Home Mathematics Conway invariant Jacobi forms on the Leech lattice
Article
Licensed
Unlicensed Requires Authentication

Conway invariant Jacobi forms on the Leech lattice

  • Kaiwen Sun and Haowu Wang ORCID logo EMAIL logo
Published/Copyright: September 30, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 6

Abstract

In this paper, we study Jacobi forms associated with the Leech lattice Λ which are invariant under the Conway group Co 0 . We determine and construct generators of modules of both weak and holomorphic Jacobi forms of integral weight and fixed index t ≤ 3 . As applications, (i) we find the modular linear differential equations satisfied by the holomorphic generators; (ii) we determine the decompositions of many products of orbits of Leech vectors; (iii) we calculate the intersections between orbits and Leech vectors; (iv) we derive some conjugate relations among orbits modulo t ⁢ Λ .

Keywords: Jacobi forms; Leech lattice; Conway group
MSC 2010: 11F50

Communicated by Jan Bruinier

Funding source: Institute for Basic Science

Award Identifier / Grant number: IBS-R003-D1

Funding statement: The first author is supported by KIAS (Grant no. QP081001). The second author thanks Max Planck Institute for Mathematics in Bonn for its hospitality, where this work was started. The second author is supported by the Institute for Basic Science (Grant no. IBS-R003-D1).

A Product decompositions of Conway orbits

We determine the following orbit product decompositions:

