb Z})$ and some applications”, Int. Math. Res. Not., 2009 (2009), 1488–1545 | DOI | MR | Zbl

[29] Stein O., “The Fourier expansion of Hecke operators for vector-valued modular forms”, Funct. Approx. Comment. Math., 52 (2015), 229–252 | DOI | MR | Zbl

[30] Stein W., Modular forms, a computational approach, Graduate Studies in Mathematics, 79, Amer. Math. Soc., Providence, RI, 2007 | DOI | MR | Zbl

[31] Vafa C., Two dimensional Yang–Mills, black holes and topological strings, arXiv: hep-th/0406058

[32] Weil A., “Sur certains groupes d'opérateurs unitaires”, Acta Math., 111 (1964), 143–211 | DOI | MR | Zbl

[33] Werner F., Vector valued Hecke theory, Ph.D. Thesis, Technische Universität, 2014

[34] Westerholt-Raum M., “Products of vector valued Eisenstein series”, Forum Math., 29 (2017), 157–186, arXiv: 1411.3877 | DOI | MR | Zbl

[35] Zhu Y., “Modular invariance of characters of vertex operator algebras”, J. Amer. Math. Soc., 9 (1996), 237–302 | DOI | MR | Zbl