https://arxiv.org/abs/math/9307231

B.Mazur. On the passage from local to global in number theory

«Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being often a stepping-stone to the more difficult global problems.

(…)

Here are two types of questions in Number Theory one might want to pursue by passing from local to global:

Type (A). Questions about rational points. Given a Diophantine equation or a system of Diophantine equations with coefficients in Q, when does knowledge about its rational solutions over the collection of local fields Q_p for all prime numbers p, and over R, give us some palpable information about its solutions over Q? (…) The question of existence of solutions of systems of Diophantine equations over R or over Q_p is “certifiably easy”; at least it is a decidable question in the sense of mathematical logic. (…)

Type (B). Passing from knowledge about local “structures” to knowledge about global “structures”. There is a wealth of literature on various aspects of questions of type (A). One cannot, really, effectively “separate” questions of type (A) from questions of type (B), but the central aim of this article is to discuss an exciting development (finiteness of certain Tate-Shafarevich groups), which is conveniently expressed in the language of (B).»

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https://arxiv.org/abs/math/9307231

B.Mazur. On the passage from local to global in number theory

«Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being often a stepping-stone to the more difficult global problems.

(…)

Here are two types of questions in Number Theory one might want to pursue by passing from local to global:

Type (A). Questions about rational points. Given a Diophantine equation or a system of Diophantine equations with coefficients in Q, when does knowledge about its rational solutions over the collection of local fields Q_p for all prime numbers p, and over R, give us some palpable information about its solutions over Q? (…) The question of existence of solutions of systems of Diophantine equations over R or over Q_p is “certifiably easy”; at least it is a decidable question in the sense of mathematical logic. (…)

Type (B). Passing from knowledge about local “structures” to knowledge about global “structures”. There is a wealth of literature on various aspects of questions of type (A). One cannot, really, effectively “separate” questions of type (A) from questions of type (B), but the central aim of this article is to discuss an exciting development (finiteness of certain Tate-Shafarevich groups), which is conveniently expressed in the language of (B).»

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3.81K viewsedited  Jun 23 at 08:43