Geodesic
Multiplicative arithmetic of theta-series of odd quadratic forms
Izvestiya. Mathematics , Tome 59 (1995) no. 3, pp. 517-578.

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We study the action of the operators of symplectic Hecke rings of arbitrary degree on the theta-series of positive definite quadratic forms in an odd number of variables with vector-valued spherical coefficients corresponding to irreducible representations of the unitary group. We find a correspondence between generators of the Hecke rings and generalized Eichler–Brandt matrices. We apply these results to obtain conditions for linear dependence of theta-series, necessary conditions for lifting automorphic eigenforms on the orthogonal group to Siegel modular eigenforms, and an Euler expansion for symmetric Dirichlet series as a product of local zeta-functions with coefficients computed explicitly in terms of Eichler–Brandt matrices. 
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V. G. Zhuravlev. Multiplicative arithmetic of theta-series of odd quadratic forms. Izvestiya. Mathematics , Tome 59 (1995) no. 3, pp. 517-578. http://geodesic.mathdoc.fr/item/IM2_1995_59_3_a3/

[1] Andrianov A. N., “Vyrozhdenie kolets operatorov Gekke na prostranstvakh teta-ryadov”, Tr. MIAN, 157, Nauka, M., 1981, 3–18 | MR | Zbl

[2] Andrianov A. N., “Deistvie operatorov Gekke na neopredelennye teta-ryady”, Matem. sb., 131 (1986), 275–292 | MR | Zbl

[3] Andrianov A. N., “Sfericheskie teta-ryady”, Matem. sb., 134(176):3(11) (1987), 291–305 | MR | Zbl

[4] Andrianov A. N., Zhuravlev V. G., Modulyarnye formy i operatory Gekke, Nauka, M., 1990 | MR

[5] Zhuravlev V. G., “Yavnye formuly dvoistvennosti simplekticheskikh i ortogonalnykh operatorov Gekke na teta-ryadakh polozhitelno opredelennykh kvadratichnykh form”, Matem. sb., 130 (1986), 413–430 | MR | Zbl

[6] Zhuravlev V. G., “Lokalnaya dvoistvennost operatorov Gekke simplekticheskoi i ortogonalnoi gruppy”, Zapiski nauch. sem. LOMI, 185, Nauka, L., 1990, 37–59 | MR

[7] Zhuravlev V. G., “Sfericheskie teta-ryady i operatory Gekke”, Tr. MIAN, 207, Nauka, M., 1994, 93–122 | MR | Zbl

[8] Zhuravlev V. G., “Obobschennye matritsy Eikhlera–Brandta, operatory Gekke i vektornye teta-ryady”, Algebra i analiz, 5:3 (1993), 143–178 | MR | Zbl

[9] Lion Zh., Vern M., Predstavlenie Veilya, indeks Maslova i teta-ryady, Mir, M., 1983 | MR

[10] Böcherer S., “Über die Fourier–Jacobi–Entwicklung Siegelscher Eisensteinreihen”, Math. Z., 183 (1983), 21–46 | DOI | MR | Zbl

[11] Gelbart S., Automorphic forms on adelic groups, Annals of Math. Series, 83, Princeton University Press, 1975 | MR | Zbl

[12] Eichler M., Quadratische Formen und orthogonale Gruppen, Springer, Berlin, 1974 | MR | Zbl

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

[14] Freitag E., “The transformation formalism of vector valued theta-functions with respect to the Siegel modular group”, J. Indian Math. Soc., 52 (1987), 185–207 | MR | Zbl

[15] Rallis S., “The Eichler commutation relation and the continuous spectrum of the Weil representation”, Lecture Notes in Math., 728, 1979, 211–244 | DOI | MR | Zbl

[16] Rallis S., “Langland's functoriality and the Weil representation”, Amer. J. Math., 104 (1982), 465–515 | DOI | MR

[17] Satake J., “Theory of spherical functions on reductive algebraic groups over $p$-adic fields”, Publ. Math. IHES, 18 (1963), 1–65

[18] Yoshida H., “Siegel's modular forms and arithmetic of quadratic forms”, Inv. math., 60 (1980), 193–248 | DOI | MR | Zbl

[19] Yoshida H., “On Siegel modular forms obtained from theta series”, J. reine und angew. Math., 352 (1984), 184–219 | MR | Zbl