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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In an unpublished manuscript [A special case of the Fundamental Lemma, Mannheim (1994)] R. Weissauer has proven this conjecture (twisted and untwisted) for the group $$G=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.

##### MSC:
 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