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Computing Hecke eigenvalues below the cohomological dimension. (English) Zbl 1037.11037

Summary: Let \(\Gamma\) be a torsion-free finite-index subgroup of \(\text{SL}_n (\mathbb{Z})\) or \(\text{GL}_n (\mathbb{Z})\), and let \(\nu\) be the cohomological dimension of \(\Gamma\). We present an algorithm to compute the eigenvalues of the Hecke operators on \(H^{\nu-1} (\Gamma; \mathbb{Z})\), for \(n=2, 3\), and 4. In addition, we describe a modification of the modular symbol algorithm of A. Ash and L. Rudolph [Invent. Math. 55, 241–250 (1979; Zbl 0426.10023)] for computing Hecke eigenvalues on \(H^\nu (\Gamma; \mathbb{Z})\).

MSC:

11F75 Cohomology of arithmetic groups
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F60 Hecke-Petersson operators, differential operators (several variables)

Citations:

Zbl 0426.10023

Software:

PARI/GP; LiDIA
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References:

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