Local epsilon factor

Definition

Let $F$ be a local field, let $V=(r,N)$ be a finite-dimensional Weil–Deligne representation, and choose a nontrivial additive character $\psi:F\to\mathbb C^\times$ . The local epsilon factor

\varepsilon(s,V,\psi)

is the nonzero elementary factor that occurs with the local $L$ -factors in the functional equation. With a self-dual Haar measure for $\psi$ , the associated gamma factor satisfies

\gamma(s,V,\psi)= \varepsilon(s,V,\psi) \frac{L(1-s,V^\vee)}{L(s,V)}.

Here $V^\vee$ is the contragredient Weil–Deligne representation. For nonarchimedean $F$ , the epsilon factor is a nonzero constant times an integral power of $q_F^{-s}$ . The exponent records the Artin conductor.

Dependence on choices

Unlike the local L-factor, $\varepsilon(s,V,\psi)$ depends on the additive character and on the Haar-measure convention. Replacing $\psi(x)$ by $\psi(ax)$ changes it by an explicit determinant and absolute-value factor. Every comparison must therefore fix:

arithmetic or geometric Frobenius;
the reciprocity-map normalization;
the additive character; and
the measure used in Fourier transform.

Representations of reductive groups

For representations of $\operatorname{GL}_n(F)$ , local Langlands defines epsilon factors through the corresponding Weil–Deligne representation and agrees with the factors constructed analytically. For a general reductive group and a representation $r$ of its $L$ -group, Langlands–Shahidi and Rankin–Selberg methods construct factors in many cases.

The unit-modulus normalization at the central point is the local root number.

References

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.
John Tate, “Number theoretic background,” in Automorphic Forms, Representations and L-Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979.