Torseurs sous une variété abélienne sur R((t))

Abstract:

Let V be an henselian discrete valuation ring with real closed residue field and let k be its quotient ring; we denote by k ⁺ and k ⁻ the two real closures of k. Consider a k-abelian variety A. We compute the Galois-cohomology group H ¹(k,A) in terms of the reduction of the dual variety of A and of the semi-algebraic topology of A(k ⁺) and A(k ⁻). The tools we need are Ogg's results concerning valuation rings with algebraically closed residue field, Hochschild–Serre spectral sequence and Scheiderer's local-global principles. At the end we study more precisely the case of an elliptic curve.