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 formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of Lfunctions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.
We determine local test vectors for Waldspurger functionals for GL 2 , in the case where both the representation of GL 2 and the character of the degree two extension are ramified, with certain restrictions. We use this to obtain an explicit version of Waldspurger's formula relating twisted central L-values of automorphic representations on GL 2 with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results. 1
Abstract. Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree 2. It is proved that if F is not in the Maaß space, then there exist infinitely many primes p for which the sequence µ(p r ), r > 0, has infinitely many sign changes.
In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J nh k,m ) of weight k ∈ Z and index m ∈ N as eigenfunctions of a degree three differential operator C k,m . We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J nh k,m . We construct new examples of cuspidal Jacobi Maaß forms F f of weight k ∈ 2Z and index 1 from weight k − 1/2 Maaß forms f with respect to 0 (4) and show that the map f → F f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J nh k,m can be "essentially" obtained from scalar or vector valued half integer weight Maaß forms.
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