[Paper Review] Some analytic aspects of automorphic forms on GL(2) of minimal type

Yueke Hu, Paul D. Nelson+1 more|arXiv (Cornell University)|Sep 29, 2017

Advanced Algebra and Geometry25 references3 citations

TL;DR

This paper studies automorphic forms on GL(2) that are not newforms—specifically, those with supercuspidal local components at primes dividing the conductor and minimal vectors at these places. It establishes a sharper sup-norm bound than known for newforms, achieving the optimal exponent in the weight aspect, and proves that these forms serve as analytic test vectors for the quantum unique ergodicity period, thereby linking the strong QUE problem and subconvexity for this class.

ABSTRACT

Let $π$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $ϕ$ be an $L^2$-normalized automorphic form in the space of $π$. The sup-norm problem asks for bounds on $\| ϕ\|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|ϕ|^2 (g) \, d g$ of $ϕ$. All previous work on these problems in the conductor-aspect has focused on the case that $ϕ$ 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 $π_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $ϕ$ for which the local components $ϕ_p \in π_p$ 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 $π_\infty$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8 -ε} k^{1/4 - ε} \ll_ε |ϕ|_\infty \ll_ε C^{1/8 + ε} k^{1/4+ε}$. 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 for newforms of powerful level.

Motivation & Objective

• To investigate the analytic behavior of automorphic forms on GL(2) that are not newforms, focusing on those with supercuspidal components and minimal vectors at finite places.
• To establish improved $ L^ atural $-norm bounds for such forms, particularly in the conductor and weight aspects.
• To demonstrate that these minimal vectors serve as analytic test vectors for the quantum unique ergodicity period, thereby linking the strong QUE problem and subconvexity.
• To contrast this behavior with the known failure of such equivalence for newforms of powerful level.

Proposed method

• The authors analyze automorphic forms $ \phi = \otimes \phi_v $ where $ \phi_\infty $ is a lowest weight vector and $ \phi_p \in \pi_p $ is a minimal vector for each supercuspidal $ \pi_p $ with conductor a fourth power.
• They derive pointwise bounds on $ |\phi(g_{\mathbf{f}}n(x)a(y))| $ using Fourier expansions, amplification, and $ L^p $-estimates via H"older's inequality.
• The method involves decomposing the automorphic form into Fourier coefficients and estimating their $ L^8 $-norms and exponential sums, particularly in the holomorphic case.
• They apply Ichino's refinement of Watson's formula to relate the period integral to $ L $-functions and local Whittaker integrals, using explicit bounds on $ I_p $ and $ I_\infty $.
• The analysis includes estimating the archimedean factor $ I_\infty = 1 $ and showing $ I_p \cdot \mathrm{Cond}(\pi_p \times \pi_p)^{1/2} \asymp 1 $ via bounds on Satake parameters and Whittaker functionals.
• The proof leverages the fact that minimal vectors satisfy a non-archimedean analogue of lowest weight vectors in holomorphic discrete series, enabling sharper estimates than for newforms.

Experimental results

Research questions

• RQ1 Can sharper sup-norm bounds be obtained for automorphic forms on GL(2) that are not newforms?
• RQ2 Do minimal vectors—non-newform vectors with special local properties—serve as effective test vectors for the quantum unique ergodicity period?
• RQ3 Is the equivalence between the strong quantum unique ergodicity problem and the subconvexity problem preserved for such minimal vectors, unlike in the newform case?
• RQ4 What is the optimal exponent in the conductor and weight aspects for the sup-norm of such non-newform automorphic forms?

Key findings

• The paper establishes the optimal sup-norm bound $ C^{1/8 - \epsilon} k^{1/4 - \epsilon} \ll_\epsilon \|\phi\|_\infty \ll_\epsilon C^{1/8 + \epsilon} k^{1/4 + \epsilon} $ for holomorphic discrete series of lowest weight $ k $, which improves upon known bounds for newforms.
• For the conductor aspect, the bound $ \|\phi\|_\infty \ll_{\epsilon} C^{1/8 + \epsilon} k^{1/4 + \epsilon} $ is sharp and matches the best-known results in the weight aspect for holomorphic forms.
• The authors prove that minimal vectors are analytic test vectors for the QUE period, meaning the period integral is non-degenerate and controls the $ L^2 $-mass distribution.
• This leads to an equivalence between the strong QUE problem and the subconvexity problem for this class of automorphic forms, in contrast to the failure of such equivalence for newforms of powerful level.
• The local period integrals $ I_p $ satisfy $ I_p \cdot \mathrm{Cond}(\pi_p \times \pi_p)^{1/2} \asymp 1 $, which is crucial for the global period formula and the main bounds.