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?

Please give me any advice!

  • $\begingroup$ 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, $\endgroup$ – Emerton Apr 30 '14 at 2:01

Yes, it is possible. For all this explained clearly and in detail, see David Rohrlich's paper "Elliptic curves and the Weil-Deligne group" along with the accompanying "Student's supplement to "Elliptic curves and the Weil-Deligne group" if needed.

  • $\begingroup$ Thanks for your comment! I will try to read the references, but firstly I want to know the case for Tate curves. How can it be done? $\endgroup$ – Hiro Jan 29 '14 at 3:54
  • $\begingroup$ 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 $-1$. $\endgroup$ – Kestutis Cesnavicius Jan 29 '14 at 13:42

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