MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This may be a silly question for experts in this area. But I am really suffering for not being able to compute local-L function of some automorphic representation.

So, I post it hoping some benevolent one shed me a light.

Let $\pi$ be a local component of some global irr.cusp.unitary automorphic representation of $U(2)$ at split finite place.(i.e. $\pi$ is $GL(2)$ reps)

Then, there are three possible candidates for $\pi$ except for supercuspidal;

1)$B(\chi_{1} , \chi_{2})$ for $\chi_{1} \cdot \chi_{2}^{-1}=1$

2)$B(\chi_{1} , \chi_{2})$ for $\chi_{1} \cdot \chi_{2}^{-1} \ne 1$

3)irreducible quotient of $B(\chi\left\vert \cdot \right\vert^{\frac{1}{2}},\chi\left\vert \cdot \right\vert^{-\frac{1}{2}})$ for unitary character $\chi$.

(here, all $\chi, \chi_{1} , \chi_{2}$ are character of $GL(1)$ )

Then, for any character $\gamma$ of $GL(1)$,

what is $L(s,\pi \otimes \gamma)$ for the above three each $\pi$?

(here, L-function is local L-function and we consider $\gamma$ as $GL(2)$ character through determinant map)

Since my main concern lies in computing the order of zero or pole of the above $L$-function at $s=0$, if it is hard to write explicitly in ramified case, would you just inform me the result for each cases? Then, I am very grateful for your kindness.

(For beginner in this area, getting used to L-function calculation is quite difficult.)

share|cite|improve this question
What is $BC$? regards – Marc Palm Feb 6 '13 at 18:03
@Marc: I guess it stands for "base change". – GH from MO Feb 7 '13 at 0:07
Sorry Marc. For I've already taken the base change into account, there does not need BC in the local L-function. Thanks for pointing it out. Regards, – anonymous Feb 7 '13 at 6:31

Since I was just asking a similar case, I can give you a partial answer. In in Definition 3.1.16

you can find the $p$-adic result, which generalize to the non-archimedean, zero-characterictic cases in the obviuous manner. References are Bump or Goldfeld-Hundley.

For the tensoring by $\gamma \circ \det$ (I think that's what you are asking about), note that $$ \gamma \circ \det \otimes B(\chi_1, \chi_2) = B(\chi_1\gamma, \chi_2 \gamma),$$ and for the Steinberg/special rep $St(\chi) \otimes \gamma \circ \det = St(\chi\gamma)$ as well. The $L$-function of a supercuspidal representation is a constant (usually chosen to be one).

The computation at the real places can be found as Lemma 5.15.1, and the complex case on page 118 ff. in Jacquet-Langlands (both with computation).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.