# Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations. However, given a galois representation, it is usually difficult to find the Weil-Deligne representations.

My questions are:

(1) Is it possible to describe explicitly the Weil-Deligne representation associated to the Tate module of an elliptic curve over a local field? How about Tate curve (corresponding to a prime element) for example?

(2) In the situation as in (1), is it possible to calculate the L function and epsilon factor associated to the Weil-Deligne representation?

(3) If these are possible, how can one do for Tate curves?

• Dear Hiro, The case of a Tate curve is particularly easy. The $\ell$-adic Tate module is the Kummer extension of $\mathbb Q_{\ell}$ by $\mathbb Q_{\ell}(1)$ corresponding to the Tate parameter $q$ defining the curve. (This follows directly from the description of the points as $\overline{\mathbb Q}_p^{\times}/q^{\mathbb Z}$.) Now just apply the standard recipe to get the Weil--Deligne representation. Regards, – Emerton Apr 30 '14 at 2:01
• For Tate curves, see section 15 of Rohrlich's paper. Namely, in the split multiplicative reduction case, the associated Weil-Deligne representation is $\omega^{-1} \otimes \mathrm{sp}(2)$, where $\mathrm{sp}(2)$ is the $2$-dimensional Steinberg representation and $\omega$ is the cyclotomic character. The $L$-factor is $(1 - q^{-s})^{-1}$ (see section 17). The epsilon factor will depend on the choices of a nontrivial additive character and a Haar measure and can be computed as in the Corollary at the end of section 12. The local root number is $-1$. – Kestutis Cesnavicius Jan 29 '14 at 13:42