O 2 ⊗ O 2 = 196560 ⁢ O 0 ⊕ 4600 ⁢ O 2 ⊕ 552 ⁢ O 3 ⊕ 46 ⁢ O 4 ⊕ 2 ⁢ O 5 ⊕ 2 ⁢ O 6 ⁢ b ⊕ O 8 ⁢ a ,
O 2 ⊗ O 3 = 47104 ⁢ O 2 ⊕ 11178 ⁢ O 3 ⊕ 2048 ⁢ O 4 ⊕ 275 ⁢ O 5 ⊕ 24 ⁢ O 6 ⁢ a ⊕ O 7 ⊕ O 8 ⁢ c ,
O 2 ⊗ O 4 = 93150 ⁢ O 2 ⊕ 48600 ⁢ O 3 ⊕ 16192 ⁢ O 4 ⊕ 4050 ⁢ O 5 ⊕ 759 ⁢ O 6 ⁢ a ⊕ 891 ⁢ O 6 ⁢ b ⊕ 100 ⁢ O 7
⊕ 8 ⁢ O 8 ⁢ b ⊕ O 9 ⁢ a ⊕ O 10 ⁢ b ,
O 3 ⊗ O 3 = 16773120 ⁢ O 0 ⊕ 953856 ⁢ O 2 ⊕ 257600 ⁢ O 3 ⊕ 64768 ⁢ O 4 ⊕ 14256 ⁢ O 5 ⊕ 2576 ⁢ O 6 ⁢ a
⊕ 2816 ⁢ O 6 ⁢ b ⊕ 352 ⁢ O 7 ⊕ 32 ⁢ O 8 ⁢ b ⊕ 2 ⁢ O 9 ⁢ b ⊕ 2 ⁢ O 10 ⁢ c ⊕ O 12 ⁢ a ,
O 2 ⊗ O 5 = 47104 ⁢ O 2 ⊕ 75900 ⁢ O 3 ⊕ 47104 ⁢ O 4 ⊕ 19450 ⁢ O 5 ⊕ 6072 ⁢ O 6 ⁢ a ⊕ 5632 ⁢ O 6 ⁢ b ⊕ 1452 ⁢ O 7
⊕ 256 ⁢ O 8 ⁢ b ⊕ 275 ⁢ O 8 ⁢ c ⊕ 23 ⁢ O 9 ⁢ a ⊕ 33 ⁢ O 9 ⁢ b ⊕ 2 ⁢ O 10 ⁢ a ⊕ O 11 ⁢ a ⊕ O 12 ⁢ b ,
O 3 ⊗ O 4 = 4147200 ⁢ O 2 ⊕ 1536975 ⁢ O 3 ⊕ 518144 ⁢ O 4 ⊕ 157950 ⁢ O 5 ⊕ 42504 ⁢ O 6 ⁢ a ⊕ 41472 ⁢ O 6 ⁢ b
⊕ 9725 ⁢ O 7 ⊕ 1792 ⁢ O 8 ⁢ b ⊕ 2025 ⁢ O 8 ⁢ c ⊕ 253 ⁢ O 9 ⁢ a ⊕ 243 ⁢ O 9 ⁢ b ⊕ 22 ⁢ O 10 ⁢ a ⊕ O 11 ⁢ b ⊕ O 12 ⁢ c ,
O 2 ⊗ O 6 ⁢ a = 48600 ⁢ O 3 ⊕ 64768 ⁢ O 4 ⊕ 44550 ⁢ O 5 ⊕ 21252 ⁢ O 6 ⁢ a ⊕ 20736 ⁢ O 6 ⁢ b ⊕ 7800 ⁢ O 7 ⊕ 2240 ⁢ O 8 ⁢ b
⊕ 2025 ⁢ O 8 ⁢ c ⊕ 506 ⁢ O 9 ⁢ a ⊕ 486 ⁢ O 9 ⁢ b ⊕ 77 ⁢ O 10 ⁢ a ⊕ 100 ⁢ O 10 ⁢ c ⊕ 8 ⁢ O 11 ⁢ b ⊕ O 12 ⁢ e ⊕ O 13 ⁢ a ,
O 2 ⊗ O 6 ⁢ b = 4600 ⁢ O 2 ⊕ 1012 ⁢ O 4 ⊕ 550 ⁢ O 5 ⊕ 276 ⁢ O 6 ⁢ a ⊕ 1782 ⁢ O 6 ⁢ b ⊕ 100 ⁢ O 7 ⊕ 4600 ⁢ O 8 ⁢ a ⊕ 28 ⁢ O 8 ⁢ b
⊕ 275 ⁢ O 8 ⁢ c ⊕ 23 ⁢ O 9 ⁢ a ⊕ O 10 ⁢ a ⊕ 44 ⁢ O 10 ⁢ b ⊕ 2 ⁢ O 11 ⁢ a ⊕ 3 ⁢ O 12 ⁢ d ⊕ O 14 ⁢ c ,
O 2 ⊗ O 8 ⁢ a = O 2 ⊕ 2 ⁢ O 6 ⁢ b ⊕ O 10 ⁢ b ⊕ O 12 ⁢ b ⊕ O 8 ⁢ c ⊕ O 14 ⁢ c ⊕ O 18 ⁢ a .

