In this paper, we show how to use a recent theorem of Nekovář [12] to produce families of examples of elliptic curves over number fields whose p-power Selmer groups grow systematically in Z_p^d-extensions. We give a somewhat different exposition and proof of Nekovář's theorem, and we show in many cases how to replace the fundamental requirement that the elliptic curve has odd p-Selmer rank by a root number calculation.

Let G be a split reductive group over a local field K, and let G((t)) be the corresponding loop group. In [1], we have introduced the notion of a representation of (the group of K-points) of G((t)) on a pro-vector space. In addition, we have defined an induction procedure, which produced G((t))-representations from usual smooth representations of G. We have conjectured that the induction of a cuspidal irreducible representation of G is irreducible. In this paper, we prove this conjecture for G=SL₂.

We describe a new method to estimate the trilinear period on automorphic representations of PGL₂ℝ. Such a period gives rise to a special value of the triple L-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of PGL₂ℝ.