标题： 显式Heegner点：Kolyvagin猜想和Shafarevich-Tate群中的非平凡元素 显示英文标题

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

摘要： 科利瓦金使用希格纳点将一个上同调类的系统与 over$\Q$的椭圆曲线联系起来，并猜想该系统包含一个非平凡类。 他的猜想对 Selmer 群的结构有深远的影响。 我们为科利瓦金的猜想提供了新的计算和理论证据。 更准确地说，我们显式计算了环类域上的希格纳点，并利用这些点验证了特定二阶椭圆曲线的猜想。 我们解释了科利瓦金的猜想如何表明，如果椭圆曲线的解析阶至少为二，则相应 Selmer 群的$\Z_p$-余秩也至少为二。 我们还使用显式计算的希格纳点在沙法列维奇-泰特群中生成非平凡类。 显示英文摘要

摘要： 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 and use these points to verify the conjecture for specific elliptic curves of rank two. We explain how Kolyvagin's conjecture implies that if the analytic rank of an elliptic curve is at least two then the $\Z_p$-corank of the corresponding Selmer group is at least two as well. We also use explicitly computed Heegner points to produce non-trivial classes in the Shafarevich-Tate group.