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A twisted fundamental lemma for \(GSp_4\). (Ein getwistetes fundamentales Lemma für die \(GSp_4\).) (German) Zbl 0899.11025
Bonner Mathematische Schriften. 303. Bonn: Univ. Bonn., Mathematisch-Naturwissenschaftliche Fakultät, 71 S. (1997).
One of the central techniques in the comparison of the Arthur-Selberg trace formula is the stabilization of the trace formula for a reductive group \(G\). Here certain summands of the trace formula are taken together to form the so-called \(\kappa\)-orbital integrals, which, according to a conjecture of Langlands, should be expressible by stable orbital integrals on smaller groups, the endoscopic groups to \(G\). The latter conjecture can be formulated in local terms, for example over a nonarchimedean local field \(F\). In this case also a twisted version can be formulated, in which the group \(G\) is replaced by its Weil restriction from an unramified extension \(E\) of \(F\) to \(F\) and everything is twisted by the action of the Frobenius automorphism.
In an unpublished manuscript [A special case of the Fundamental Lemma, Mannheim (1994)] R. Weissauer has proven this conjecture (twisted and untwisted) for the group \(G= GSp_4\). The idea is to use the trace formula, i.e. global methods, and some explicit calculations.
In the present doctoral thesis the twisted version is proven for \(F=\mathbb{Q}_p\), \(p\neq 2\) and a special spherical function \(\phi_m\) by direct, i.e. local methods. The function \(\phi_m\) is such that it suffices for the stabilization of the Lefschetz trace formula for the Hecke \(\times\) Galois-action on the cohomology of the corresponding Shimura variety.

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11G35 Varieties over global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings