Averages of fractional exponential sums weighted by Maass forms

<p>The purpose of this study is to investigate the oscillatory behavior of the fractional exponential sum weighted by certain automorphic forms for GL(2) x GL(3) case. Automorphic forms are complex-values functions defined on some topological groups which satisfy a number of applicable properties. One nice property that all automorphic forms admit is the existence of Fourier series expansions, which allows us to study the properties of automorphic forms by investigating their corresponding Fourier coefficients. The Maass forms is one family of the classical automorphic forms, which is the major focus of this study.</p> <p>Let f be a fixed Maass form for SL(3, Z) with Fourier coefficients Af(m, n). Also, let {gj} be an orthonormal basis of the space of the Maass cusp form for SL(2, Z) with corresponding Laplacian eigenvalues 1/4+kj^2, kj>0. For real α be nonzero and β>0, we considered the asymptotics for the sum in the following form Sx(f x gj, α, β) = ∑Af(m, n)λgj(n)e(αn^β)φ(n/X) where φ is a smooth function with compactly support, λgj(n) denotes the nth Fourier coefficient of gj, and X is a real parameter that tends to infinity. Also, e(x) = exp(2πix) throughout this thesis.</p> <p>We proved a bound of the weighted average sum of Sx(f x gj, α, β) over all Laplacian eigenvalues, which is better than the trivial bound obtained by the classical Rankin-Selberg method. In this case, we allowed the form varies so that we can obtain information about their oscillatory behaviors in a different aspect. Similar to the proofs of the subconvexity bounds for Rankin-Selberg L-functions for GL(2) x GL(3) case, the method we used in this study includes several sophisticated techniques such as weighted first and second derivative test, Kuznetsov trace formula, and Voronoi summation formula.</p>