Hasse principle

2020 Mathematics Subject Classification: Primary: 11Dxx Secondary: 11Gxx [MSN][ZBL]

The Hasse principle is 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 $M$ be a class of algebraic varieties over a global field $K$. The Hasse principle holds in $M$ if for any $X$ in $M$ such that for all non-trivial absolute valuations $\nu$ on $K$ the set of $K_\nu$-rational points $X(K_\nu)$ of $X$ is non-empty, the set of $K$-rational points $X(K)$ is also not empty (where $K_\nu$ is the completion of $K$ relative to $\nu$). In particular, if $K$ is the field $\Q$ of rational numbers, then if the set of real points $X(\R)$ and the set of $p$-adic points $\Q_p$, for all primes $p$, are not empty, it follows that the set of rational points $X(\Q)$ is also not empty. The Hasse principle holds for quadrics [CaFr], and so it is valid for algebraic curves of genus 0 (see [Ca]). For quadrics over a number field the Hasse principle was stated and proved by H. Hasse in [Ha]. For cubic hypersurfaces the Hasse principle is not true, in general (see [Ca], [Ma]); a counterexample (over $\Q$) is the projective curve $3x^3+4y^3+5z^3 = 0$ or the projective surface $5x^3+12y^3+9z^3+10t^3=0$.

Let $G$ be an algebraic group over $K$ and let $M(G)$ be the class of algebraic varieties consisting of all principal homogeneous spaces over $G$ (see Galois cohomology; Weil–Châtelet group, and also [CaFr], [Ca], [Se]). One says that the Hasse principle holds for $G$ if it holds for $M(G)$. The Hasse principle holds for simply-connected and adjoint semi-simple algebraic groups over number fields ([Se], [Ch]). If $G$ is an Abelian variety, then the Hasse principle holds for $G$ if and only if the Shafarevich–Tate group (cf. Galois cohomology) of $G$ vanishes (see the examples in [Ru], [Ko]).

References

[Ca]	J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc., 41 (1966) pp. 193–291
[CaFr]	J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967)
[Ch]	V. Chernusov, "The Hasse principle for groups of type $E_8$", Minsk (1988) (In Russian)
[Ha]	H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" J. Reine Angew. Math., 153 (1924) pp. 113–130
[Ko]	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)
[Ma]	Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic", North-Holland (1974) (Translated from Russian)
[Ru]	K. Rubin, "Tate–Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" Invent. Math., 89 (1987) pp. 527–560
[Se]	J.-P. Serre, "Cohomologie Galoisienne", Springer (1964)