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In [2], J. Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their implication to the structure of the cuspidal spectrum for symplectic groups ([16] and [20]). As result, we obtain certain characterization and construction of small cuspidal automorphic representations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak in [43].

Thesis (Ph. D.)--Columbia University, 1991. Includes bibliographical references (leaf 48).

We will show that for any noncompact arithmetic hyperbolic m-manifold with m>3, and any compact arithmetic hyperbolic m-manifold with m > 4 that is not a 7-dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic 3-manifold groups. We will also show that a compact orientable irreducible 3-manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.

In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\R^2\times H^1(\R^3,\R^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\R^3\\ -\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\R^3\\ u,v>0&\hbox{in}\;\R^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\R^3}u^2=a^2\quad\hbox{and}\;\int_{\R^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the case of fixed frequencies $\lambda_1,\lambda_2$. When the masses are prescribed, the standard approach to this problem is variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist for all $\beta>0$ and all $a,b>0$.