Some analytic aspects of automorphic forms on GL(2) of minimal type

Abstract

Let $\pi$ be a cuspidal automorphic representation of PGL${}_2({\mathbb{A}}_{\mathbb{Q}})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $\varphi$ be an $L^2$-normalized automorphic form in the space of $\pi$. The sup-norm problem asks for bounds on $\Vert \varphi {\Vert }_{\infty }$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|\varphi {|}^2(g)dg$ of $\varphi$. Previous work on these problems in the conductor-aspect has focused on the case that $\varphi$ is a newform.

In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor $p$ of $C$, the local component ${\pi }_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $\varphi$ for which the local components ${\varphi }_p\in {\pi }_p$ are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL${}_2(\mathbb{R})$.

For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if ${\pi }_{\infty }$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8-ϵ}{k}^{1/4-ϵ}{\ll }_{ϵ}|\varphi {|}_{\infty }{\ll }_{ϵ}{C}^{1/8+ϵ}{k}^{1/4+ϵ}$. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.