Igusa modular forms and 'the simplest' Lorentzian Kac–Moody algebras

V. A. Gritsenko   ; V. V. Nikulin

Sbornik. Mathematics, Tome 187 (1996) no. 11, pp. 1601-1641

Résumé

Automorphic corrections for the Lorentzian Kac–Moody algebras with the simplest generalized Cartan matrices of rank 3, $$ A_{1,0}=\begin{pmatrix} \hphantom{-}{2}&\hphantom{-}{0}&{-1} \\ \hphantom{-}{0}&\hphantom {-}{2}&{-2} \\ {-1}&{-2}&\hphantom {-}{2} \end{pmatrix} \quad\text{and}\quad A_{1,\mathrm {I}}=\begin {pmatrix} \hphantom {-}{2}&{-2}&{-1} \\ {-2}&\hphantom {-}{2}&{-1} \\ {-1}&{-1}&\hphantom {-}{2} \end{pmatrix} $$ are found. For $A_1,0$ this correction, which is a generalized Kac–Moody Lie super algebra, is delivered by $\chi_{35}(Z)$, the Igusa $\operatorname{Sp}_4(\mathbb Z)$-modular form of weight $35$, while for $A_{1,\mathrm{I}}$ it is given by some Siegel modular form $\widetilde \Delta_{30}(Z)$ of weight 30 with respect to a 2-congruence subgroup of $\operatorname{Sp}_4(\mathbb Z)$. Expansions of $\chi_{35}(Z)$ and $\widetilde\Delta_{30}(Z)$ in infinite products are obtained and the multiplicities of all the roots of the corresponding generalized Lorentzian Kac–Moody superalgebras are calculated. These multiplicities are determined by the Fourier coefficients of certain Jacobi forms of weight 0 and index 1. The method adopted for constructing $\chi_{35}(Z)$ and $\widetilde\Delta_{30}(Z)$ leads in a natural way to an explicit construction (as infinite products or sums) of Siegel modular forms whose divisors are Humbert surfaces with fixed discriminants. A geometric construction of these forms was proposed by van der Geer in 1982. To show the prospects for further studies, the list of all hyperbolic symmetric generalized Cartan matrices $A$ with the following properties is presented: $A$ is a matrix of rank 3 and of elliptic or parabolic type, has a lattice Weyl vector, and contains a parabolic submatrix $\widetilde{\mathbb A}_1$.

Bibliographie

[1] Baily W. L., “Fourier–Jacobi series”, Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math., 9, eds. A. Borel, G. D. Mostow, Amer. Math. Soc., Providence, RI, 1966, 296–300 | MR

[2] Borcherds R., “Generalized Kac–Moody algebras”, J. Algebra, 115 (1988), 501–512 | DOI | MR | Zbl

[3] Borcherds R., “The monster Lie algebra”, Adv. Math., 83 (1990), 30–47 | DOI | MR | Zbl

[4] Borcherds R., “The monstrous moonshine and monstrous Lie superalgebras”, Invent. Math., 109 (1992), 405–444 | DOI | MR | Zbl

[5] Borcherds R., “Sporadic groups and string theory”, Proc. European Congress of Mathem., 1992, 411–421 | MR

[6] Borcherds R., “Automorphic forms on $\operatorname{O}_{s+2,2}(\mathbb R)$ and infinite products”, Invent. Math., 120 (1995), 161–213 | DOI | MR | Zbl

[7] Borcherds R., The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Preprint, 1994 | MR | Zbl

[8] Eichler M., Zagier D., The theory of Jacobi forms, Progress in Math., 55, Birkhäuser, 1985 | MR

[9] Feingold A. J., Frenkel I. B., “A hyperbolic Kac–Moody algebra and the theory of Siegel modular forms of genus 2”, Math. Ann., 263:1 (1983), 87–144 | DOI | MR | Zbl

[10] Freitag E., Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983 | MR

[11] van der Geer G., Hilbert modular surfaces, Erg. Math. Grenzgeb., 3. Folge, 16, Springer-Verlag, Berlin, 1988 | Zbl

[12] van der Geer G., “On the geometry of a Siegel modular threefold”, Math. Ann., 260 (1982), 317–350 | DOI | MR | Zbl

[13] Gritsenko V. A., “Irrationality of the moduli spaces of polarized abelian surfaces”, Internat. Math. Res. Notices, 6 (1994), 235–243 ; “In full form in Abelian varieties”, Proc. of the Egloffstein conference (1993), de Gruyter, Berlin, 1995, 63–81 | DOI | MR | Zbl | MR | Zbl

[14] Gritsenko V. A., “Modulformen zur Paramodulgruppe und Modulräume der Abelschen Varietäten”, Mathematica Gottingensis Schrift. des SFB. Geometrie und Analysis, 12 (1995), 1–89 | MR

[15] Gritsenko V. A., “Induktsiya v teorii dzeta-funktsii”, Algebra i analiz, 6:1 (1994), 3–63 | MR | Zbl

[16] Gritsenko V. A., “Funktsii Yakobi i eilerovy proizvedeniya dlya ermitovykh modulyarnykh form”, Zapiski nauch. sem. LOMI, 183, Nauka, L., 1990, 77–123 | MR

[17] Gritsenko V. A., “Deistvie modulyarnykh operatorov na koeffitsienty Fure–Yakobi modulyarnykh form”, Matem. sb., 119 (1982), 248–277 | MR | Zbl

