## Hecke operators for non-congruence subgroups of Bianchi groups

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**139**(2011), 3853-3865 Request permission

## Abstract:

We prove that the action of the Hecke operators on the cohomology of a finite index non-congruence subgroup $\Gamma$ of a Bianchi group is essentially the same as the action of Hecke operators on the cohomology groups of $\hat {\Gamma }$, the congruence closure of $\Gamma$. This is a generalization of Atkin’s conjecture, first confirmed in a special case by Serre in $1987$ and proved in general by Berger in $1994$.## References

- Avner Ash and Glenn Stevens,
*Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues*, J. Reine Angew. Math.**365**(1986), 192–220. MR**826158** - A. O. L. Atkin, Wen-Ching Winnie Li, and Ling Long,
*On Atkin and Swinnerton-Dyer congruence relations. II*, Math. Ann.**340**(2008), no. 2, 335–358. MR**2368983**, DOI 10.1007/s00208-007-0154-7 - Hyman Bass,
*Algebraic $K$-theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR**0249491** - Gabriel Berger,
*Hecke operators on noncongruence subgroups*, C. R. Acad. Sci. Paris Sér. I Math.**319**(1994), no. 9, 915–919 (English, with English and French summaries). MR**1302789** - Frank Calegari and Barry Mazur,
*Nearly ordinary Galois deformations over arbitrary number fields*, J. Inst. Math. Jussieu**8**(2009), no. 1, 99–177. MR**2461903**, DOI 10.1017/S1474748008000327 - P. M. Cohn,
*On the structure of the $\textrm {GL}_{2}$ of a ring*, Inst. Hautes Études Sci. Publ. Math.**30**(1966), 5–53. MR**207856**, DOI 10.1007/BF02684355 - Tobias Finis, Fritz Grunewald, and Paulo Tirao,
*The cohomology of lattices in $\textrm {SL}(2,\Bbb C)$*, Experiment. Math.**19**(2010), no. 1, 29–63. MR**2649984**, DOI 10.1080/10586458.2010.10129067 - Fritz Grunewald and Joachim Schwermer,
*On the concept of level for subgroups of $\textrm {SL}_2$ over arithmetic rings*, Israel J. Math.**114**(1999), 205–220. MR**1738680**, DOI 10.1007/BF02785578 - S. K. Gupta and M. P. Murthy,
*Suslin’s work on linear groups over polynomial rings and Serre problem*, ISI Lecture Notes, vol. 8, Macmillan Co. of India, Ltd., New Delhi, 1980. MR**611151** - G. Harder,
*Eisenstein cohomology of arithmetic groups. The case $\textrm {GL}_2$*, Invent. Math.**89**(1987), no. 1, 37–118. MR**892187**, DOI 10.1007/BF01404673 - Min Ho Lee,
*Hecke operators on cohomology*, Rev. Un. Mat. Argentina**50**(2009), no. 1, 99–144. MR**2643521** - A. W. Mason,
*Anomalous normal subgroups of $\textrm {SL}_2(K[x])$*, Quart. J. Math. Oxford Ser. (2)**36**(1985), no. 143, 345–358. MR**800766**, DOI 10.1093/qmath/36.3.345 - Jean-Pierre Serre,
*Le problème des groupes de congruence pour SL2*, Ann. of Math. (2)**92**(1970), 489–527 (French). MR**272790**, DOI 10.2307/1970630 - Jean-Pierre Serre,
*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504**, DOI 10.1007/978-3-642-61856-7 - J. G. Thompson,
*Hecke operators and noncongruence subgroups*, Group theory (Singapore, 1987) de Gruyter, Berlin, 1989, pp. 215–224. Including a letter from J.-P. Serre. MR**981844** - Klaus Wohlfahrt,
*Über Dedekindsche Summen und Untergruppen der Modulgruppe*, Abh. Math. Sem. Univ. Hamburg**23**(1959), 5–10 (German). MR**102559**, DOI 10.1007/BF02941021 - Klaus Wohlfahrt,
*An extension of F. Klein’s level concept*, Illinois J. Math.**8**(1964), 529–535. MR**167533**

## Additional Information

**Saeid Hamzeh Zarghani**- Affiliation: Department of Mathematics, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany
- Email: zarghani@math.uni-duesseldorf.de, zarghani.s@gmail.com
- Received by editor(s): May 13, 2010
- Received by editor(s) in revised form: September 21, 2010
- Published electronically: March 25, 2011
- Additional Notes: The author was supported in part by Graduirtenkolleg Homotopie und Kohomologie (GRK1150)
- Communicated by: Kathrin Bringmann
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 3853-3865 - MSC (2010): Primary 11F03, 11F25, 11F75, 20G30; Secondary 19B37
- DOI: https://doi.org/10.1090/S0002-9939-2011-10807-4
- MathSciNet review: 2823032

Dedicated: Before the first draft of this work was completed, Fritz Grunewald tragically passed away. Indeed, without his guidance and support this work would never have been done. I dedicate this paper to his memory with admiration, gratitude and love.