Mazur [8] has introduced the concept of visible elements in the Tate-Shafarevich group of optimal modular elliptic curves. We generalized the notion to arbitrary abelian subvarieties of abelian varieties and found, based on calculations that assume the BirchSwinnerton-Dyer conjecture, that there are elements of the Tate-Shafarevich group of certain sub-abelian varieties of J0(p) and J1(p) that ...

Meal machine, experimental laboratory, status symbol, domestic prison, or the creative and spiritual heart of the home? Over the course of the past century no other room has been the focus of such intensive aesthetic and technological innovation, or as loaded with cultural significance. Kitchen design has been both a central concern of modernism and fundamental to our concept of modern life (Mo...

We improve Kolyvagin’s upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.

Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over Q and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer groups. We provide new computational and theoretical evidence for Kolyvagin’s conjecture. More precisely, we explicitly compute Heegner points over ring class fields...

We classify the possible Mumford-Tate groups of polarizable rational Hodge structures.

We prove that for abelian varieties with semistable ordinary reduction the p-adic Mazur-Tate height pairing is induced by the unit root splitting of the Hodge filtration on the first deRham cohomology.