[18] Gritsenko V., Hulek K., Minimal Siegel modular threefolds, Prépublication de l'Institut Fourier, No 312, 1995 ; arXiv: alg-geom/9506017 | MR

[19] Gritsenko V. A., Nikulin V. V., “Siegel automorphic form correction of some Lorentzian Kac–Moody Lie algebras”, Amer. J. Math., 1996 (to appear) ; E-print alg-geom/9504006 | MR

[20] Gritsenko V. A., Nikulin V. V., “Siegel automorphic form correction of a Lorentzian Kac–Moody algebra”, C. R. Acad. Sci. Paris Sér. A–B, 321 (1995), 1151–1156 | MR | Zbl

[21] Gritsenko V. A., Nikulin V. V., “$K3$ surfaces, Lorentzian Kac–Moody algebras and mirror symmetry”, Math. Res. Lett., 3:2 (1996), 211–229 ; arXiv: alg-geom/9510008 | MR | Zbl

[22] Gritsenko V. A., Nikulin V. V., The Igusa modular forms and “the simplest” Lorentzian Kac–Moody algebras, Preprint No 3 Forschergruppe Automorphe Formen Universität Mannheim und Universität Heidelberg, 1996, p. 1–40 ; arXiv: alg-geom/9603010 | MR

[23] Igusa J., “On Siegel modular forms of genus two, II”, Amer. J. Math., 84:2 (1964), 392–412 | DOI | MR | Zbl

[24] Igusa J., “Ring of modular forms of degree two over $\mathbb Z$”, Amer. J. Math., 101 (1979), 132–148 | DOI | MR

[25] Kac V., Infinite dimensional Lie algebras, Cambridge Univ. Press, Cambridge, 1990 | MR

[26] Kac V., “Lie superalgebras”, Adv. Math., 26 (1977), 8–96 | DOI | MR | Zbl

[27] Kac V., “Infinite-dimensional algebras, Dedekind's $\eta$-function, classical Möbius function and the very strange formula”, Adv. Math., 30 (1978), 85–136 | DOI | MR | Zbl

[28] Maass H., “Die Multiplikatorsysteme zur Siegelschen Modulgruppe”, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1964, no. 11, 125–135 | MR | Zbl

[29] Maass H., “Über ein Analogon zur Vermutung von Saito-Kurokawa”, Invent. Math., 60 (1980), 85–104 | DOI | MR | Zbl

[30] Nikulin V. V., “Konechnye gruppy avtomorfizmov kelerovykh poverkhnostei tipa $K3$”, Tr. MMO, 38, URSS, M., 1979, 75–137 | MR | Zbl

[31] Nikulin V. V., “Tselochislennye simmetricheskie bilineinye formy i nekotorye ikh geometricheskie prilozheniya”, Izv. AN SSSR. Ser. matem., 43:1 (1979), 111–177 | MR | Zbl

[32] Nikulin V. V., “O faktor-gruppakh grupp avtomorfizmov giperbolicheskikh form po podgruppam, porozhdennym 2-otrazheniyami. Algebro-geometricheskie prilozheniya”, Itogi nauki i tekhn. Sovr. probl. matem., 18, VINITI, M., 1981, 3–114 | MR

[33] Nikulin V. V., “Ob arifmeticheskikh gruppakh, porozhdennykh otrazheniyami, v prostranstvakh Lobachevskogo”, Izv. AN SSSR. Ser. matem., 44:3 (1980), 637–668 | MR

[34] Nikulin V. V., “O klassifikatsii arifmeticheskikh grupp, porozhdennykh otrazheniyami, v prostranstvakh Lobachevskogo”, Izv. AN SSSR. Ser. matem., 45:1 (1981), 113–142 | MR | Zbl

[35] Nikulin V. V., “Involyutsii tselochislennykh kvadratichnykh form i ikh prilozheniya k veschestvennoi algebraicheskoi geometrii”, Izv. AN SSSR. Ser. matem., 47:1 (1983), 109–188 | MR

[36] Nikulin V. V., “Poverkhnosti tipa $K3$ s konechnoi gruppoi avtomorfizmov i gruppoi Pikara ranga tri”, Tr. MIAN, 165, Nauka, M., 1984, 119–142 | MR | Zbl

[37] Nikulin V. V., “Discrete reflection groups in Lobachevsky spaces and algebraic surfaces”, Proc. Int. Congr. Math., v. 1 (Berkeley, 1986), 654–669 | MR

[38] Nikulin V. V., A lecture on Kac–Moody Lie algebras of the arithmetic type, Preprint 1994-16, Queen's University, Canada, 1995 ; arXiv: alg-geom/9412003 | Zbl

[39] Nikulin V. V., “Gruppy otrazhenii v prostranstvakh Lobachevskogo i tozhdestvo dlya znamenatelya lorentsevykh algebr Katsa–Mudi”, Izv. AN SSSR. Ser. matem., 60:2 (1996), 73–106 | MR | Zbl

[40] Nikulin V. V., The remark on discriminants of $K3$ surfaces moduli as sets of zeros of automorphic forms, arXiv: alg-geom/9512018

[41] Ray U., “A character formula for generalized Kac–Moody superalgebras”, J. Algebra, 177 (1995), 154–163 | DOI | MR | Zbl