Abstract. Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk → ∞. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime.We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions of Watson's formula to certain triple product integrals involving forms of non-squarefree level. By a theorem of Ichino and a lemma of Michel-Venkatesh, this amounts to a detailed study of Rankin-Selberg integrals |f | 2 E attached to newforms f of arbitrary level and Eisenstein series E of full level.We find that the local factors of such integrals participate in many amusing analogies with global L-functions. For instance, we observe that the mass equidistribution conjecture with a power savings in the depth aspect is equivalent to knowing either a global subconvexity bound or what we call a "local subconvexity bound"; a consequence of our local calculations is what we call a "local Lindelöf hypothesis".

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson's formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

Abstract. We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N , with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.

Let f be an L 2 -normalized Hecke-Maass cuspidal newform of level N , character χ and Laplace eigenvalue λ. Let N1 denote the smallest integer such that N |N 2 1 and N0 denote the largest integer such that N 2 0 |N . Let M denote the conductor of χ and define M1 = M/ gcd(M, N1). We prove the bound, which generalizes and strengthens previously known upper bounds for f ∞ . This is the first time a hybrid bound (i.e., involving both N and λ) has been established for f ∞ in the case of non-squarefree N . The only previously known bound in the non-squarefree case was in the N -aspect; it had been shown by the author that f ∞ ≪ λ,ε N 5/12+ε provided M = 1. The present result significantly improves the exponent of N in the above case. If N is a squarefree integer, our bound reduces to f ∞ ≪ε N 1/3+ε λ 5/24+ε , which was previously proved by Templier.The key new feature of the present work is a systematic use of p-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for GL2(F ) where F is a local field.

Abstract. We give a lower bound for the sup-norm of an L 2 -normalized newform in an irreducible, unitary, cuspidal representation π of GL2 over a number field. When the central character of π is sufficiently ramified, this bound improves upon the trivial bound by a positive power of N where N is the norm of the conductor of π. This generalizes a result of Templier, who dealt with the special case when the conductor of the central character equals the conductor of the representation. We also make a conjecture about the true size of the sup-norm in the N -aspect that takes into account this central character phenomenon. Our results depend upon some explicit formulas and bounds for the Whittaker newvector over a non-archimedean local field, which may be of independent interest.