An invariant of a local Galois representation or of a finite extension of local fields that measures the depth of wild (non‑tame) ramification; it is the contribution to the conductor coming from higher ramification groups beyond the tame part.

Principle

Quantify the size and complexity of wild ramification by summing weighted jumps in the upper numbering filtration of inertia; the Swan conductor isolates the purely wild contribution to the Artin conductor.

Demonstration

For a finite extension of p‑adic fields with nontrivial higher ramification groups, compute the breaks in the upper numbering filtration for inertia and form the weighted sum of their dimensions in a Galois representation; the resulting nonzero integer (or rational in some normalizations) is the Swan conductor, distinguishing wild from tame behaviour.

Misapplication

Treating the Swan conductor as equal to the full Artin conductor or using it to measure tame ramification only; similarly, applying its usual definition blindly to representations without control of wild inertia (e.g., poorly defined infinite image of wild inertia) leads to incorrect values.

Consequence

A positive Swan conductor forces the presence of wild phenomena: it affects epsilon factors and local constants in functional equations, contributes to global conductor exponents, and controls growth of discriminants and bounds in lifting problems.

Reversal

When the Swan conductor is zero the representation/extension is tamely ramified and no wild contribution appears; inverting the concept yields a criterion that wild behaviour vanishes precisely when the Swan term disappears.

Boundary

A local invariant for representations of the absolute Galois group of a local field (or for finite extensions); it is not a direct global invariant unless summed over places, and standard definitions require finite image on wild inertia or l‑adic continuity to ensure well‑posedness.

Semantic Tension

Confusion often arises with the Artin conductor (which contains both tame and wild parts) or with discriminant exponents; the Swan conductor specifically isolates wild contribution, whereas nearby notions mix tame and wild data.

Synthesis

The Swan conductor is the numerical gauge of wild ramification: compute jumps in inertia, weight them by dimension, and the nonzero remainder pinpoints exactly how and how much an extension or representation departs from tame behaviour.