The action of modular operators on the Fourier–Jacobi coefficients of modular forms

V. A. Gritsenko

Sbornik. Mathematics, Tome 47 (1984) no. 1, pp. 237-268

Résumé

The author studies the imbedding of the Hecke $p$-ring $L_p^{n+1}$ of the modular group $\mathrm{Sp}_{n+1}(\mathbf{Z})$ of genus $n+1$ in the Hecke ring $L_p^{n,1}$ of the group $\Gamma_{n,1}$ given by $$ \Gamma_{n,1}=\left\{\begin{pmatrix} A&0&B&*\\ *&*&*&*\\ C&0&D&*\\ 0&0&0&* \end{pmatrix}\in\mathrm{Sp}_{n+1}(\mathbf{Z})\right\}. $$ It is proved that the Hecke polynomial $Q_{n,1}^{(n+1)}(z)$ of $L_p^{n+1}$ splits over $L_p^{n,1}$, and the coefficients of the factors can be written explicitly in terms of the coefficients of the Hecke polynomial $Q^{(n)}(z)$ of genus $n$ and “negative” powers of a particular element $\Lambda$ of $L_p^{n,1}$. The "$-1$ power" of $\Lambda$ is computed and a formula for $\Lambda^{-2}$ is presented. The results that are obtained permit one to describe a large class of power series constructed from the Fourier–Jacobi coefficients by means of eigenfunctions with denominators depending only on the eigenvalues. Bibliography: 19 titles.

Bibliographie

[1] Andrianov A. N., “Multiplikativnaya arifmetika zigelevykh modulyarnykh form”, UMN, XXXIV:1 (1979), 67–135 | MR

[2] Andrianov A. N., “O razlozhenii mnogochlenov Gekke dlya simplekticheskoi gruppy roda $n$”, Matem. sb., 104(146) (1977), 390–427 | MR | Zbl

[3] Andrianov A. N., “Eilerovy razlozheniya teta-preobrazovanii zigelevykh modulyarnykh form roda $n$”, Matem. sb., 105(147) (1978), 291–341 | MR | Zbl

[4] Andrianov A. N., “Eilerovy proizvedeniya, otvechayuschie modulyarnym formam Zigelya roda 2”, UMN, XXIX:3 (1974), 43–110 | MR

[5] Andrianov A. N., Kalinin V. L., “Ob analiticheskikh svoistvakh standartnykh dzeta-funktsii zigelevykh modulyarnykh form”, Matem. sb., 106(148) (1972), 323–339 | MR

[6] Gritsenko V. A., “Analiticheskoe prodolzhenie simmetrichnykh kvadratov”, Matem. sb., 107(149) (1978), 323–346 | MR | Zbl

[7] Evdokimov S. A., “Analiticheskie svoistva eilerovykh proizvedenii dlya kongruents-podgrupp $\mathrm{Sp}_2(\mathbf Z)$”, Matem. sb., 110(152) (1979), 369–398 | MR | Zbl

[8] Evdokimov S. A., “Bazis iz sobstvennykh funktsii operatorov Gekke v teorii modulyarnykh form roda $n$”, Matem. sb., 115(157) (1981), 337–363 | MR | Zbl

[9] Pyatetskii-Shapiro I. I., Geometriya klassicheskikh oblastei i teoriya avtomorfnykh funktsii, Fizmatgiz, M., 1961 | MR

[10] Shimura G., Vvedenie v arifmeticheskuyu teoriyu avtomorfnykh funktsii, Mir, M., 1973 | MR | Zbl

[11] Seminaive H., Cavtan (10e annee, 1957–1958. Fonctions Automorphes), Paris, 1958

[12] Eichler M., “Über die Anzahl der linear unabhängigen Siegelschen Modulformen von gegebenem Gewicht”, Math. Ann., 213 (1975), 281–291 | DOI | MR | Zbl

[13] Maass H., “Über eine Spezialschar von Modulformen zweiten Grades”, Invent. math., 52 (1979), 95–104 ; “II”, 53 (1979), 249–253 ; “III”, 53, 255–265 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR

[14] Eichler M., Über Jacobische Formen, preprint, 1980

[15] Eichler M., Introduction to the theory of algebraic numbers and functions, New–York–London, 1966 | MR | Zbl

[16] Andrianov A. N., Maloletkin G. N., “Povedenie teta-ryadov roda $n$ pri modulyarnykh podstanovkakh”, Izv. AN. Seriya matem., 39 (1975), 243–258 | MR | Zbl

[17] H. Maass, Siegel's mosulav forms and Dirichlet series, Lecture notes in Math., 216, Springer-Verlag, Berlin, 1971 | MR | Zbl

[18] Andrianov A. N., “Sfericheskie funktsii dlya $\mathrm{GL}_n$ nad lokalnymi polyami i summirovanie ryadov Gekke”, Matem. sb., 83 (125) (1970), 429–451 | MR | Zbl

[19] Satake I., “Theory of spherical functions on reductive algebraic group over p-adic fields”, Publ. Math., IHES, 1963, no. 18 | MR