In algebraic geometry and related areas of mathematics, local analysis is the practice of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture.
[1] In number theory one may study a Diophantine equation, for example, modulo p for all primes p, looking for constraints on solutions.
[2] The next step is to look modulo prime powers, and then for solutions in the p-adic field.
The point of view that one would like to understand what extra conditions are needed has been very influential, for example for cubic forms.
Some form of local analysis underlies both the standard applications of the Hardy–Littlewood circle method in analytic number theory, and the use of adele rings, making this one of the unifying principles across number theory.