Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL 1 and GL 2 automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino-Ikeda.

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.

In this paper we calculate the asymptotics of various moments of the central values of Rankin-Selberg convolution L-functions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexity-breaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators.

A new protein fractionation technique based on off-gel isoelectric focusing (IEF) is presented, where the proteins are separated according to their isoelectric point (pI) in a multiwell device with the advantage to be directly recovered in solution for further analysis. The protein fractions obtained with this technique have then been characterized with polymer nanoelectrospray for mass spectrometry (MS) analyses or with Bioanalyzer for mass identification. This methodology shows the possibility of developing alternatives to the classical two-dimensional (2-D) gel electrophoresis. One species numerical simulation of the electric field distribution during off-gel separation is also presented in order to demonstrate the principle of the purification. Experiments with pI protein markers have been carried out in order to highlight the kinetics and the efficiency of the technique. Moreover, the resolution of the fractionation was shown to be 0.1 pH unit for the separation of beta-lactoglobulin A and B. In addition, the isoelectric fractionation of an Escherichia coli extract was performed in standard solubilization buffer to demonstrate the performances of the technique, notably for proteomics applications.