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Abstract. Let f be an L 2 -normalized weight zero Hecke-Maaß cusp form of square-free level N , character χ and Laplacian eigenvalue λ 1/4. It is shown that f ∞ ≪ λ N −1/37 , from which the hybrid bound f ∞ ≪ λ 1/4 (N λ) −δ (for some δ > 0) is derived. The first bound holds also for f = y k/2 F where F is a holomorphic cusp form of weight k with the implied constant now depending on k.

We approach the holomorphic analogue to the Quantum Unique Ergodicity conjecture through an application of the Large Sieve. We deal with shifted convolution sums as in [Ho], with various simplifications in our analysis due to the knowledge of the Ramanujan-Petersson conjecture in this holomorphic case.

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on SL2( )\À. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result which suggests that the average size of the summation terms should be sufficient in application to Quantum Unique Ergodicity. In other words, cancellations among the summation terms, although welcomed, may not be required. Furthermore, the sieve method may be applied to shifted sums of other multiplicative functions with similar results under suitable conditions.

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Let 1 N < M with N and M coprime and square-free. Through classical analytic methods we estimate the first moment of central L-values L( 1 2 , f × g) where f ∈ S * k (N ) runs over primitive holomorphic forms of level N and trivial nebentypus and g is a given form of level M . As a result, we recover the bound L( 1The first moment method also applies to the special derivative L ′ ( 1 2 , f × g) under the assumption that it is non-negative for all f ∈ S * k (N ).

Abstract In this paper, we introduce a simple Bessel $\delta $-method to the theory of exponential sums for $\textrm{GL}_2$. Some results of Jutila on exponential sums are generalized in a less technical manner to holomorphic newforms of arbitrary level and nebentypus. In particular, this gives a short proof for the Weyl-type subconvex bound in the $t$-aspect for the associated $L$-functions.

For a fixed cusp form π on GL 3 (Z) and a varying Dirichlet character χ of prime conductor q, we prove that the subconvex bound2 ) ≪ q 3/4−δ holds for any δ < 1/36. This improves upon the earlier bounds δ < 1/1612 and δ < 1/308 obtained by Munshi using his GL 2 variant of the δ-method. The method developed here is more direct. We first express χ as the degenerate zero-frequency contribution of a carefully chosen summation formula à la Poisson. After an elementary "amplification" step exploiting the multiplicativity of χ, we then apply a sequence of standard manipulations (reciprocity, Voronoi, Cauchy-Schwarz and the Weil bound) to bound the contributions of the nonzero frequencies and of the dual side of that formula. Contents 1. Introduction 2. Preliminaries 3. Division of the proof 4. Estimates for F 5. Estimates for O Appendix A. Correlations of Kloosterman sums Appendix B. Comparison with Munshi's approach References