×

zbMATH — the first resource for mathematics

Computations of cuspidal cohomology of congruence subgroups of SL(3,\({\mathbb{Z}})\). (English) Zbl 0552.10015
In the classical case of congruence subgroups \(\Gamma\) of \(SL_ 2({\mathbb{Z}})\) the space of cusp forms with respect to \(\Gamma\) is well understood. Via the Eichler-Shimura isomorphism the cusp cohomology of \(\Gamma\) isolates special types (depending on the chosen coefficients) of cusp forms. This relation between an analytically defined subspace of the cohomology of \(\Gamma\) and cuspidal automorphic forms with respect to \(\Gamma\) is also valid more generally for an arithmetic subgroup \(\Gamma\) of a semi-simple algebraic group G over \({\mathbb{Q}}\). It is this cohomological interpretation of cusp forms which allows one to introduce geometrical methods in the study of cusp forms.
Using this approach the paper under review contains explicit computations of the dimension of the cusp cohomology of the congruence groups \(\Gamma_ 0(p)=\{(a_{ij})\in SL_ 3({\mathbb{Z}})| a_{i1}\equiv 0 mod p,\quad 2\leq i\leq 3\}\) of \(SL_ 3({\mathbb{Z}})\) for primes \(p\leq 113\). For all but four primes \(\leq 113\), this dimension turns out to be zero. For \(p=53\), 61, 79, 89 it is two. In these cases also the action of the Hecke algebra is considered.
Since the cusp cohomology of \(\Gamma\) can be interpreted in this case as the image of the cohomology of the arithmetic quotient \(\Gamma\) \(\setminus X\) \((X=SL_ 3({\mathbb{R}})/SO(3))\) with compact supports in the usual cohomology the computational problem is converted into finite- dimensional linear algebra. The authors present explicit algorithms to perform these computations. [A general non-vanishing result for the cusp cohomology of arithmetic subgroups of \(SL_ 3({\mathbb{Z}})\) may be found in R. Lee and the reviewer, Invent. Math. 73, 189-239 (1983; Zbl 0525.10014)].
Reviewer: J.Schwermer

MSC:
11F27 Theta series; Weil representation; theta correspondences
22E40 Discrete subgroups of Lie groups
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
68W99 Algorithms in computer science
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Borel, A.; Serre, J.-P., Corners and arithmetic groups, Comm. math. helv., 48, 436-491, (1973) · Zbl 0274.22011
[2] Kazhdan, D., Connection of the dual space of a group with the structure of its closed subgroups, Funktsional anal i prilozhen, 1, 71-74, (1967) · Zbl 0168.27602
[3] Ash, A., Cohomology of congruence subgroups of SL_n(\(Z\)), Math. ann., 249, 55-73, (1980) · Zbl 0438.20035
[4] Borel, A., Cohomology of arithmetic groups, (), 435-442
[5] Wallach, N., On the constant term of a square integrable automorphic form, (), Monographs and Studies in Mathematics No. 18 · Zbl 0554.22004
[6] Lee, R.; Schwermer, J., Cohomology of arithmetic subgroups of SL3 at infinity, Crelle’s J., 330, 100-131, (1982) · Zbl 0463.57014
[7] Ash, A.; Rudolph, L., The modular symbol and continued fractions in higher dimensions, Invent. math., 55, 241-250, (1979) · Zbl 0426.10023
[8] Shimura, G., ()
[9] Ash, A., On the top Betti number of subgroups of SL(n, \(Z\)), Math. ann., 264, 277-281, (1983) · Zbl 0505.20035
[10] Satake, I., Spherical functions and Ramanujan conjecture, (), 258-264, Providence, R. I.
[11] Dold, A.; Eckmann, B., (), Lecture Notes in Mathematics No. 476
[12] Wada, H., A table of Hecke operators, II, (), 380-384 · Zbl 0273.10019
[13] Gelbart, S.; Jacquet, H., A relation between automorphic forms on GL(2) and GL(3), (), 3348-3350 · Zbl 0373.22008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.