\(\ell\)-modular zeta functions. (Fonctions zêta \(\ell\)-modulaires.) (French. English summary) Zbl 1282.11157
Let \(D\) be a division algebra over a non-archimedean local field with residual characteristic \(p\). The author generalizes the theory of zeta functions of \(\mathrm{GL}_m(D)\) of Godement and Jacquet for complex representations to the case of \(\overline{\mathbb F}_{\ell}\)-representations, \(\ell \neq p\). He shows that the proof by Godement and Jacquet remains valid for any algebraically closed field of characteristic 0. The result for \(\overline{\mathbb F}_{\ell}\)-representations is proved using reduction modulo \(\ell\) from \(\overline{\mathbb Q}_{\ell}\)-representations supplied with a \(\overline{\mathbb Z}_{\ell}\)-structure.
The \(L\)-function is explicitly computed for certain representations (for all representations in case \(\ell\) does not divide the number of elements of \(\mathrm{GL}_m(k_D)\), \(k_D\) being the residue field of \(D\)).
The \(L\)-function is explicitly computed for certain representations (for all representations in case \(\ell\) does not divide the number of elements of \(\mathrm{GL}_m(k_D)\), \(k_D\) being the residue field of \(D\)).
Reviewer: J. G. M. Mars (Utrecht)
MSC:
| 11S40 | Zeta functions and \(L\)-functions |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
Citations:
Zbl 0244.12011References:
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