Abstract
It is proved that, for n = 8, 9, 10, the natural algebra of automorphic forms of the group O+2,n(ℤ) acting on the n-dimensional symmetric domain of type IV is free, and the weights of generators are found. This extends results obtained in the author’s previous paper for n ≤ 7. On the other hand, as proved in a recent joint paper of the author and O. V. Shvartsman, similar algebras of automorphic forms cannot be free for n > 10.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. C. Shephard and J. A. Todd, “Finite unitary reflection groups,” Canad. J. Math., 6 (1954), 274–304.
C. Chevalley, “Invariants of finite groups generated by reflections,” Amer. J. Math., 77 (1955), 778–782.
O. V. Shvartsman, “Cocycles of complex reflection groups and the strong simple-connectedness of quotient spaces,” in: Problems in Group Theory and Homological Algebra [in Russian], Yaroslav. Gos. Univ., Yaroslavl, 1991, 32–39.
E. B. Vinberg and V. L. Popov, “Invariant Theory,” in: Algebraic Theory–4, Itogi Nauki i Tekhniki. Sovrem. Probl. Mat. Fund. Napr., vol. 55, VINITI, Moscow, 1989, 137–309; English transl.: in: Encycl. Math. Sci., vol. 55, Algebrai Geometry–IV, Springer-Verlag, Berlin, 123–278.
E. B. Vinberg, “Some free algebras of automorphic forms on symmetric domains of type IV,” Transform. Groups, 15:3 (2010), 701–741.
W. L. Baily and A. Borel, “Compactification of arithmetic quotients of bounded symmetric domains,” Ann. of Math. (2), 84 (1966), 442–528.
J. Igusa, “On Siegel modular forms of genus two,” Amer. J. Math., 84 (1962), 175–200.
E. B. Vinberg, “On the algebra of Siegel modular forms of genus 2,” Trudy Moskov. Mat. Obshch., 74:1, 1–16; English transl.: Trans. Moscow Math. Soc., 74 (2013), 1–13.
E. B. Vinberg and O. V. Shvartsman, “A criterion of smoothness at infinity for an arithmetic quotient of the future tube,” Funkts. Anal. Prilohzen., 51:1 (2017), 40–59; English transl.: Functional Anal. Appl., 51:1 (2017), 32–47.
V. V. Nikulin, “Integral symmetric bilinear forms and some of their applications,” Izv. Akad. Nauk SSSR, Ser. Mat., 43:1 (1979), 111–177; English transl.: Math. USSR Izv., 14:1 (1980), 103–167.
E. B. Vinberg, “On groups of unit elements of certain quadratic forms,” Mat. Sb., 87(129):1 (1972), 18–36; English transl.: Math. USSR Sb., 16:1 (1972), 17–35.
B. Saint-Donat, “Projective models of K3 surfaces,” Amer. J. Math., 96 (1974), 602–639.
E. B. Vinberg, “The two most algebraic K3 surfaces,” Math. Ann., 265:1 (1983), 1–21.
T. Johnsen and A. L. Knutsen, K3 Projective Models in Scrolls, Lecture Notes in Math., Springer-Verlag, Berlin, 2004.
V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps, vol. 1, Classification of Critical Points, Caustics and Wave Fronts, Birkhäuser, Boston–Basel–Stuttgart, 1985.
Vik. S. Kulikov and P. F. Kurchanov, “Complex algebraic manifolds: periods of integrals, Hodge structures,” in: Algebraic geometry–3, Itogi Nauki i Tekhniki. Sovrem. Probl. Mat. Fund. Napr., vol. 36, VINITI, Moscow, 1989, 5–231; English transl.: in: Encycl. Math. Sci., vol. 36, Algebraic Geometry–III, Springer-Verlag, Berlin, 1998, 1–218.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional′nyi Analiz i Ego Prilozheniya, Vol. 52, No. 4, pp. 38–61, 2018
This work was supported by RFBR grant no. 16-01-00818.
Rights and permissions
About this article
Cite this article
Vinberg, E.B. On Some Free Algebras of Automorphic Forms. Funct Anal Its Appl 52, 270–289 (2018). https://doi.org/10.1007/s10688-018-0237-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-018-0237-0