Abstract. Let f be a fixed self-contragradient Hecke-Maass form for SL(3, Z), and u an even Hecke-Maass form for SL(2, Z) with Laplace eigenvalue 1/4 + k 2 , k ≥ 0. A subconvexity bound O (1 + k) 4/3+ε in the eigenvalue aspect is proved for the central value atMeanwhile, a subconvexity bound O (1 + |t|) 2/3+ε in the t aspect is proved for L(1/2 + it, f ). These bounds improved corresponding subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The main techniques in the proofs, other than those used by Li, are nth-order asymptotic expansions of exponential integrals in the cases of the explicit first derivative test, the weighted first derivative test, and the weighted stationary phase integral, for arbitrary n ≥ 1. These asymptotic expansions sharpened the classical results for n = 1 by Huxley.

The subject matter of this paper is an integral with exponential oscillation of phase f (x) weighted by g(x) on a finite interval [α, β]. When the phase f (x) has a single stationary point in (α, β), an nth-order asymptotic expansion of this integral is proved for n ≥ 2. This asymptotic expansion sharpens the classical result for n = 1 by M.N. Huxley. A similar asymptotic expansion was proved by Blomer, Khan and Young under the assumptions that f (x) and g(x) are smooth and g(x) is compactly supported on R.In the present paper, however, these functions are only assumed to be continuously differentiable on [α, β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory and other fields.2010 Mathematics Subject Classification. 41A60.

We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic C. This can be expressed as a relation between the period spectrum and the ortholength spectrum of C. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along C as well estimates on the lengths of geodesic segments which start and end orthogonally on C. Variant trace formulas also lead to several simultaneous nonvanishing results for different periods.

Abstract. We prove a Weyl Harish-Chandra integration formula for the action of a reductive dual pair on the corresponding symplectic space W. As an intermediate step, we introduce a notion of a Cartan subspace and a notion of an almost semisimple element in the symplectic space W. We prove that the almost semisimple elements are dense in W. Finally, we provide estimates for the orbital integrals associated with the different Cartan subspaces in W.

Let π be a unitary automorphic cuspidal representation of GL2(Q A ) with Fourier coefficients λπ(n).Asymptotic expansions of certain sums of λπ(n) are proved using known functorial liftings from GL2, including symmetric powers, isobaric sums, exterior square from GL4 and base change. These asymptotic expansions are manifestation of the underlying functoriality and reflect value distribution of λπ(n) on integers, squares, cubes and fourth powers.