5
$\begingroup$

I ran into this question when trying to compute the Atkin-Lehner pseudo-eigenvalue of newforms. Let $k \geq 2$, let $\omega$ be a Dirichlet character modulo $N$ and let $f \in S_k(N,\omega)$ be a newform. We use $\pi_f$ to denote the corresponding representation of $\text{GL}_2(\mathbb{A}_\mathbb{Q})$, and we have $\pi_f = \prod_v \pi_{f,v}$. Then there should be a function $\epsilon(\pi, s)$ such that

$L(\pi, s) = \epsilon(\pi,s)L(\pi^{\vee}, 1-s)$. (1)

Further, fix an additive character $\psi$ of $\mathbb{A}$, then we have the decomposition

$\epsilon(\pi,s) = \prod_v \epsilon(\pi_v,s,\psi_v)$. (2)

Now I am aware of two set of conventions of the local epsilon factors, one from Gelbart and the other from Bushnell-Henniart, but I wasn't able to use them to compute $\epsilon$, and their definitions are different. I'll explain what I tried:

1.I started from GL(1). If $\chi$ is an unramified character of $(\mathbb{Q}_p)^{\times}$, then Gelbart defined $\epsilon(\chi,s) = 1$, where as [BH] defines $\epsilon(\chi,s) = p^{1/2-s}\chi(p)^{-1}$. When $p \nmid N$, the local representation $\pi_{f,p}$ is unramified principal series and the first definition would yield $\epsilon(\pi_p) = 1$ and second one would yield $\epsilon_p = \omega(p)$. This makes me confused about the [BH] definition, since there will be infinitely many $p$ with $\epsilon_p \neq 1$, so I don't know how to make sense of the infinite product over all primes. Also it seems to me that [BH] did not treat the case $v = \infty$.

  1. I was not able to find the formula for the local constants in Gelbart when the local represenation is special or supercuspidal.

Overall, I'm asking for a definition of these epsilon constants that would make (1) and (2) hold. I've already spent several hours online looking for references without success, so I'd much appreciate experts' help and guidance!

$\endgroup$
4
  • $\begingroup$ You should look at Chapter 11 of Goldfeld and Hundley, where they work out the local functional equation for representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Or look at the paper "Some Remarks on Local Newforms for $\mathrm{GL}(2)$" by Ralf Schmidt. $\endgroup$ Jun 18, 2015 at 21:56
  • $\begingroup$ If it's any consolation, I think that the issue of the compatibilities of various notions of epsilon-factor are not easy to certify. I seem to recall reading that Dwork and Langlands and others decided to not publish (!?!) certain of their investigations into such compatibilities because the write-up would simply "go on too long". In particular, any arrangement of an argument that would show that almost-all local epsilons are $1$, and certify the notion that a global epsilon-factor is well-defined, and so on, is apparently ... expensive. $\endgroup$ Jun 19, 2015 at 0:16
  • $\begingroup$ You can also look at Gealy's thesis thesis.library.caltech.edu/5020 where he recalls the normalization for the global L-function and epsilon-factor of a modular form (see Section 5.4 but also Chapter 8 for the finite epsilon factors). In any case, I think it's reasonable to ask that the epsilon factor should be 1 whenever the representation is unramified (in GL(1) as well as GL(2) case). $\endgroup$ Jun 19, 2015 at 13:41
  • 1
    $\begingroup$ Thanks everyone for your helpful comments! I think I'm able to figure it out for my case: the formula in [BH] assumes that the additive character has level one, while the classical choice of the additive character has level zero. [BH] also provides a formula on what happens when one changes the additive character, so I was able to get the global epsilon constant (finally)! $\endgroup$
    – Hao Chen
    Jun 20, 2015 at 4:57

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.