B Intersection between Conway orbits and Leech vectors

For the intersection between O * and any Leech vector of type 2, we have

O 2 : u ± 4 + 4600 ⁢ u ± 2 + 47104 ⁢ u ± 1 + 93150 ,
O 3 : 512 ⁢ ( 92 ⁢ u ± 3 + 1863 ⁢ u ± 2 + 8100 ⁢ u ± 1 + 12650 ) ,
O 4 : 4050 ⁢ ( 23 ⁢ u ± 4 + 1024 ⁢ u ± 3 + 8096 ⁢ u ± 2 + 23552 ⁢ u ± 1 + 32890 ) ,
O 5 : 23552 ⁢ ( 2 ⁢ u ± 5 + 275 ⁢ u ± 4 + 4050 ⁢ u ± 3 + 19450 ⁢ u ± 2 + 45100 ⁢ u ± 1 + 58806 ) ,
O 6 ⁢ a : 518400 ⁢ ( 8 ⁢ u ± 5 + 253 ⁢ u ± 4 + 2024 ⁢ u ± 3 + 7176 ⁢ u ± 2 + 14352 ⁢ u ± 1 + 17894 ) ,
O 6 ⁢ b : 2300 ⁢ ( 2 ⁢ u ± 6 + 891 ⁢ u ± 4 + 5632 ⁢ u ± 3 + 22518 ⁢ u ± 2 + 41472 ⁢ u ± 1 + 55530 ) ,
O 7 : 953856 ⁢ ( u ± 6 + 100 ⁢ u ± 5 + 1452 ⁢ u ± 4 + 7900 ⁢ u ± 3 + 22825 ⁢ u ± 2 + 41152 ⁢ u ± 1 + 49700 ) ,
O 8 ⁢ a : u ± 8 + 4600 ⁢ u ± 4 + 47104 ⁢ u ± 2 + 93150 ,
O 8 ⁢ b : 16394400 ⁢ ( 2 ⁢ u ± 6 + 64 ⁢ u ± 5 + 567 ⁢ u ± 4 + 2368 ⁢ u ± 3 + 5902 ⁢ u ± 2 + 9856 ⁢ u ± 1 + 11622 ) ,
O 8 ⁢ c : 47104 ⁢ ( u ± 7 + 275 ⁢ u ± 5 + 2300 ⁢ u ± 4 + 9153 ⁢ u ± 3 + 24576 ⁢ u ± 2 + 37675 ⁢ u ± 1 + 48600 ) ,
O 9 ⁢ a : 4147200 ⁢ ( u ± 7 + 23 ⁢ u ± 6 + 529 ⁢ u ± 5 + 3059 ⁢ u ± 4 + 10879 ⁢ u ± 3 + 24035 ⁢ u ± 2 + 37743 ⁢ u ± 1 + 44022 ) ,
O 9 ⁢ b : 32972800 ⁢ ( 11 ⁢ u ± 6 + 162 ⁢ u ± 5 + 1053 ⁢ u ± 4 + 3586 ⁢ u ± 3 + 8019 ⁢ u ± 2 + 12636 ⁢ u ± 1 + 14586 ) ,
O 10 ⁢ a : 47692800 ⁢ ( 2 ⁢ u ± 7 + 78 ⁢ u ± 6 + 814 ⁢ u ± 5 + 3993 ⁢ u ± 4 + 11882 ⁢ u ± 3 + 24266 ⁢ u ± 2 + 36454 ⁢ u ± 1 + 41582 ) .

For the intersection between O * and any Leech vector of type 3, we have

O 2 : 6 ⁢ ( 92 ⁢ u ± 3 + 1863 ⁢ u ± 2 + 8100 ⁢ u ± 1 + 12650 ) ,
O 3 : u ± 6 + 11178 ⁢ u ± 4 + 257600 ⁢ u ± 3 + 1536975 ⁢ u ± 2 + 3934656 ⁢ u ± 1 + 5292300 ,
O 4 : 6075 ⁢ ( 8 ⁢ u ± 5 + 253 ⁢ u ± 4 + 2024 ⁢ u ± 3 + 7176 ⁢ u ± 2 + 14352 ⁢ u ± 1 + 17894 ) ,
O 5 : 276 ⁢ ( 275 ⁢ u ± 6 + 14256 ⁢ u ± 5 + 157950 ⁢ u ± 4 + 743600 ⁢ u ± 3 + 1986525 ⁢ u ± 2 + 3434400 ⁢ u ± 1 + 4099108 ) ,
O 6 ⁢ a : 16200 ( 3 u ± 7 + 322 u ± 6 + 5313 u ± 5 + 33396 u ± 4 + 115115 u ± 3 + 256542 u ± 2
+ 403857 u ± 1 + 467544 ) ,
O 6 ⁢ b : 6900 ⁢ ( 11 ⁢ u ± 6 + 162 ⁢ u ± 5 + 1053 ⁢ u ± 4 + 3586 ⁢ u ± 3 + 8019 ⁢ u ± 2 + 12636 ⁢ u ± 1 + 14586 ) ,
O 7 : 11178 ( u ± 8 + 352 u ± 7 + 9725 u ± 6 + 84800 u ± 5 + 376750 u ± 4 + 1053504 u ± 3 + 2075603 u ± 2
+ 3053600 u ± 1 + 3464450 ) ,
O 8 ⁢ a : 6 ⁢ ( 92 ⁢ u ± 6 + 1863 ⁢ u ± 4 + 8100 ⁢ u ± 1 + 12650 ) ,
O 8 ⁢ b : 1536975 ( u ± 8 + 56 u ± 7 + 728 u ± 6 + 4264 u ± 5 + 14924 u ± 4 + 36024 u ± 3 + 64744 u ± 2
+ 90728 u ± 1 + 101222 ) ,
O 8 ⁢ c : 552 ( u ± 9 + 2025 u ± 7 + 22528 u ± 6 + 137700 u ± 5 + 476928 u ± 4 + 1151700 u ± 3 + 2073600 u ± 2
+ 2902878 u ± 1 + 3238400 ) .

