A phenomenon in arithmetic geometry and number theory where a Diophantine equation or arithmetic variety has a solution in every completion of a global field (for example over every p-adic field and the real/complex completions) but admits no rational point over the global field itself.

Principle

Global solvability is not guaranteed by local solvability; global arithmetic invariants and reciprocity laws can produce obstructions invisible to completion-by-completion checks.

Demonstration

A genus-one curve that has points over R and over Q_p for every prime p, yet no rational point over Q; local checks at all completions succeed while a global cohomological invariant obstructs existence of a rational solution.

Misapplication

Assuming that verifying solutions in all completions of a number field suffices to conclude the existence of a global rational solution, and therefore stopping all further global checks.

Consequence

Detection of a local–global obstruction forces the use of global tools (cohomology, descent, reciprocity pairings) to explain failure of the Hasse principle and to characterize rational points or their absence.

Reversal

If a Diophantine equation has a global rational solution then it necessarily has solutions in every completion; the reverse implication fails in the presence of an obstruction.

Boundary

Applies to arithmetic problems over global fields (number fields, function fields of curves over finite fields); does not concern purely local analytic approximations or problems over non-global fields.

Semantic Tension

Competes with the Hasse principle and local-to-global heuristics: both appeal to local data, but a local–global obstruction emphasizes hidden global invariants that break the heuristic.

Synthesis

Local–Global Obstruction names the mismatch between completion-wise solvability and global solvability: local information can be consistent everywhere while a global invariant (often cohomological) prohibits a global rational solution, forcing global methods to explain and remedy the failure.