We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan-Gross-Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding L-functions. Ratner’s results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher-rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino-Ikeda conjecture.

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂P^{n+1} is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank n⩾3 is not generated by linear and Jonquières elements.