For the intersection between O * and any Leech vector of type 4, we have

O 2 : 2 ⁢ ( 23 ⁢ u ± 4 + 1024 ⁢ u ± 3 + 8096 ⁢ u ± 2 + 23552 ⁢ u ± 1 + 32890 )
O 3 : 256 ⁢ ( 8 ⁢ u ± 5 + 253 ⁢ u ± 4 + 2024 ⁢ u ± 3 + 7176 ⁢ u ± 2 + 14352 ⁢ u ± 1 + 17894 )
O 4 : u ± 8 + 16192 ⁢ u ± 6 + 518144 ⁢ u ± 5 + 4595032 ⁢ u ± 4 + 19171328 ⁢ u ± 3 + 47829696 ⁢ u ± 2
+ 79794176 ⁢ u ± 1 + 94184862
O 5 : 23552 ⁢ ( 2 ⁢ u ± 7 + 78 ⁢ u ± 6 + 814 ⁢ u ± 5 + 3993 ⁢ u ± 4 + 11882 ⁢ u ± 3 + 24266 ⁢ u ± 2 + 36454 ⁢ u ± 1 + 41582 )
O 6 ⁢ a : 256 ( 253 u ± 8 + 14168 u ± 7 + 184368 u ± 6 + 1078792 u ± 5 + 3779498 u ± 4 + 9114072 u ± 3
+ 16396432 u ± 2 + 22954184 u ± 1 + 25634466 )
O 6 ⁢ b : 92 ( 11 u ± 8 + 512 u ± 7 + 6864 u ± 6 + 39936 u ± 5 + 139854 u ± 4 + 337920 u ± 3 + 606832 u ± 2
+ 850432 u ± 1 + 949278 )
O 7 : 11776 ( 4 u ± 9 + 389 u ± 8 + 6776 u ± 7 + 48532 u ± 6 + 200772 u ± 5 + 564135 u ± 4 + 1181756 u ± 3
+ 1943044 u ± 2 + 2592004 u ± 1 + 2846536 )
O 8 ⁢ a : 2 ⁢ ( 23 ⁢ u ± 8 + 1024 ⁢ u ± 6 + 8096 ⁢ u ± 4 + 23552 ⁢ u ± 2 + 32890 )
O 8 ⁢ b : 4048 ( 4 u ± 10 + 896 u ± 9 + 23011 u ± 8 + 209664 u ± 7 + 1038804 u ± 6 + 3398784 u ± 5 + 8194512 u ± 4
+ 15480192 u ± 3 + 23860008 u ± 2 + 30652288 u ± 1 + 33300674 )
O 8 ⁢ c : 47104 ( u ± 9 + 22 u ± 8 + 209 u ± 7 + 1024 u ± 6 + 3356 u ± 5 + 8096 u ± 4 + 15292 u ± 3 + 23552 u ± 2
+ 30294 u ± 1 + 32868 )

Acknowledgements

The authors thank Daniel Allcock and Kimyeong Lee for useful discussions, and Brandon Williams for pointing out [19] and for fruitful discussions on Borcherds’ thesis. The authors also thank Richard Borcherds for valuable comments. The authors also thank the referee for many valuable comments which improve this paper.

References

[1] R. E. Borcherds, The Leech lattice and other lattices, Ph.D. dissertation, Cambridge University, Cambridge, 1985. Search in Google Scholar

[2] R. E. Borcherds, The monster Lie algebra, Adv. Math. 83 (1990), no. 1, 30–47. 10.1016/0001-8708(90)90067-WSearch in Google Scholar

[3] R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), no. 2, 405–444. 10.1007/BF01232032Search in Google Scholar

[4] R. E. Borcherds, Automorphic forms on O s + 2 , 2 ⁢ ( 𝐑 ) and infinite products, Invent. Math. 120 (1995), no. 1, 161–213. 10.1007/BF01241126Search in Google Scholar

[5] R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. 10.1007/s002220050232Search in Google Scholar

[6] R. E. Borcherds, The Gross–Kohnen–Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), no. 2, 219–233. 10.1215/S0012-7094-99-09710-7Search in Google Scholar

