[2][3] Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial.
The wild invariant[3] or Swan conductor[4] of the character is in other words, the sum of the higher order terms with i > 0.
of the Galois group G of a finite extension L/K of global fields is an ideal of K, defined to be where the product is over the primes p of K, and f(χ,p) is the local Artin conductor of the restriction of
It cannot in general be realized over the rationals or over the local field Qp, suggesting that there is no easy way to construct the Artin representation explicitly.
[5] The optimal level in the Serre modularity conjecture is expressed in terms of the Artin conductor.
The Artin and Swan representations are used to define the conductor of an elliptic curve or abelian variety.