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.)

  • $\begingroup$ What is $BC$? regards $\endgroup$ – Marc Palm Feb 6 '13 at 18:03
  • $\begingroup$ @Marc: I guess it stands for "base change". $\endgroup$ – GH from MO Feb 7 '13 at 0:07
  • $\begingroup$ 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, $\endgroup$ – anonymous Feb 7 '13 at 6:31

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

http://www.math.ou.edu/~kmartin/autreps/ch3.pdf 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).


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