Symplectic Automorphic Forms and Kloosterman Sums

Abstract

In this thesis, we study automorphic forms on the rank 2 symplectic group Sp(4), in the context of analytic number theory. While much of the abstract theory is described in Langlands’ theory, one needs more explicit formulae for applications in analytic number theory. The thesis consists of three parts. <br /> In the first part of the thesis, we first give explicit formulations for Sp(4) Eisenstein series. Then we compute explicit formulae for constant terms and Fourier coefficients of Sp(4) Eisenstein series, in terms of Whittaker functions. <br /> In the second part of the thesis, we study Sp(4) Kloosterman sums, and evaluate non-trivial bounds for these sums, using a stratification argument, and p-adic stationary phase method. We also compute explicitly the Fourier coefficients of Sp(4) Poincaré series, using Kloosterman sums. <br /> In the third part of the thesis, we construct an Sp(4) analogue of the Kuznetsov trace formulae. We also obtain explicit relations between Fourier coefficients of Sp(4) automorphic forms and Hecke eigenvalues. Using these results, and estimates of Sp(4) Kloosterman sums, we establish strong bounds for the number of automorphic forms of level q violating the Ramanujan conjecture at any given unramified place, which go beyond Sarnak’s density hypothesis.