A prime (or place) of a number field at which an algebraic variety, scheme, or model acquires singularities or otherwise degenerates after reduction modulo that prime, so the special fiber is not smooth and arithmetic properties that rely on smoothness may fail.

Principle

A variety has good reduction at a prime if it admits a smooth proper model over the local ring at that prime; bad reduction is the negation, occurring when the special fiber is singular, reducible, or has other degeneracies that alter the arithmetic and geometric structure modulo the prime.

Demonstration

For an elliptic curve given by a Weierstrass equation, the discriminant Δ vanishing modulo a prime p indicates bad reduction at p; the reduced curve modulo p is singular (nodal or cuspidal) or otherwise degenerate, and invariants like the local Tamagawa number reflect this degeneration.

Misapplication

Assuming bad reduction implies no points modulo p or no global points; bad reduction concerns geometry of the fiber and can coexist with many local points or with global rational points — it does not by itself imply arithmetic emptiness.

Consequence

Bad reduction affects local Galois representations, alters inertia action, changes local factors of L-functions and Tamagawa numbers, and complicates descent, integral points, and deformation problems; it often necessitates more delicate local analysis.

Reversal

Good reduction: the variety has a smooth proper model at the prime, the special fiber is smooth, and many arithmetic invariants behave well and can be controlled by reduction modulo the prime.

Boundary

The notion applies to primes of the ring of integers of a number field and to integral models; it excludes generic fiber phenomena over characteristic zero and requires specification of a chosen model — different models can present different reduction behaviors if not minimal.

Semantic Tension

Tension exists between bad reduction as a geometric singularity of the special fiber and arithmetic degeneration: some arithmetic pathologies (e.g., wild ramification in Galois action) correlate with bad reduction but are not equivalent; conversely, singular reduction does not always imply severe arithmetic failure.

Synthesis

Bad reduction is the placewise degeneration of a variety's geometry upon reduction modulo a prime — the failure of a smooth integral model at that prime — and it signals the need for refined local arithmetic tools while not by itself ruling out local or global points.