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Over a global field   K   (number field, or function field of a curve over a finite field   F  ), arithmetic duality theorems for the Galois cohomology of tori and finite Galois modules have long been known. More recent work investigates the case where   K   is the function field of a curve over a   p  ‐adic field.  For   K   the function field of a curve over the formal series field    F  =  C  (  (  t  )  )   , we establish analogous duality theorems. We thus control the obstruction to the local–global principle and to weak approximation for homogeneous spaces of tori.  There are differences with the afore described cases. For example, the Hasse principle need not hold for principal homogeneous spaces of a   K  ‐rational torus.

Dualité et principe local‐global pour les tores sur une courbe au‐dessus de ℂ(( t ))