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

Abstract

Let π be a cuspidal automorphic representation of PGL 2 (A 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 ∞ in terms of C and T . The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L 2 -mass |φ| 2 (g) dg 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 πp is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms φ for which the local components φp ∈ πp 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−ǫ k 1/4−ǫ ≪ǫ |φ|∞ ≪ǫ C 1/8+ǫ k 1/4+ǫ . 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.

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces

“…7 Finally, we remark that the results of this paper appear to be the first time that the local bound in the conductor aspect has been improved upon for squarefull conductors, for any kind of automorphic form on a compact domain. (In the noncompact case, this had been achieved in our previous paper [11].) It seems also worth mentioning here the very recent work of Hu [9] which generalizes [11] and proves a sub-local bound in the depth aspect for automorphic forms of minimal type on GL n under the assumption that the corresponding local representations have "generic" induction datum.…”

“…(In the noncompact case, this had been achieved in our previous paper [11].) It seems also worth mentioning here the very recent work of Hu [9] which generalizes [11] and proves a sub-local bound in the depth aspect for automorphic forms of minimal type on GL n under the assumption that the corresponding local representations have "generic" induction datum.…”

“…A key class of φ p for which this is true is the family of minimal vectors. Minimal vectors may be viewed as p-adic analogues of holomorphic vectors at infinity and have several remarkable properties, which were proved in our recent work [11] (where the analytic properties of the corresponding global automorphic forms of minimal type were studied for the first time in the setting of GL 2 ). The main result of [11] proved an optimal sup-norm bound for such forms in the setting of GL 2 .…”

“…However, the techniques used in [11] relied on a very special property of the Whittaker/Fourier expansion of φ which only works in the non-compact setting. Therefore, the proof cannot carry over to the compact case, i.e., to our case of indefinite quaternion division algebras D, as no Whittaker/Fourier expansions exist here.…”

“…We remark that the condition on the conductor being the fourth power of an odd integer is a convenient one that was assumed for the definition of minimal vectors in our work [11]. However, this restriction has been partially removed in more recent work of Hu and Nelson [10] where they define and study properties of minimal vectors for all supercuspidal representations of D × where D is a (split or division) quaternion algebra over a p-adic field with p = 2.…”