Geodesic
On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 5-17 
A. N. Andrianov. On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 5-17. http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a0/
@article{ZNSL_2012_404_a0,
     author = {A. N. Andrianov},
     title = {On interaction of symplectic and orthogonal {Hecke{\textendash}Shimura} rings of one-class quadratic forms},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--17},
     year = {2012},
     volume = {404},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a0/}
}
TY  - JOUR
AU  - A. N. Andrianov
TI  - On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2012
SP  - 5
EP  - 17
VL  - 404
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a0/
LA  - en
ID  - ZNSL_2012_404_a0
ER  - 
%0 Journal Article
%A A. N. Andrianov
%T On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms
%J Zapiski Nauchnykh Seminarov POMI
%D 2012
%P 5-17
%V 404
%U http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a0/
%G en
%F ZNSL_2012_404_a0

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Transformation formulas of theta-series with harmonic polynomials of one-class quadratic forms under Hecke operators are interpreted as a result of interaction of standard representation of symplectic Hecke–Shimura ring on theta-series with natural representation of orthogonal Hecke–Shimura ring on the same theta-series considered as invariants of quadratic forms. Properties of the interaction maps and their relations with action of Hecke operators are considered.

[1] A. N. Andrianov, “Deistvie operatorov Gekke na neodnorodnye teta-ryady.”, Mat. sb., 131(173):3 (1986), 275–292 ; “The action of Hecke operators on nonhomogeneous theta series”, Math. USSR–Sb., 59:2 (1988), 269–285 | MR | Zbl | DOI | Zbl

[2] A. N. Andrianov, “Simmetrii garmonicheskikh teta-funktsii tselochislennykh kvadratichnykh form”, Uspekhi mat. nauk, 50:4 (1995), 3–44 | MR | Zbl

[3] A. N. Andrianov, “Garmonicheskie teta-funktsii i operatory Gekke”, Algebra i analiz, 8:5 (1996), 1–31 ; “Harmonic theta-functions and Hecke operators”, St. Petersburg Math. J., 8:5 (1997), 695–720 | MR | Zbl

[4] A. N. Andrianov, “Vokrug pod'ema Ikedy”, Funkts. analiz i ego prilozh., 35:2 (2001), 70–74 | DOI | MR | Zbl

[5] A. N. Andrianov, “O dzeta-funktsiyakh ortogonalnykh grupp odnoklassnykh polozhitelno opredelennykh kvadratichnykh form”, Algebra i analiz, 17:4 (2005), 3–41 ; “On zeta-functions of orthogonal groups of single-class positive definite quadratic forms”, St. Petersburg Math. J., 17:4 (2006), 553–579 | MR | Zbl

[6] A. N. Andrianov, “Dzeta-funktsii ortogonalnykh grupp tselochislennykh polozhitelno opredelennykh kvadratichnykh form”, Uspekhi mat. nauk, 61:6 (2006), 3–44 | DOI | MR | Zbl

[7] A. N. Andrianov, “Interaction sums and action of Hecke operators on theta-series”, Acta Arithm., 140:3 (2009), 271–304 | DOI | MR | Zbl

[8] J. W. S. Cassels, Rational Quadratic Forms, Academic Press, 1978 | MR | Zbl

[9] A. N. Andrianov, “On interaction of Hecke–Shimura rings: symplectic versus orthogonal”, Acta Arithm., 155:1 (2012), 97–107 | DOI | MR | Zbl