4
$\begingroup$

Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi_0$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{f} =\langle \varphi_0 \rangle$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}_2(\mathbb{Z})$) generated by $\varphi_0$. To each smooth vector $\varphi \in \pi$ we can associate its Whittaker function

$$W_{\varphi}(g) = \int_{\mathbb{Z} \backslash \mathbb{R}}{\varphi(n(x)g) e(-x)}dx\qquad n(x) = \begin{pmatrix} 1 & x\\ 0 & 1 \end{pmatrix} $$ and the Mellin-transform thereof, known as the local zeta integral attached to $\varphi$ : $$ Z(W_{\varphi}, s) = \int_{\mathbb{R}^{\times}}{W_{\varphi}(a(y)) |y|^{s-1/2} d^{\times}y} \qquad a(y) = \begin{pmatrix} y & 0\\ 0 & 1 \end{pmatrix} $$

It is known (in my case, from Proposition 1.5 in Cogdell Piatetski-Shapiro "The arithmetic and spectral analysis of Kloosterman sums") that the assignment $\varphi \mapsto Z(W_{\varphi},s)$ is continuous for the smooth topology on $\pi^{\infty}$ and even uniformly so for $s$ varying in a compact subset of some right half-plane.

Does this uniform continuity remain true for the meromorphic continuation of the local local zeta integral? In particular, I'd like to know whether this holds for compact subsets of $\Re(s) = 1/2$.

The local functional equation gives uniform continuity for $s$ varying in compact subsets of $\{|\Re(s)-1/2| > A\} - \{\text{poles}\}$ for some $A > 0$ (I think $A = 5/2$ is valid).

$\endgroup$
3
  • $\begingroup$ You may have already tried, but doesn't proposition 3.2.3 of Michel-Venkatesh's subconvexity paper give what you want? $\endgroup$ Apr 26, 2018 at 18:15
  • $\begingroup$ Hey Subhajit, it looks related, but I don't see how it would imply what I want. Maybe we can talk about it tomorrow ;) $\endgroup$
    – m.s
    Apr 26, 2018 at 19:52
  • $\begingroup$ See also my comment to Subhajit's answer. $\endgroup$
    – GH from MO
    Apr 27, 2018 at 23:51

1 Answer 1

3
$\begingroup$

As we are considering continuity of the local zeta integral it is sufficient to pose the problem locally. Thus we think the Archimedean local component of $\pi$ is the principal series $\pi_\mu$ where $\mu\in \mathbb{C}$ such that the Laplacian eigenvalue of $f$ is $1/4-\mu^2$.

Let $\{\mathcal{D}_i\}$ be the set of invariant differential operators of degree $\le 2$. We define a Sobolev norm (semi-norm) on $\pi_\mu$ by $$S(v):=\sum_{i}\|\mathcal{D}_iv\|_\pi.$$ Thus to prove the uniform continuity it is enough to show the following: For every compact set $K\subseteq \{s\mid\Re(s)> 0\}$, $$|Z(W_{v_1},s)-Z(W_{v_2},s)|\ll_K S(v_1-v_2),$$ whenever $s\in K$. Using linearity of the zeta integral it is enough to show that $$Z(W_v,s)\ll_K S(v).$$ But the LHS is $$\int_{\mathbb{R}^\times}W_v(a(y))|y|^{s-1/2}d^\times y\ll_KS(v)\int_{\mathbb{R}^\times}\min(|y|^{1/2-\eta},|y|^{-2})|y|^{\Re(s)-1/2}\frac{dy}{|y|}.$$ We have used Proposition 3.2.3 of Michel-Venkatesh in a slightly modified form (their seminorm is only on the Kirillov model of $\pi_\mu$). If $\pi_\mu$ is tempered, i.e. $\mu$ is purely imaginary then $\eta$ can be replaced by any $\epsilon > 0$ (with $\ll_K$ replaced by $\ll_{K,\epsilon}$). As $\Re(s)> 0$, the last integral is $\ll_K 1$ choosing $\epsilon$ small enough.

$\endgroup$
1
  • 1
    $\begingroup$ The last bound also follows (with a slightly different Sobolev norm) from (29) in Blomer-Harcos: The spectral decomposition of shifted convolution sums (see arxiv.org/abs/math/0703246). $\endgroup$
    – GH from MO
    Apr 27, 2018 at 23:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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