Maximal discrete subgroups of 𝑆𝑂⁺(2,𝑛+2)

Krieg, Aloys; Schaps, Felix

doi:10.1090/proc/15889

Maximal discrete subgroups of $SO^+(2,n+2)$
HTML articles powered by AMS MathViewer

by Aloys Krieg and Felix Schaps PDF

Proc. Amer. Math. Soc. 150 (2022), 2357-2365 Request permission

Abstract:

We characterize the maximal discrete subgroups of $SO^+(2,n+2)$, which contain the discriminant kernel of an even lattice with two hyperbolic planes over $\mathbb {Z}$. They coincide with the normalizers in $SO^+(2,n+2)$ and are given by the group of all integral matrices inside $SO^+(2,n+2)$, whenever the underlying lattice is maximal even. Finally we deal with the irreducible root lattices as examples.

References

Richard E. Borcherds, Automorphic forms on $\textrm {O}_{s+2,2}(\textbf {R})$ and infinite products, Invent. Math. 120 (1995), no. 1, 161–213. MR 1323986, DOI 10.1007/BF01241126
Jan H. Bruinier, Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR 1903920, DOI 10.1007/b83278
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
C. Dieckmann, A. Krieg, and M. Woitalla, The graded ring of modular forms on the Cayley half-space of degree two, Ramanujan J. 48 (2019), no. 2, 385–398. MR 3911795, DOI 10.1007/s11139-017-9970-x
Wolfgang Ebeling, Lattices and codes, 3rd ed., Advanced Lectures in Mathematics, Springer Spektrum, Wiesbaden, 2013. A course partially based on lectures by Friedrich Hirzebruch. MR 2977354, DOI 10.1007/978-3-658-00360-9
Jonas Gallenkämper and Aloys Krieg, The Hecke algebras for the orthogonal group $\rm SO(2,3)$ and the paramodular group of degree 2, Int. J. Number Theory 14 (2018), no. 9, 2409–2423. MR 3855466, DOI 10.1142/S1793042118501464
V. A. Gritsenko, Fourier-Jacobi functions in $n$ variables, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), no. Anal. Teor. Chisel i Teor. Funktsiĭ. 9, 32–44, 187–188 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 243–252. MR 982481, DOI 10.1007/BF01303648
V. A. Gritsenko, K. Hulek, and G. K. Sankaran, The Kodaira dimension of the moduli of $K3$ surfaces, Invent. Math. 169 (2007), no. 3, 519–567. MR 2336040, DOI 10.1007/s00222-007-0054-1
A. Hauffe-Waschbüsch, Verschiedene Aspekte von Modulformen in mehreren Variablen, Ph.D. thesis, RWTH Aachen, 2021.
Adrian Hauffe-Waschbüsch and Aloys Krieg, Congruence subgroups and orthogonal groups, Linear Algebra Appl. 618 (2021), 22–36. MR 4213701, DOI 10.1016/j.laa.2021.01.011
Martin Kneser, Quadratische Formen, Springer-Verlag, Berlin, 2002 (German). Revised and edited in collaboration with Rudolf Scharlau. MR 2788987, DOI 10.1007/978-3-642-56380-5
Aloys Krieg, Jacobi forms of several variables and the Maaßspace, J. Number Theory 56 (1996), no. 2, 242–255. MR 1373550, DOI 10.1006/jnth.1996.0016
Aloys Krieg, Integral orthogonal groups, Dynamical systems, number theory and applications, World Sci. Publ., Hackensack, NJ, 2016, pp. 177–195. MR 3444246
A. Krieg, M. Raum, and A. Wernz, The maximal discrete extension of the Hermitian modular group, Doc. Math. 26 (2021), 1871–1888.
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
K. G. Ramanathan, Discontinuous groups. II, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1964 (1964), 145–164. MR 171781
H. Wang, Reflective modular forms on lattices of prime level, Trans. Amer. Math. Soc. (2022), DOI 10.1090/tran/8604.
Haowu Wang, The classification of free algebras of orthogonal modular forms, Compos. Math. 157 (2021), no. 9, 2026–2045. MR 4297064, DOI 10.1112/S0010437X21007429
Haowu Wang and Brandon Williams, On some free algebras of orthogonal modular forms, Adv. Math. 373 (2020), 107332, 22. MR 4130466, DOI 10.1016/j.aim.2020.107332
H. Wang and B. Williams, Projective spaces as orthogonal modular varieties, arXiv:2008.08392, 2020.
H. Wang and B. Williams, Simple lattices and free algebras of modular forms, arXiv:2009.13343, 2020.
Annalena Wernz, Hermitian theta series and Maaßspaces under the action of the maximal discrete extension of the Hermitian modular group, Results Math. 75 (2020), no. 4, Paper No. 163, 10. MR 4160944, DOI 10.1007/s00025-020-01286-1