Hasse principle

One of the central principles of Diophantine geometry, which reduces the problem of the existence of rational points on an algebraic variety over a global field to the analogous problem over local fields.

Let be a class of algebraic varieties over a global field . The Hasse principle holds in if for any in such that for all non-trivial absolute valuations on the set of -rational points of is non-empty, the set of -rational points is also not empty (where is the completion of relative to ). In particular, if is the field of rational numbers, then if the set of real points and the set of -adic points , for all primes , are not empty, it follows that the set of rational points is also not empty. The Hasse principle holds for quadrics [2], and so it is valid for algebraic curves of genus 0 (see [3]). For quadrics over a number field the Hasse principle was stated and proved by H. Hasse in [1]. For cubic hypersurfaces the Hasse principle is not true, in general (see [3], [4]); a counterexample (over ) is the projective curve or the projective surface .

Let be an algebraic group over and let be the class of algebraic varieties consisting of all principal homogeneous spaces over (see Galois cohomology; Weil–Châtelet group, and also [2], [3], [5]). One says that the Hasse principle holds for if it holds for . The Hasse principle holds for simply-connected and adjoint semi-simple algebraic groups over number fields ([5], [6]). If is an Abelian variety, then the Hasse principle holds for if and only if the Shafarevich–Tate group (cf. Galois cohomology) of vanishes (see the examples in [7], [8]).

References

[1]	H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" J. Reine Angew. Math. , 153 (1924) pp. 113–130
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967)
[3]	J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 193–291
[4]	Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian)
[5]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964)
[6]	V. Chernusov, "The Hasse principle for groups of type " , Minsk (1988) (In Russian)
[7]	K. Rubin, "Tate–Shafarevich groups and -functions of elliptic curves with complex multiplication" Invent. Math. , 89 (1987) pp. 527–560
[8]	V. Kolyvagin, "The Mordell–Weil groups and the Shafarevich–Tate groups of Weil's elliptic curves" Izv. Akad. Nauk. SSSR Ser. Mat. , 52 : 6 (1988)