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

A subscription is required to access this article.


Let π\pi be a cuspidal automorphic representation of PGL2(AQ)_2(\mathbb A_\mathbb Q) of arithmetic conductor CC and archimedean parameter TT, and let ϕ\phi be an L2L^2-normalized automorphic form in the space of π\pi. The sup-norm problem asks for bounds on ϕ\| \phi \|_\infty in terms of CC and TT. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L2L^2-mass ϕ2(g)dg|\phi|^2 (g) \, d g of ϕ\phi. Previous work on these problems in the conductor-aspect has focused on the case that ϕ\phi 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 pp of CC, the local component πp\pi_p is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms ϕ\phi for which the local components ϕpπp\phi_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 PGL2(R)_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\pi_\infty is a holomorphic discrete series of lowest weight kk, we obtain the optimal bound C1/8ϵk1/4ϵϵϕϵC1/8+ϵk1/4+ϵC^{1/8 -\epsilon} k^{1/4 - \epsilon} \ll_{\epsilon} |\phi|_\infty \ll_{\epsilon} C^{1/8 + \epsilon} k^{1/4+\epsilon}. 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.

Cite this article

Yueke Hu, Paul D. Nelson, Abhishek Saha, Some analytic aspects of automorphic forms on GL(2) of minimal type. Comment. Math. Helv. 94 (2019), no. 4, pp. 767–801

DOI 10.4171/CMH/473