Motivated by the conjecture of Birch and Swinnerton-Dyer, I will explain a strategy for bounding Mordell-Weil ranks of elliptic curves in certain abelian towers of number fields.This strategy comes down to techniques in analysis, including bounds for certain Fourier coefficients of automorphic forms, and more generally the limiting behaviours of certain "automorphic periods".

Let W be a reflection group in a vector space and P a rational polytope that is invariant under the W-action. On the one hand, the action of W on P induces a W-action on the toric variety X_P associated with P. On the other hand, W-action on P defines the region R of P such that each point x in R represents the W-orbit of x. Since the closure \overline{R} of R is again a rational polytope, we have the toric variety X_{\overline{R}} associated to it. It is conjectured that X_\overline{R} and the quotient variety {X_P}/W are isomorphic. In this talk, we shall see various examples of toric varieties and cohomological evidences supporting this conjecture. We also see its application to the cohomological study of regular Hessenberg varieties. This talk is based on the joint work with T. Horiguchi, M. Masuda and J. Shareshian.