On the passage from local to global in number theory. (English) Zbl 0813.14016

Mazur, B.

Bull. Am. Math. Soc., New Ser. 29, No. 1, 14-50 (1993).

In this survey article a comprehensive exposition of various local-global principles in number theory is given. Starting with the classical theorem of Hasse-Minkowski i.e. with the question of rational zeros of quadratic forms defined over the rational numbers a more geometric approach is emphasized, dealing with the general problem of passing from knowledge about local structures to knowledge about global structures. While the answer in the case of curves of genus 0 is quite satisfactory, it is known that the local-to-global principle does not always hold for curves of genus 1. The probably most popular example, Selmer’s curve is discussed in some detail.
It is an open question, whether or not for all smooth projective varieties the local-to-global principle holds up to finite obstruction.
Specially for abelian varieties \(A\) over \(\mathbb{Q}\) it is an open conjecture of Tate and Shafarevich, that the Tate-Shafarevich group \(\text{ Ш} (A/ \mathbb{Q})\) – which measures the deviation in the passage from local to global – is a finite group. It is known in general, that \(\text{ Ш} (A/ \mathbb{Q})\) is an abelian torsion group with the property that the kernel of the homomorphism given by multiplication by any nonzero integer \(n\) on it is finite.
In the paper under review it is shown that the Tate-Shafarevich conjecture together with some technical extra conditions implies the more general local-to-global principle for smooth projective varieties up to finite obstruction (theorem 2). Proofs of this theorem and related facts are given in part III, proofs which as far as I know are not yet explicitly in the literature. The proofs use the cohomology of locally algebraic group schemes (for a projective variety \(V\) over a number field \(K\) one takes the locally constant group of components of the automorphism group \(\operatorname{Aut}(V/K))\). Connections with the Hasse-Weil \(L\)- function of modular elliptic curves are also mentioned (theorem 3). Part II of the paper is written in more technical language and deals with Selmer groups of elliptic curves, class field theory and Kolyvagin test classes. They provide under good circumstances bounds on the size of the Selmer group.

Reviewer: H.-J.Bartels (Mannheim)

MSC:

14G25	Global ground fields in algebraic geometry
14G20	Local ground fields in algebraic geometry
11-02	Research exposition (monographs, survey articles) pertaining to number theory
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
11Gxx	Arithmetic algebraic geometry (Diophantine geometry)
14G05	Rational points
14J20	Arithmetic ground fields for surfaces or higher-dimensional varieties
14K15	Arithmetic ground fields for abelian varieties
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

rationality questions; rational points; Hasse-Weil \(L\)-function of modular elliptic curves; local-global principles; Selmer’s curve; smooth projective varieties; Tate-Shafarevich group; Tate-Shafarevich conjecture; Selmer groups of elliptic curves; class field theory; Kolyvagin test classes

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