Abstract. We study the average of the product of the central values of two L-functions of modular forms f and g twisted by Dirichlet characters to a large prime modulus q. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q, and we use the theory of Deligne and Katz to prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of q. When at least one of the forms f and g is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted L-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M. Young's asymptotic evaluation of the fourth moment of Dirichlet L-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.

We extend to an arbitrary number field the best known bounds towards Ramanujan for the group GLn, n = 2, 3, 4. In particular, we present a technique which overcomes the analytic obstacles posed by the presence of an infinite group of units.

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

The aim of this paper is to derive bounds on the critical line

Let $q$ be a large prime and $\chi$ the quadratic character modulo $q$. Let $\phi$ be a self-dual cuspidal Hecke eigenform for ${\rm SL}(3, \Bbb{Z})$, and $f$ a Hecke-Maa{\ss} cusp form for $\Gamma_0(q) \subseteq {\rm SL}_2(\Bbb{Z})$. We consider the twisted $L$-functions $L(s, \phi \times f \times \chi)$ and $L(s, \phi \times \chi)$ on ${\rm GL}(3)\times {\rm GL}(2)$ and ${\rm GL}(3)$ with conductors $q^6$ and $q^3$, respectively. We prove the subconvexity bounds $$ L(1/2, \phi \times f \times \chi) \ll_{\phi, f, \varepsilon} q^{5/4+\varepsilon},\quad L(1/2+it, \phi \times \chi) \ll_{\phi, t, \varepsilon} q^{5/8+\varepsilon} $$ for any $\varepsilon > 0$.