Generating Elements of the Annihilating Ideal for Modular Symbols
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 4, pp. 27-38.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper, the concrete generators are found for the annihilating ideal of the modular symbols associated with $SL(s;\mathbb{Z})$. Using this result, the (co)homology of subgroups of finite index is evaluated for $SL(s;\mathbb{Z})$.
Keywords: modular form, modular element, modular symbol, Eichler–Shimura module, annihilator ideal, generators of an ideal.
@article{FAA_2003_37_4_a7,
     author = {V. A. Bykovskii},
     title = {Generating {Elements} of the {Annihilating} {Ideal} for {Modular} {Symbols}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {27--38},
     publisher = {mathdoc},
     volume = {37},
     number = {4},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2003_37_4_a7/}
}
TY  - JOUR
AU  - V. A. Bykovskii
TI  - Generating Elements of the Annihilating Ideal for Modular Symbols
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2003
SP  - 27
EP  - 38
VL  - 37
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2003_37_4_a7/
LA  - ru
ID  - FAA_2003_37_4_a7
ER  - 
%0 Journal Article
%A V. A. Bykovskii
%T Generating Elements of the Annihilating Ideal for Modular Symbols
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2003
%P 27-38
%V 37
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2003_37_4_a7/
%G ru
%F FAA_2003_37_4_a7
V. A. Bykovskii. Generating Elements of the Annihilating Ideal for Modular Symbols. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 4, pp. 27-38. http://geodesic.mathdoc.fr/item/FAA_2003_37_4_a7/

[1] Eichler M., “Eine Verallgemeinerung der Abelschen Integrale”, Math. Z., 67 (1957), 267–298 | DOI | MR | Zbl

[2] Shimura G., Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, NJ, 1971 ; Shimura G., Vvedenie v arifmeticheskuyu teoriyu avtomorfnykh funktsii, Mir, M., 1973 | MR | Zbl | MR | Zbl

[3] Manin Yu. I., “Parabolicheskie formy i dzeta-funktsii modulyarnykh krivykh”, Izv. AN SSSR, ser. matem., 36 (1972), 19–66 | MR | Zbl

[4] Manin Yu. I., “Periody parabolicheskikh form i $p$-adicheskie ryady Gekke”, Matem. sb., 92:3 (1973), 378–401 | MR | Zbl

[5] Manin Yu. I., “Explicit formulas for the eigenvalues of Hecke operators”, Acta Arith., 24 (1973), 239–249 | DOI | MR | Zbl

[6] Ash A., Rudolph L., “The modular symbol and continued fractions in higher dimensions”, Invent. Math., 55 (1979), 241–250 | DOI | MR | Zbl

[7] Ash A., “Unstable cohomology of $\mathrm{SL}(n,\mathscr{O})$”, J. Algebra, 167:2 (1994), 330–342 | DOI | MR | Zbl

[8] Ash A., Sinnott W., “An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the $\mod p$ cohomology of $\mathrm{GL}(n,\mathbb{Z})$”, Duke Math. J., 105:2 (2000), 1–24 | DOI | MR | Zbl

[9] Brown K., Cohomology of Groups, Springer-Verlag, New York–Berlin, 1982 ; Braun K., Kogomologii grupp, Nauka, M., 1987 | MR | Zbl | MR

[10] Borel A., Serre J.-P., “Corners and arithmetic groups”, Comm. Math. Helv., 48 (1973), 436–491 | DOI | MR | Zbl

[11] Ash A., “Cohomology of congruence subgroups of $\mathrm{SL}(n,\mathbb{Z})$”, Math. Ann., 249:1 (1980), 55–73 | DOI | MR | Zbl

[12] Reeder M., Modular Symbols and Steinberg representation., Preprint, 1990 | MR | Zbl

[13] Reeder M., “The Steinberg Module and the Cohomology of Arithmetic Groups”, J. Algebra, 141:2 (1991), 287—315 | DOI | MR | Zbl

[14] Soule C., “The cohomology of $\mathrm{SL}_3(\mathbb{Z})$”, Topology, 17 (1978), 1–22 | DOI | MR | Zbl