[7] Y. Choie and H. Kim, Differential operators on Jacobi forms of several variables, J. Number Theory 82 (2000), no. 1, 140–163. 10.1006/jnth.1999.2491Search in Google Scholar

[8] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033. 10.4007/annals.2017.185.3.8Search in Google Scholar

[9] J. H. Conway, A characterisation of Leech’s lattice, Invent. Math. 7 (1969), 137–142. 10.1007/BF01389796Search in Google Scholar

[10] J. H. Conway, A group of order 8 , 315 , 553 , 613 , 086 , 720 , 000 , Bull. Lond. Math. Soc. 1 (1969), 79–88. 10.1112/blms/1.1.79Search in Google Scholar

[11] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, 𝔸 ⁢ 𝕋 ⁢ 𝕃 ⁢ 𝔸 ⁢ 𝕊 of Finite Groups, Oxford University, Eynsham, 1985. Search in Google Scholar

[12] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss. 290, Springer, New York, 1999. 10.1007/978-1-4757-6568-7Search in Google Scholar

[13] L. Dixon, P. Ginsparg and J. Harvey, Beauty and the beast: Superconformal symmetry in a Monster module, Comm. Math. Phys. 119 (1988), no. 2, 221–241. 10.1007/BF01217740Search in Google Scholar

[14] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progr. Mathematics 55, Birkhäuser, Boston, 1985. 10.1007/978-1-4684-9162-3Search in Google Scholar

[15] V. Gritsenko, Modular forms and moduli spaces of abelian and K ⁢ 3 surfaces, Algebra i Analiz 6 (1994), no. 6, 65–102. Search in Google Scholar

[16] V. A. Gritsenko, Fourier-Jacobi functions in n variables, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), no. 9, 32–44, 187–188. Search in Google Scholar

[17] V. A. Gritsenko, Reflective modular forms and their applications, Uspekhi Mat. Nauk 73 (2018), no. 5(443), 53–122. 10.1070/RM9853Search in Google Scholar

[18] J. Leech, Notes on sphere packings, Canad. J. Math. 19 (1967), 251–267. 10.4153/CJM-1967-017-0Search in Google Scholar

[19] J. Martinet, Reduction modulo 2 and 3 of Euclidean lattices, J. Algebra 251 (2002), no. 2, 864–887. 10.1006/jabr.2001.9018Search in Google Scholar

[20] K. Sakai, Topological string amplitudes for the local 1 2 ⁢ K3 surface, PTEP. Prog. Theor. Exp. Phys. (2017), no. 3, Article ID 033B09. 10.1093/ptep/ptx027Search in Google Scholar

[21] N. R. Scheithauer, Automorphic products of singular weight, Compos. Math. 153 (2017), no. 9, 1855–1892. 10.1112/S0010437X17007266Search in Google Scholar

[22] K. Sun and H. Wang, Weyl invariant E 8 Jacobi forms and E-strings, preprint (2021), https://arxiv.org/abs/2109.10578. Search in Google Scholar

[23] H. Wang, Reflective modular forms: A Jacobi forms approach, Int. Math. Res. Not. IMRN 2021 (2021), no. 3, 2081–2107. 10.1093/imrn/rnz070Search in Google Scholar

[24] H. Wang, Weyl invariant E 8 Jacobi forms, Commun. Number Theory Phys. 15 (2021), no. 3, 517–573. 10.4310/CNTP.2021.v15.n3.a3Search in Google Scholar

[25] H. Wang, Weyl invariant Jacobi forms: a new approach, Adv. Math. 384 (2021), Paper No. 107752. 10.1016/j.aim.2021.107752Search in Google Scholar

[26] H. Wang and B. Williams, On weak Jacobi forms of rank two, preprint (2021), https://arxiv.org/abs/2102.11190. Search in Google Scholar

[27] K. Wirthmüller, Root systems and Jacobi forms, Compos. Math. 82 (1992), no. 3, 293–354. Search in Google Scholar
Received: 2022-03-10
Revised: 2022-08-11
Published Online: 2022-09-30
Published in Print: 2022-11-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
https://doi.org/10.1515/forum-2022-0077
Keywords for this article
Jacobi forms; Leech lattice; Conway group

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 20.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0077/html
Scroll to top button