##  [Swan Conductor](/swan-conductor-0) 

 Definition

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

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

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

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

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

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

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

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

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.