8
$\begingroup$

Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.

I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$ is absolutely convergent in $\{s\in\mathbb{C}:\operatorname{Re}(s)>1\}$. I can only convince myself for the special case that $\pi$ and $\pi'$ are contragredient.

Why is this true for other cases?

$\endgroup$
4
  • 2
    $\begingroup$ For what range of $s$? The Dirichlet series is clearly divergent for $\operatorname{Re}(s) \ll 0$. $\endgroup$ Jun 6, 2014 at 20:23
  • 6
    $\begingroup$ Essentially Cauchy-Schwarz allows you to get the absolute convergence for $\pi \times \pi^{\prime}$ from knowing $\pi \times \overline{\pi}$ and ${\pi^{\prime}} \times \overline{\pi^{\prime}}$. See Molteni's paper in Duke Math. J. $\endgroup$
    – Lucia
    Jun 6, 2014 at 22:46
  • $\begingroup$ @Lucia: My response below is an elaboration of your comment. Thanks! $\endgroup$
    – GH from MO
    Jun 7, 2014 at 13:08
  • 3
    $\begingroup$ In fact absolute convergence was already proved by Jacquet-Shalika in general: see (5.3) Theorem on page 555 in Amer. J. of Math. 103 (1981). $\endgroup$
    – GH from MO
    Jun 9, 2014 at 6:39

1 Answer 1

11
$\begingroup$

This is an elaboration of Lucia's comment. Let us consider the Dirichlet coefficients of $L(s,\pi\times\pi')$, $L(s,\pi\times\tilde\pi)$, $L(s,\pi'\times\tilde\pi')$ at a prime power $p^k$. Following the proof of Proposition 6 in Molteni's paper Upper and lower bounds at $s = 1$ for certain Dirichlet series with Euler product (Duke Math. J. 111 (2002), 133-158), we see that there exists a polynomial $P_k\in\mathbb{N}[x_1,\dots,x_k]$ such that \begin{gather*} k!\ a_{\pi\times\pi'}(p^k) = P_k(\sigma_1\tau_1,\dotsc,\sigma_k\tau_k), \\ k!\ a_{\pi\times\tilde\pi}(p^k) = P_k(\lvert\sigma_1\rvert^2,\dotsc,\lvert\sigma_k\rvert^2), \\ k!\ a_{\pi'\times\tilde\pi'}(p^k) = P_k(\lvert\tau_1\rvert^2,\dotsc,\lvert\tau_k\rvert^2), \end{gather*} where $\sigma_h$ (resp. $\tau_h$) denotes the $h$-th power sum of the Langlands parameters of $\pi$ (resp. $\pi'$) at $p$. Using Cauchy–Schwarz, it follows that $$ \lvert a_{\pi\times\pi'}(p^k)\rvert\leq \sqrt{a_{\pi\times\tilde\pi}(p^k)a_{\pi'\times\tilde\pi'}(p^k)}.$$ Note that the coefficients on the right are nonnegative. Hence for all positive integers $n$, we have $$ \lvert a_{\pi\times\pi'}(n)\rvert\leq \sqrt{a_{\pi\times\tilde\pi}(n)a_{\pi'\times\tilde\pi'}(n)}\leq (a_{\pi\times\tilde\pi}(n)+a_{\pi'\times\tilde\pi'}(n))/2.$$ It follows that the absolute Dirichlet series for $L(s,\pi\times\pi')$ is majorized by the average of the Dirichlet series for $L(\sigma,\pi\times\tilde\pi)$ and $L(\sigma,\pi'\times\tilde\pi')$. The latter converges for $\sigma>1$ by Landau's lemma, hence $L(s,\pi\times\pi')$ converges absolutely for $\Re(s)=\sigma>1$.

Added 1. The above argument requires that $p$ is an unramified prime for $\pi$ and $\pi'$, hence also that $n$ is only divisible by such primes. This does not alter the final conclusion though, because omitting or including finitely many Euler factors does not alter absolute convergence of the corresponding Dirichlet series in the half-plane $\Re(s)>1$.

Added 2. For a related and more complete result concerning the logarithmic derivative of Rankin–Selberg $L$-functions, see Brumley's Appendix to the exciting new preprint Weak subconvexity without a Ramanujan hypothesis of Soundararajan and Thorner.

Added 3. The above bounds have been extended to all prime powers $p^k$ and all positive integers $n$. See Lemma 3.1 in the nice paper Exponential sums with multiplicative coefficients without the Ramanujan conjecture of Jiang–Lü–Wang. The proof relies on Brumley's Appendix that I emphasized in the previous section.

$\endgroup$
5
  • 1
    $\begingroup$ A bit off-topic, but if I'm not mistaken, I was told that the non vanishing on the $\sigma=1$ line of these L functions was needed to ensure unique factorization. Has it been proved that these L functions don't vanish on the considered line? I can ask this question in a new thread if needed. $\endgroup$ Jun 7, 2014 at 19:39
  • 2
    $\begingroup$ @Sylvian: Automorphic forms factor into local representations over the primes, which yields the Euler product factorization of $L(s,\pi)$ and $L(s,\pi\times\pi')$. This has nothing to do with non-vanishing on $\sigma=1$. Regarding non-vanishing, it was proved by Jacquet--Shalika in 1976 that $L(s,\pi)\neq 0$ on $\sigma=1$. I think the same is not known for $L(s,\pi\times\pi')$, but of course by the Langlands conjectures these $L$-functions are also of the form $L(s,\Pi)$ with a single $\Pi$, so non-vanishing should hold for them as well. $\endgroup$
    – GH from MO
    Jun 7, 2014 at 20:12
  • $\begingroup$ Ok, thanks. My question originated in the possible proof of the PNT for the Selberg class by Yoshikatsu Yashiro, as I wanted to know if it was sufficient to ensure unique factorization in this class but M. Ram Murty told me that one needed the existence of some kind of "Rankin-Selberg" L-functions in it. $\endgroup$ Jun 7, 2014 at 20:26
  • 2
    $\begingroup$ @Sylvain: Factoring an automorphic $L$-function or a Rankin-Selberg $L$-function into primitive elements of the Selberg class is off-topic here, especially that these $L$-functions themselves are not known to belong to the Selberg class. $\endgroup$
    – GH from MO
    Jun 7, 2014 at 20:58
  • 3
    $\begingroup$ @SylvainJULIEN: I told you a while ago that $L(s,\pi\times\pi')\neq 0$ on $\sigma=1$ is not known. I was wrong, Shahidi proved this nonvanishing, see Theorem 5.2 on page 353 in Amer. J. of Math. 103 (1981). $\endgroup$
    – GH from MO
    Nov 2, 2016 at 7:30

Your Answer

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

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