Artin conductor - Knowlpedia

Definition

Let $F$ be a nonarchimedean local field, and let $r$ be a finite-dimensional representation of its absolute Galois group or Weil group with finite inertia image. Choose a finite Galois extension through which the inertia action factors, and write $G_i$ for its lower-numbered ramification groups, with $G_0$ the inertia group. The Artin conductor exponent is

a(r)= \sum_{i\geq 0} \frac{|G_i|}{|G_0|} \operatorname{codim} V^{G_i}.

This integer is independent of the chosen finite extension. It is zero exactly when $r$ is unramified.

Tame and wild parts

The term for $i=0$ is $\dim V-\dim V^{I_F}$ and measures tame failure of inertia invariance. The remaining sum is the Swan conductor, which measures wild ramification. Thus

a(r)=\operatorname{codim}V^{I_F}+\operatorname{Swan}(r).

The conductor is additive in short exact sequences and direct sums.

Weil–Deligne form

For a Weil–Deligne representation $(r,N)$ , the conductor also detects monodromy:

a(r,N) = a(r)+\dim V^{I_F}-\dim(\ker N)^{I_F}.

This is the exponent that appears in the power of $q_F^{-s}$ in a local epsilon factor. It also records the ramification contribution of a local parameter to its local $L$ -function.

Global conductor

For a global representation unramified outside finitely many finite places, the local exponents assemble into the conductor ideal

\mathfrak f(r)=\prod_v\mathfrak p_v^{a(r_v)}.

The global conductor is therefore arithmetic data assembled from local ramification, not a separate choice of normalization.

References

Jean-Pierre Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, 1979, Chapter VI, §2. Springer.
Pierre Deligne, “Les constantes des équations fonctionnelles des fonctions $L$ ,” in Modular Functions of One Variable II, Lecture Notes in Mathematics 349, Springer, 1973, 501–597. IAS copy.