Geodesic
Spherical theta series
Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 289-304 
A. N. Andrianov. Spherical theta series. Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 289-304. http://geodesic.mathdoc.fr/item/SM_1989_62_2_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper explicit formulas are obtained which express the images under the action of the Hecke operators of theta series with spherical polynomials and rational translations which correspond to integral quadratic forms in an even number of variables, in terms of linear combinations of theta series of the same type. Transformation formulas are also obtained for such theta series under the action of the integral symplectic group. Bibliography: 7 titles.

[1] Andrianov A. N., Maloletkin G. N., “Povedenie teta-ryadov roda $n$ pri modulyarnykh podstanovkakh”, Izv. AN SSSR. Ser. matem., 39:2 (1975), 243–258 | MR | Zbl

[2] Andrianov A. N., Maloletkin G. N., “Povedenie teta-ryadov roda $n$ neopredelennykh kvadratichnykh form pri modulyarnykh podstanovkakh”, Tr. MIAH SSSR, 148, 1978, 5–15 | MR | Zbl

[3] Andrianov A. N., “Multiplikativnaya arifmetika zigelevykh modulyarnykh form”, UMN, 34:1 (1979), 67–135 | MR | Zbl

[4] Freitag E., “Die Wirkung von Heckeoperatoren auf Thetareihen mit harmonischen Koeffizienten”, Math. Ann., 258 (1982), 419–440 | DOI | MR | Zbl

[5] Andrianov A. N., Quadratic forms and Hecke operators, Grundlehren Math. Wiss, B. 286, Springer, Berlin–Heidelberg–New York, 1987 | MR | Zbl

[6] Andrianov A. N., “Deistvie operatorov Gekke na neodnorodnye teta-ryady”, Matem. sb., 135(173) (1986), 275–292 | MR

[7] Andrianov A. N., “Vychislenie koeffitsientov v formulakh preobrazovaniya neodnorodnykh teta-ryadov pod deistviem operatorov Gekke”, 3ap. nauchn. seminarov LOMI, 160 (1987), 5–12