We prove an upper bound for the L^4-norm and for the L^2-norm restricted to
the vertical geodesic of a holomorphic Hecke cusp form of large weight. The
method is based on Watson's formula and estimating a mean value of certain
L-functions of degree 6. Further applications to restriction problems of Siegel
modular forms and subconvexity bounds of degree 8 L-functions are given.Comment: 23 page
A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ℓ-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ℓ twisted by the q-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.
We prove two results on arithmetic quantum chaos for dihedral Maaß forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level 1 forms, these results were previously known for Eisenstein series and conditionally on the generalised Lindelöf hypothesis for Hecke-Maaß eigenforms. A key aspect of the proofs is bounds for certain mixed moments of L-functions that imply hybrid subconvexity.
Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic Lfunctions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet L-functions modulo q weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length q 1/4 . An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic Lfunctions, which we also use to improve the best known subconvexity bounds for automorphic L-functions in the level aspect. R |Λ(1/2 + it, E)| 2 dt ≈ ∞ 0 |E(iy)| 2 dy.
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