For SL(2) over local fields of any characteristic, including p=2 where Shalika's expansion fails, the paper proves an endoscopic germ expansion for orbital integrals near the center and states a conjecture for arbitrary groups.
Stabilisation et germes pour $SL(2)$ en toutes caract\'eristiques
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abstract
We give the stabilisation of local orbital integrals and the trace formula over a global field for $SL(2)$ with proofs valid in any characteristic. New features appear in characteristic 2. We obtain, via the stabilisation, an asymptotic expansion near the identity of local orbital integrals which is equivalent, up to a Fourier transform, to the standard germ expansion due to Shalika when the characteristic is not 2 but it is new in characteristic 2. Similarly the fine expansion of the unipotent contribution to the trace formula cannot be obtained using Arthur's techniques and a pre-stabilisation is necessary.
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Germ expansion for SL(2) in arbitrary characteristics
For SL(2) over local fields of any characteristic, including p=2 where Shalika's expansion fails, the paper proves an endoscopic germ expansion for orbital integrals near the center and states a conjecture for arbitrary groups.