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

Yueke Hu ETH Zürich, Switzerland Paul D. Nelson ETH Zürich, Switzerland Abhishek Saha Queen Mary University of London, UK

Number Theory mathscidoc:2206.24008

Commentarii Mathematici Helvetici, 94, (4), 767-801, 2019.12
Let π be a cuspidal automorphic representation of PGL_2(AQ) of arithmetic conductor C and archimedean parameter T, and let ϕ be an L2-normalized automorphic form in the space of π. The sup-norm problem asks for bounds on ∥ϕ∥_∞ in terms of C and T. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L2-mass |ϕ|^2(g)dg of ϕ. 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∈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(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_∞ is a holomorphic discrete series of lowest weight k, we obtain the optimal bound C^{1/8−ϵ} k^{1/4−ϵ} ≪_ϵ |ϕ|_∞ ≪_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.
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  title={Some analytic aspects of automorphic forms on GL(2) of minimal type},
  author={Yueke Hu, Paul D. Nelson, and Abhishek Saha},
  booktitle={Commentarii Mathematici Helvetici},
Yueke Hu, Paul D. Nelson, and Abhishek Saha. Some analytic aspects of automorphic forms on GL(2) of minimal type. 2019. Vol. 94. In Commentarii Mathematici Helvetici. pp.767-801.
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