Artin Conductor

The Artin conductor measures the ramification of a finite-dimensional Galois representation. Let be a finite Galois extension of local fields, with Galois group and lower ramification groups , and let be such a representation. Its Artin conductor exponent is the nonnegative integer

Here is the subspace fixed by . The term with is the contribution from tame ramification. The sum over is the Swan conductor, which measures the contribution of wild ramification.

The local number is called a conductor exponent because, for a global representation, these exponents form the prime-power factors of its Artin conductor. The global conductor records the primes at which the representation is ramified (LMFDB). Related conductor invariants include the Swan conductor, elliptic curve conductor, and local conductor exponent.