Arithmetical Torelli theorems for K3 surfaces and for curves (Q1103682)

Let (X,L) and (X',L') be polarized \(K_ 3\)-surfaces and let \(f: H^ 2(X',{\mathbb{Z}})(1)\to H^ 2(X,{\mathbb{Z}})\) (1) be a given isomorphism of \({\mathbb{Z}}\)-modules satisfying the following conditions: \((1)\quad f([L'])=[L];\) (2) f is compatible with the intersection forms; (3) f preserves the Hodge structures. - Then there exists an isomorphism \(\phi: X\overset \sim \rightarrow X'\) inducing f. This is the global Torelli theorem established by \textit{I. I. Pyatetskij- Shapiro} and \textit{I. R. Shafarevich} [Math. USSR, Izv. 5(1971), 547-588 (1972); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 35, 530-572 (1971; Zbl 0219.14021)]. Here the author proves an arithmetical analogue of that version for \(K_ 3\)-surfaces defined over K, a subfield finitely generated over \({\mathbb{Q}}\). In order to do so, he replaces condition (3) by (3') \(f\otimes {\mathbb{Q}}_{\ell}: H^ 2(X'_{\bar K},{\mathbb{Q}}_{\ell})(1)\to H^ 2(X_{\bar K},{\mathbb{Q}}_{\ell})(1)\) preserves the action of \(Gal(\bar K/K)\) for some prime \(\ell\). - Also he uses extensively the technique of \textit{N. O. Nygaard} [in Arithmetic and geometry, Vol. 1: Arithmetic, Prog. Math. 35, 267-276 (1983; Zbl 0574.14031)]. Furthermore an arithemetical Torelli theorem for non singular projective curves over K is given using results of \textit{G. Faltings} [Invent. Math. 73, 349-366 (1983; Zbl 0588.14026)].