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.
It was shown in previous works that the measure associated to holomorphic newforms of weight k and level q will tend weakly to the Haar measure on modular curve of level 1, as qk → ∞. In this paper we proved that this phenomenon is also true on modular curves of general level N.3
In this paper we generalized Venkatesh and Woodbury's work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors. In particular we derived a nice general formula for the local integrals whenever one of the representations has sufficiently higher level than the other two.
We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.
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