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. To illustrate what the earthy geometric terms local and global signify in Number Theory, let us consider any Diophantine equation, say,