Chow groups of châtelet surfaces over dyadic fields (Q290933)

Let \(X\) be a cubic Châtelet surface over a field \(K\), i.e. a smooth projective model of the surface \(y^2-dx^2=f(x)\) in \(A^3_K\), where \(d\in K^\ast\) and \(f(x)\in K[x]\) is a monic cubic separable polynomial. Denote by \(A_0(X)\) the degree-zero part of the Chow group of zero-cycles on \(X\) modulo rational equivalence. Suppose moreover that \(K\) is perfect. Then \(A_0(X)\) is a 2-torsion group, and if \(d\in K^{\ast 2}\), \(A_0(X)\cong A_0 (P^2_K)=(0)\) [\textit{J.-L. Colliot-Thelene} and \textit{D. Coray}, Compos. Math. 39, 301--332 (1979; Zbl 0386.14003)]. The subject of the present paper is the determination of \(A_0(X)\) when \(K\) is a finite extension of \(Q_p\) and \(d\notin K^{\ast 2}\). The answer depends on the factorization of \(f(x)\) in \(K[x]\) and is completely known except when \(f(x)=x(x^2-e)\), with \(e\in K^\ast\backslash K^{\ast 2}\). In this case, put \(L=K(\sqrt d)\) and \(E=K(\sqrt e)\). If \(L=E\), then \(A_0(X)\) is trivial. Henceforward, suppose \(L\neq E\). The cases: (1) \(p\neq 2\), (2) \(p=2\) and \(L/K\) is unramified, (3) \(K=Q_2\), \(L/K\) is ramified, or \(L/K\) is unramified and \(v_K(e)\equiv 2\pmod{4}\), have been solved by \textit{S. Pisolkar} [Indag. Math., New Ser. 19, No. 3, 427--439 (2008; Zbl 1245.14026)]. Here the author deals with some of the remaining cases : (4) \(L/K\) is unramified and \(v_K(e)\equiv 2\pmod{4}\), (5) \(p=2\), \(L/Q_2\) is totally ramified, and \(v_K(e)\) and the conductor of \(L/K\) have different parities, (6) \(K=Q_2(\sqrt 2)\), \(L/K\) is ramified and \(v_K(d)\) is even. The whole approach rests on a number-theoretic description due to \textit{J. L. Colliot-Thelene} and \textit{J. J. Sansuc} [in: Journees de geometrie algebrique, Angers/France 1979, 223--237 (1980; Zbl 0451.14018)], who showed that \(A_0(X)\) is isomorphic to the image of a certain map \(M:=\{x\in K; x(x^2- e)\in N_{L/K}L\}\to (K^\ast/N_{L/K} L^\ast)^2\). It remains to determine this image by normic computations in local fields.