Heegner points on Mumford–Tate curves

Heegner points on Mumford-Tate urves

Heegner points, p-adic L-functions, and the Cerednik-Drinfeld uniformization

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 1 Quaternion algebras, upper half planes, and trees . . . . . . . . . . . . . . . . . . . . . . . . . 456 2 The p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 3 Generalities on Mumford curves . . . . . . . ....

Explicit Heegner Points: Kolyvagin’s Conjecture and Non-trivial Elements in the Shafarevich-Tate Group

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...

On the Computation of the Cassels Pairing for Certain Kolyvagin Classes in the Shafarevich-Tate Group

Kolyvagin has shown how to study the Shafarevich-Tate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit non-trivial elements of the Shafarevich-Tate group is by proving that a locally trivial Kolyvagin class is globally non-trivial, which is difficult in practice. We provide a method for testing whether an...

Computing the Cassels Pairing on Kolyvagin Classes in the Shafarevich-Tate Group

Kolyvagin has shown how to study the Shafarevich-Tate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit non-trivial elements of the Shafarevich-Tate group is by proving that a locally trivial Kolyvagin class is globally non-trivial, which is difficult in practice. We provide a method for testing whether an...